Paper Title: Theoretical study on the possibility of high $T_c$ s$\pm$-wave superconductivity in the heavily hole-doped infinite layer nickelates.



APS/123-QED

 

sakakibara@tottori-u.ac.jp
Advanced Mechanical and Electronic System Research Center(AMES), Faculty of Engineering, Tottori University, 4-10 Koyama-cho, Tottori, Tottori 680-8552, Japan

Forefront Research Center, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan

Forefront Research Center, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan
Department of Physics, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan

Department of Applied Physics, Shimane University, 1060 Nishikawatsu-cho, Matsue, Shimane 690-8504, Japan

Department of Physics, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan


             

§ ABSTRACT 
We theoretically propose a possibility of realizing high T_c superconductivity having s±-wave symmetry in the heavily hole-doped infinite layer nickelates La_1-xSr_xNiO_2.  We consider situations where the original P4/mmm symmetry of LaNiO_2 is maintained even for a significant amount of Sr substitution by growing thin films on substrates having tetragonal symmetry. Considering such cases is indeed justified by our phonon calculations. For electron configurations somewhat close to d^8, the interaction between the d_x^2-y^2 band and the other 3d bands that lie just below the Fermi level results in an enhancement of superconductivity where the sign of the gap function is reversed between the former and the latter bands. The strong enhancement of superconductivity can be attributed to the large energy level offset between d_x^2-y^2 and other orbitals due to the absence of the apical oxygens, as has been pointed out in previous studies.
 
 

74.20.Mn,74.70.−b
                             

                              

Theoretical study on the possibility of high T_c s±-wave superconductivity in the heavily hole-doped infinite layer nickelates 

    Kazuhiko Kuroki
    August 5, 2024
=================================================================================================================================

Introduction.—

The discovery of superconductivity in thin films of lightly hole doped  infinite layer nickelates RE_1-xAE_xNiO_2 (RE=La, Pr, Nd, AE=Ca, Sr) <cit.> has now led to the “nickel age”<cit.> of unconventional superconductivity. In fact, last year, superconductivity with a maximum T_c exceeding 80 K was discovered in a bilayer Ruddlesden-Popper nickelate La_3Ni_2O_7 <cit.> under pressure, confirmed by the experiments that followed <cit.>.  Subsequently, superconductivity with T_c= 20∼ 30 K was discovered in a trilayer version of the nickelate La_4Ni_3O_10 also under pressure <cit.>, which was further confirmed by other experiments <cit.>. It is worth mentioning that, before entering the nickel age, theoretical studies on the mechanism of unconventional high T_c superconductivity in specific materials have been performed only after their experimental discoveries, but for the bilayer nickelate La_3Ni_2O_7, two of the present authors have discussed the possibility of superconductivity in this material even before its experimental discovery <cit.>, and for the trilayer La_3Ni_2O_7, the structural transition to tetragonal symmetry under pressure as well as the occurrence of superconductivity with a T_c comparable to those of the relatively low T_c cuprates was predicted theoretically by the present authors <cit.> also before the experimental observations.

As for the infinite layer nickelates, several authors, including us, have theoretically proposed a cuprate-like d-wave pairing scenario, since the electron configuration is considered to be close to d^9 as in the cuprates <cit.>.
On the other hand, in a previous study, two of the present authors proposed a possibility of realizing s±-wave superconductivity in the infinite layer nickelates with possibly an even higher T_c than for the d-wave pairing in the low doping regime <cit.>. Here, s±-wave means that the signs of the superconducting gap function between 3d_x^2-y^2 and other four 3d orbitals are reversed. In Ref. <cit.>, a possible existence of residual hydrogen was assumed, as suggested theoretically <cit.>, so that the electron configuration is closer to d^8 than in the absence of the hydrogen. When the energy level offsets Δ E between 3d_x^2-y^2 orbitals and the other 3d (especially 3d_3z^2-r^2) orbitals are large, as in the infinite-layer case, where there are no apical oxygens, the electron configuration somewhat close to d^8 would result in a situation where the four 3d bands are incipient (i.e., they lie close to, but do not intersect, the Fermi level),  which leads to a high T_c s±-wave superconductivity, as suggested previously by two of the present authors <cit.>. The basic idea comes from the mathematical equivalence between two-orbital and bilayer Hubbard models <cit.>, where the interorbital energy level offset Δ E in the former is transformed to the interlayer hopping t_⊥ in the latter. The experimental circumstances regarding the possibility of residual hydrogen is controversial : there is a study which suggests the residual hydrogen playing an important role <cit.>, while several studies rule out such a possibility <cit.>.

In the present study, we theoretically seek for another possibility of attaining s±-wave high T_c superconductivity in the infinite layer nickelates by a realization of electron configuration somewhat close to d^8. Namely, we consider heavily doping holes in the original material (say LaNiO_2) by largely substituting the rare earth elements (such as La) with alkaline-earth elements (such as Sr or Ba). 

The end materials SrNiO_2 and BaNiO_2 are known to have orthorhombic crystal structures <cit.>, different from the one with tetragonal (P4/mmm) symmetry of LaNiO_2, but if the P4/mmm symmetry can be maintained even for (La,Sr)NiO_2 with significant Sr content by growing thin films on substrates having tetragonal symmetry,  there may be a chance of realizing s±-wave superconductivity with a T_c possibly higher than in the low doping regime. Phonon calculations in the present study indeed suggest such a situation is plausible.

In Ref. <cit.>, a rigid band was assumed in the entire doping regime, and also a fixed set of electron-electron interaction parameters was used. However, since the interorbital energy level offset Δ E and the interorbital interactions are both crucial in the s±-wave superconductivity, in the present study we employ the virtual crystal approximation (VCA) to take into account the variance of bands, especially Δ E, upon increasing the substitution content x, and also determine the electron-electron interaction values by using constrained random phase approximation (cRPA) <cit.> for each x and substrate combination. Our analysis using fluctuation exchange approximation suggests that superconductivity with relatively high T_c is possible, especially when substrates with small lattice constants are used.


Method.—

First, we perform first-principles calculation to obtain the band structure of La_1-xAE_xNiO_2(AE=Sr, Ba) using the QUANTUM ESPRESSO code <cit.>. We assume cases where the in-plane lattice constants are determined by the substrate. We consider three possibilities for the substrate : SrTiO_3, (LaAlO_3)_0.3(Sr_2TaAlO_6)_0.35 (LSAT), and LaAlO_3, whose lattice constants a are 3.905 Å, 3.868 Å, and 3.790 Å, respectively.  We use the crystal structure optimized within this restriction. 
Here we briefly comment on the lattice mismatch for the cases considered. For the non-doped bulk LaNiO_2, the experimental value of the lattice constant is a=3.958 Å <cit.>, and a  obtained by fully-relaxing the lattice structure within the first principles calculation is a=3.937 Å, in nice agreement (the error is less than 1 %).
The lattice mismatches between the experimental value of the bulk lattice constant and the substrates are 1.3, 2.3 and 4.2 % for SrTiO_3, LSAT, and LaAlO_3 cases, respectively.
Moreover, the fully-relaxed first principles value of a tends to be reduced by Sr doping
(e.g., a=3.910 Å at x=0.5 and a=3.855 Å at x=1.0).
Hence, growing thin films on these substrates can be considered as realistic from the viewpoint of the lattice mismatch, especially  in the heavily doped regime.
As for the comparison between BaNiO_2 and SrNiO_2, the former has a slightly larger value of the fully-relaxed  lattice constant (a=3.916 Å).


We also perform phonon calculation using the density functional perturbation theory.  We take a 3 × 3 × 3 q-mesh.  We also adopt VCA to take into account the effect of the partial substitution. 

The Perdew-Burke-Ernzerhof parametrization of the generalized gradient approximation (PBE-GGA) <cit.>
and the scalar-relativistic version of the optimized norm-conserving Vanderbilt pseudopotentials <cit.> taken from PseudoDojo <cit.> are used.
We take 100 Ry plane-wave cutoff energy, a 12 × 12 × 12 k-mesh, and an energy width of 0.02 Ry for Gaussian smearing.

We then extract maximally localized Wannier functions <cit.> using the RESPACK code <cit.>, 
by which we also obtain the hopping parameters among the Wannier functions. 
We have constructed a five-orbital model consisting of all the Ni 3d orbitals
and performed cRPA calculation to determine the Coulomb and exchange interaction parameter of the model,
where the doping effects on these parameters are small.





We explore the possibility of superconductivity 
for the obtained low-energy four-orbital model within the fluctuation-exchange (FLEX) approximation <cit.>. 

As the interaction term of the Hamiltonian, we only take the on-site interactions, namely, intraorbital(interorbital) Coulomb interactions U(U'), 
Hund's coupling J, and pair hopping J'.
We calculate the self-energy induced by the spin-fluctuation formulated as shown in the literatures <cit.> in a self-consistent calculation.
The explicit formulae of the irreducible, spin, and charge susceptibilities 
describing the fluctuations are shown in Eqs.(2)-(4) of Ref. <cit.>.
The real part of the self-energy at the lowest Matsubara frequency is subtracted in the same manner with Ref. <cit.>
to maintain the band structure around the Fermi level obtained by first-principles calculation.


The obtained Green's function and the pairing interaction, mediated mainly by spin fluctuations, are plugged into the linearized Eliashberg equation.
Since the eigenvalue λ of the linearized Eliashberg equation reaches unity at T=T_c, 
we adopt it as a measure of superconductivity at a fixed temperature, T=0.01 eV. 
For convenience, we will call the eigenfunction (with the largest eigenvalue) of the linearized Eliashberg equation at the lowest Matsubara frequency iω(=iπ k_ BT) the “superconducting gap function”. We take a 16×16×4 k-point mesh and 4096 Matsubara frequencies for the FLEX calculation.













Results and Discussions.—


As for the stability of the lattice structure of La_1-xSr_xNiO_2, our phonon calculations show that, interestingly, the P4/mmm symmetry of the original LaNiO_2 is dynamically stable for the entire Sr content range of 0≤ x ≤ 1 for all the three substrates considered here. In the top left panel of Fig. <ref>, we present the phonon dispersion for the case of x=0.5 assuming a SrTiO_3 substrate, where imaginary phonon frequencies do not appear. The rest of the phonon calculations are presented in the supplemental material <cit.>. 

Our phonon calculations justify our adopting the P4/mmm symmetry of the lattice structure for the electronic band structure calculation and model construction for all the cases considered in the present study. In Fig. <ref>(c)–(e), we show the band structure of La_1-xSr_xNiO_2 adopting the in-plane lattice constant of the SrTiO_3 substrate for  various Sr content x. While the Fermi level intersects the d_x^2-y^2 band, it stays somewhat away from the  d_3z^2-r^2 band for all x because the apical oxygen is absent. On the other hand, the Fermi level approaches the top of the t_2g bands as x is increased, and eventually intersects those bands for too large x. 

In Fig. <ref>, we plot the number of electrons and Δ E (the energy level offset from the d_x^2-y^2 orbital) of each orbital against the Sr content x for various substrates. It can be seen that Δ E depends on the substrate with a larger Δ E for a shorter lattice constant. This is because when the in-plane lattice constant is shorter, the energy level of the d_x^2-y^2 orbital, elongated toward the in-plane oxygens, is pushed up. 
Also, the relatively large variation of Δ E against x indicates that the rigid band picture does not hold.

As for the number of electrons in each orbital, obviously, that of the d_x^2-y^2 orbital decreases as x is increased, but a more important point is that at certain point that of the d_yz (and equivalently d_xz) orbital starts to decrease, indicating that the Fermi level has started intersecting these bands. This critical point of x is larger for substrates with shorter lattice constant, reflecting the larger values of Δ E between the d_x^2-y^2 and d_xz/yz orbitals.

In Fig. <ref>, we plot the eigenvalue of the linearized Eliashberg equation against the Sr content x  for the three substrates. For comparison, we also show the range of the eigenvalue calculated for various cuprates within the same formalism. For x≥ 0.4, the dominant pairing symmetry becomes s±-wave, and the eigenvalue is maximized around x≃ 0.5 – 0.7, depending on the substrate. As mentioned before, by s±-wave, we mean that the sign of the gap function is reversed between the d_x^2-y^2 and the other orbitals, as is depicted in Fig. <ref> for the case of x=0.6 and the LSAT substrate. Only for the SrTiO_3 substrate, we calculate the eigenvalue at a low Sr content of x=0.2, for which the dominant pairing symmetry is d-wave. We note here that for small x (where d-wave dominates), the rare earth 5d bands are known to intersect the Fermi level, and in such a case, a more accurate evaluation of the electron-electron interaction by cRPA requires including those bands <cit.>. Since this is not done in the present study, the interaction values are underestimated in the low doping regime, which results in an overestimation of λ. 
The difference in λ between the cases of AE= Sr and Ba is small, which may be due to the fact that the in-plane lattice constant is fixed at the value of the substrate. 


The relatively large λ values in the heavily doped regime is qualitatively consistent with the previous study <cit.>, but quantitatively there are differences. For substrates with shorter lattice constants, λ is maximized at a larger x with a larger maximum value. This is because for shorter lattice constants, Δ E is larger  (Fig. <ref>) so that the Fermi level starts to intersect the d_xz/yz bands at a larger x, for which the superconductivity is degraded due to strong renormalization effects, and also the large Δ E itself favors high T_c <cit.>.

Summary.—
We have theoretically examined the possibility of realizing high T_c superconductivity having s±-wave symmetry in the heavily hole-doped infinite layer nickelates La_1-xSr_xNiO_2. We have considered situations where the original P4/mmm symmetry of LaNiO_2 is maintained for  significant amount of Sr substitution by growing thin films on substrates having tetragonal symmetry. We have shown that considering such cases is justified by performing phonon calculations for the crystal structures obtained by fixing the in-plane lattice constants to those of several substrates.

For electron configurations somewhat close to d^8, the interaction between the d_x^2-y^2 band and the other 3d incipient bands that lie just below the Fermi level results in an enhancement of superconductivity where the sign of the gap function is reversed between the d_x^2-y^2 and the incipient bands. The strong enhancement of superconductivity can be attributed to the large energy level offset between the d_x^2-y^2 and other orbitals due to the absence of the apical oxygens, as has been pointed out in previous studies <cit.>. To investigate this effect in a quantitative manner, we have employed VCA to take into account the variance of bands upon increasing the substitution content x, and have also determined the electron-electron interaction values by cRPA for each x and substrate combination. Our FLEX analysis has shown that superconductivity with relatively high T_c is possible, especially when substrates with small lattice constants are adopted.



We are supported by JSPS KAKENHI Grant No. JP22K03512 (H. S.), JP22K04907 (K. K.), JP24K01333. The computing resource is supported by 
the supercomputer system (system-B) in the Institute for Solid State Physics, the University of Tokyo, 
and the supercomputer of Academic Center for Computing and Media Studies (ACCMS), Kyoto University.






 Paper Title: Energy-lowering symmetry breaking creates a flat-band insulator in paramagnetic Nb3Cl8.





Renewable and Sustainable Energy Institute, University of Colorado, Boulder, Colorado 80309, USA

Renewable and Sustainable Energy Institute, University of Colorado, Boulder, Colorado 80309, USA

alex.zunger@colorado.edu
Renewable and Sustainable Energy Institute, University of Colorado, Boulder, Colorado 80309, USA



§ ABSTRACT 
Ordinary band structure calculations of quantum materials often incorrectly predicted metallic, instead of insulating electronic structure, motivating Mott-Hubbard strong electron correlation as a gapping mechanism. More recently, allowing the formation of local structural symmetry breaking motifs in otherwise ordinary band theory was shown to lower the total energy while predicting insulating gaps when they are observed. An important counter example was recently pointed out whereby the flat band formed in Nb_3Cl_8 by symmetry breaking is by itself partially occupied, thus failing to account for the observed insulating state. It is shown here that a generalized symmetry breaking involving the cooperative structural and magnetic effects produces in mean-field-like density functional theory an energy lowering insulating phase.


Energy-lowering symmetry breaking creates a flat-band insulator in paramagnetic Nb_3Cl_8
    Alex Zunger
    August 5, 2024
========================================================================================



Gap anomaly in Nb_3Cl_8: A range of quantum materials manifest local microscopic degrees of freedom nested within the average crystallographic structure. Such local motifs are experimentally observable by local structural probes [1,2] and calculable quantum mechanically as energy-lowering symmetry breaking motifs [3,4]. Symmetry breaking can split off narrow flat bands [5–8] from a broad conduction band (as in LaTiO_3 and VO_2), or from the broad valence band (as in LaFeO_3 and SrBiO_3), leading in a single step to the observed insulating phases in these quantum oxides. This offers a link between energy gap formation and the local structural motifs  traditional in solid-state chemistry, such as Peierls instabilities, Jahn-Teller distortions, or bond disproportionation, both in d-electron (such as LaMnO_3 [9] and NbO_2 [10]) or in non-d-electron compounds (such as BaBiO_3 [11]). It was recently pointed out [12,13] that α-phase Nb_3Cl_8 is special among other quantum materials in that symmetry breaking (here, Nb-Nb-Nb trimer formation) creates a partially occupied band in the ground state, hence the system remains metallic whereas the observed α-phase [12,13] is insulating. This discrepancy opens the possibility that α-phase Nb_3Cl_8 (see Supplemental Section A on the structures) is the long-sought example where purely electronic strong correlation, encoded by the Mott-Hubbard approach [14] in a fixed crystallographic structure, is not just a textbook illustration, but is required for explaining gaping unobtainable by uncorrelated physics.

The significance of this discovery: This triumph [12] is far from being trivial. It requires avoidance of numerous alternatives explainable via conventional band structure such as density functional theory (DFT). Had the composition of the measured sample been Cl-richer (such as Nb_3Cl_9) or one where one Cl atom would be replaced by Te (such as in Nb_3Cl_7Te [15]), the correct insulating phases would be guaranteed by the trivial closed-shell configuration. Structural and magnetic symmetry breaking must be examined, although they were rejected to establish a compelling alternative scenario for insulation via strong correlation. For example, the sister d-electron compound NbO_2 is predicted by DFT to be a metal without any structural symmetry breaking, but once Peierls-like symmetry breaking of Nb-Nb dimerization is examined [16], it shows total energy stabilization, concomitantly with becoming an insulator, in agreement with experiments without need for strong correlation Hubbard description [Figure 1(a, b)]. Also, other perovskites like SrVO_3 [17] and LaNiO_3 [18] are predicted by DFT to be metals, and structural or magnetic symmetry breaking is too weak to convert them to insulators. However, the predicted metallic states in the former oxides are the correct experimental reality for these two compounds [19–21], unlike the case for the insulating α-phase of Nb_3Cl_8.

The question posed: Can symmetry breaking—either structural or magnetic, or both—lower the total energy, while also opening a band gap? Here we examine the possible role of magnetism in explaining the insulation of Nb_3Cl_8. We use the DFT exchange-correlation functional [22] meta-generalized gradient approximation (meta-GGA) to calculate the electronic structures without Coulomb repulsion U. The DFT method details are provided in Supplemental Section B.





Structural symmetry breaking in nonmagnetic NbO_2 suffices to correctly describe it as an insulator: Before considering Nb_3Cl_8, we first glance again at its simpler sister compound NbO_2. This compound has a metal-insulator transition with two nonmagnetic phases whose crystal structures are both tetragonal but different in formation of Nb-Nb dimers. As shown in Fig. 1(a, b), the high-temperature (high-T) phase has the Fermi level residing in the broad principal conduction band and is metallic because it has no Nb-Nb dimers [Fig. 1(a)]. As shown in Fig. 1(c, d) the low-temperature (low-T) phase has Nb-Nb dimerization [Fig. 1(c)]. This dimerization constitutes structural symmetry breaking which stabilizes the total energy by 152 meV/formula unit relative to the metallic phase and splits off occupied flat bands from the principal conduction band down into the internal energy gap, thereby opening a band gap [Fig. 1(b)]. The states of the split-off flat bands in the low-T insulating phase are mainly localized on the Nb-Nb dimers [Fig. 1(d)], while the states near the Fermi level in the high-T metallic phase surround the Nb atoms [Fig. 1(b)].

The structural symmetry breaking in Nb_3Cl_8 retains the incorrect metallic phase: The compound Nb_3Cl_8 is more complex than its sister compound NbO_2 in terms of the structural symmetry breaking, paramagnetism and thus band gap opening mechanism. Nb_3Cl_8 is a layered material where each layer has Nb trimers forming the distorted Kagome lattice with a breathing mode [12,23]. Each Nb trimer is surrounded by 13 Cl ions, forming an Nb_3Cl_13 cluster [Nb_3Cl_8]^5+[Cl_5]^5-. Here a Nb trimer has 15 electrons but 8 of them are transferred to Cl_8, so 7 electrons are left per cluster. They occupy the molecular orbitals configuration (1a_1)^2(1e)^4(2a_1)^1 so that the highest occupied molecular orbital (HOMO) is singly occupied for the nonmagnetic Nb trimer cluster. By comparing the band structures of monolayer and bilayer structures for the α-phase Nb_3Cl_8, Supplemental Section C shows that the interlayer coupling does not influence the metallic state of the non-magnetic phase. Thus, subsequent calculations are performed by using the monolayer structure. The Nb-Nb-Nb trimerization in α-phase Nb_3Cl_8 is observed experimentally, showing short intracluster Nb-Nb bond length d^short(Nb-Nb) of 2.834 Åand long intercluster Nb-Nb bond length d^long(Nb-Nb) of 3.998 Å[24]. The DFT structural relaxed calculations giving intracluster distance of 2.85 Åand intercluster of 4.02 Åare close to the experimental measurement [24]. We define the “degree of trimerization” (DOT) as:

    DOT=d^long(Nb-Nb)-d^short(Nb-Nb)/d^long(Nb-Nb)+d^short(Nb-Nb)× 100%.

The DOT for the DFT relaxed structure and for the  experimental structure [24] are both estimated to be 17.0%. The calculated total energy gives an enormous stabilization of 3365 meV/f.u. but the system is a band structure metal.

Paramagnetism in α-phase Nb_3Cl_8: The paramagnetic (PM) α-phase of Nb_3Cl_8 was observed experimentally by measuring the magnetic susceptibility as a function of temperature, which was well fitted to the Curie-Weiss function giving rise to an effective PM Bohr magneton of 1.65 μ_B [12]. This paramagnetism is not considered relevant to gapping in the strongly correlated approach. In such Mott model, magnetic exchange J deciding magnetism is modeled perturbatively in terms of Mott-ness strong Coulomb repulsion J1/U, leading to the expected weak or no magnetism when a large Coulomb U is selected. Here, we do not exclude the PM configuration.

Description of paramagnetic symmetry breaking: We compute PM non-perturbatively using magnetic exchange interactions J obtained from spin polarized DFT. The PM configuration is not replaced a priori by the nonmagnetic structure where each site has zero moment. The latter view assumes that the local symmetry equals the average crystal structure of global symmetry. Instead, a PM configuration can be described more generally as a distribution of non-zero local magnetic moment. We use for this purpose a “specially constructed” [25] supercell consisting of n Nb_3Cl_8 formula units (where n is a size convergence parameter and we use n=16 here) with zero global magnetization. Much like “special k-points for Brillouin zone integrating” [26,27], the spin-special quasirandom structure (spin-SQS) method is not a single snapshot but represents a configurational average. Indeed, the spin-SQS searches for a magnetic configuration that best matches the target multisite spin-spin correlation function with the given short-range order. The local magnetic moments are decided by DFT energy minimization and charge self-consistency starting from the optimal spin-SQS magnetic structure. Since the band structure of PM phase in a small Brillouin zone (corresponding to a real space supercell) is inconveniently dense to observe important information such as band dispersion, we unfold the band structure into the original Brillouin zone thereby obtaining effective spectral functions [28] that have, unlike conventional band structure, natural energy-dependent intensities and “fuzziness”.

The important observation regarding magnetic symmetry breaking: Once structural symmetry breaking (Nb trimerization) is accounted for, we find that magnetic symmetry breaking opens a gap in the PM phase, explaining the insulating phase without recourse to strong correlation or to dynamic effects. This is described next by breaking the process into steps that clarify the individual contributions.

Hierachy of symmetry breaking: Figure 2 demonstrates the three successive levels of theoretical description of monolayer Nb_3Cl_8 based on the existence or absence of two symmetry breaking modes—Nb trimerization and magnetic symmetry breaking.



Level 1 (symmetry unbroken) is for the hypothetical structure without Nb trimerization and magnetism. As shown in Fig. 2(a), the Fermi level lies within a broad principal conduction band with its charge density distributed equally on the Nb atoms. This configuration is taken as the total energy reference Δ E=0.

In Level 2 (structural symmetry breaking) with the Nb trimerization, we find that the Nb trimerization stabilizes the total energy by 3365 meV/f.u. relative to Level 1 and induces a single half-occupied isolated flat band at the Fermi level. This isolated flat band with band width of 0.3 eV emerges from the broad principal conduction band of Level 1 and its charge density is dominantly located around the Nb trimers, as shown in Fig. 2(b). For the bilayer structure of nonmagnetic α-phase Nb_3Cl_8, there are two split-off flat bands crossing with the Fermi level that are degenerate at H point (See Supplemental Section B). Evidently, in both monolayer and bilayer structures of α-phase Nb_3Cl_8, the Nb trimerization with the formation of partially occupied split-off flat bands cannot open a band gap.

In Level 3 (structural and magnetic symmetry breaking) that considers both (i) Nb trimers due to structural symmetry breaking and (ii) magnetic symmetry breaking. The latter is depicted as PM spin-SQS with self-consistently determined distribution of local moments, as shown in Fig. 2(c). This shows that the half-filled isolated flat band of Level 2 split forming two isolated flat bands and creating a band gap of 0.3 eV. It increases to 1.2 eV when using the HSE06 exchange correlation functional instead of SCAN. This process from Level 2 to Level 3 reduces the total energy by 105 meV/f.u. The band gap is opened by interaction of (i) structural symmetry breaking (forming trimers) and (ii) magnetic symmetry breaking. We find that the band gap cannot be opened if there is only one of the two symmetry breaking modes. This shows that one can predict the insulating phase of such quantum materials without Hubbard U.

Distribution of local motifs: To further indicate the difference between PM and nonmagnetic phases, Figure 3 shows the distribution of local magnetic moments in PM Nb_3Cl_8 (Level 3). It is  observed that: (i) All of these results present a distribution of non-zero local magnetic moments and an average zero net magnetic moment, which is independent of the exchange correlation functionals and supercell sizes; (ii) The distribution of magnetic moments for SCAN and HSE06 is very different partially due to the strong exchange interaction in HSE06, but the choice of different exchange correlation functionals do not qualitatively change our main point that PM is essential to open the band gap.

Discussion: The PM phase lacks the long-range order of the AFM phase but has short-range order that can still be essential to open the band gap. Indeed long-range antiferromagnetic order is well known [29] to enable insulating gaps. However, the absence of long-range magnetic order as the case in PM phase can, in the presence of short-range order lead either (i) to metallic phase if symmetry breaking is weak, as in SrVO_3 [3,17] and LaNiO_3 [30,31], or (ii) to an insulating PM phase, as in MnO, FeO, CoO, and NiO. Furthermore, symmetry breaking can induce other effects such as effective mass enhancement [17] without strong correlation.

The present results indicate that approximating the PM as nonmagnetic configuration and using primitive unit cell as input to do the calculations is not appropriate, missing energy lowering and gapping. Compared to nonmagnetic NbO_2 that only needs structural symmetry breaking (Nb-Nb dimerization) to open a band gap (Fig. 1), α-phase Nb_3Cl_8 is more complicated because it needs both structural (Nb-Nb-Nb trimerization) and magnetic (PM) symmetry breaking modes (Fig. 2).

The insulating bilayer nonmagnetic β-phase Nb_3Cl_8: Besides the high-T PM α-phase, Nb_3Cl_8 has a low-T nonmagnetic β-phase, which is measured also as an insulator [12]. Because of the proximity of the layers in the β-phase, we allow in this case interlayer interaction via adding the Van der Waals (VdW) dispersive forces to the regular DFT. These dispersive forces have no influence on the α-phase but shorten the interlayer distance and increase the coupling within the bilayer β-phase. We find that in nonmagnetic β-phase Nb_3Cl_8 with two chemical formulae in the primitive cell, there are two split-off flat bands crossing with the Fermi level that are not degenerate at any k-points (see Supplemental Section C). This energy splitting is induced by the interlayer interaction. Therefore, in the β-phase the Nb trimerization and coupling of Nb trimers from adjacent layers have the potential to open a band gap, even without magnetism or strong correlation. Supplemental Section C provides a few tests of the insulating gap under different crystal structural parameters. A detailed study of the insulating gap is β-phase Nb_3Cl_8 is however beyond our scope as the details of the interlayer geometry are unknown.

Many-body physics is certainly to be preferred over single particle theory for understanding multiplet effects in electronically isolated ions in the gas phase [32,33], or for d-electron ions in wide-gap solids [34,35], etc. Many-body effects certainly exist as spectroscopic multiplet effects [36,37] and in multi excitons. Still, multi-electron effects that are often apparent in isolated ions such as carbon [38] often “melt away” when such atoms bond in condensed phases—as is the case when diamond or graphite are formed from carbon atoms. So, the question is: when is a many-electron treatment unescapable? But we see that the outlier phenomenology of gapping, the unusual case of Nb_3Cl_8 is describable by conventional DFT while reproducing also observable structural (Nb-Nb-Nb trimerization) and magnetic (the stabilization of a PM) symmetry breaking. This leads to an interesting duality: How will the band gap change if both the observable symmetry breaking (discussed in the present work) and the strong correlation Mott-Hubbard treatment used successfully in Ref. [12,23] to obtain gapping (without magnetic symmetry breaking) have important influences in producing an insulating phase? This remained question will inspire future research.


    We thank Dr. Shunye Gao for discussions of his paper denoted here as Ref. [12]. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under Grant No. DE-SC0010467. This work used resources from the National Energy Research Scientific Computing Center (NERSC), which was supported by the Office of Science of the U.S. Department of Energy.







§ REFERENCE

[1]	A. N. Beecher, O. E. Semonin, J. M. Skelton, J. M. Frost, M. W. Terban, H. Zhai, A. Alatas, J. S. Owen, A. Walsh, and S. J. L. Billinge, Direct observation of dynamic symmetry breaking above room temperature in methylammonium lead iodide perovskite, ACS Energy Lett. 1, 880 (2016).

[2]	E. S. Božin, C. D. Malliakas, P. Souvatzis, T. Proffen, N. A. Spaldin, M. G. Kanatzidis, and S. J. L. Billinge, Entropically stabilized local dipole formation in lead chalcogenides, Science 330, 1660 (2010).

[3]	O. I. Malyi and A. Zunger, False metals, real insulators, and degenerate gapped metals, Appl. Phys. Rev. 7, 041310 (2020).

[4]	X.-G. Zhao, Z. Wang, O. I. Malyi, and A. Zunger, Effect of static local distortions vs. dynamic motions on the stability and band gaps of cubic oxide and halide perovskites, Materials Today 49, 107 (2021).

[5]	Z. Zhu, H. Peelaers, and C. G. Van De Walle, Electronic and protonic conduction in LaFeO3, J. Mater. Chem. A 5, 15367 (2017).

[6]	M. Eaton, A. Catellani, and A. Calzolari, VO2 as a natural optical metamaterial, Opt. Express 26, 5342 (2018).

[7]	F. El-Mellouhi, E. N. Brothers, M. J. Lucero, I. W. Bulik, and G. E. Scuseria, Structural phase transitions of the metal oxide perovskites SrTiO3, LaAlO3, and LaTiO3 studied with a screened hybrid functional, Phys. Rev. B 87, 035107 (2013).

[8]	J. Varignon, J. Santamaria, and M. Bibes, Electrically switchable and tunable Rashba-type spin splitting in covalent perovskite oxides, Phys. Rev. Lett. 122, 116401 (2019).

[9]	H. Sawada, Y. Morikawa, K. Terakura, and N. Hamada, Jahn-Teller distortion and magnetic structures in LaMnO3, Phys. Rev. B 56, 12154 (1997).

[10]	A. O’Hara and A. A. Demkov, Nature of the metal-insulator transition in NbO2, Phys. Rev. B 91, 094305 (2015).

[11]	S. Sarkar, R. Raghunathan, S. Chowdhury, R. J. Choudhary, and D. M. Phase, The mystery behind dynamic charge disproportionation in BaBiO3, Nano Lett. 21, 8433 (2021).

[12]	S. Gao, S. Zhang, C. Wang, S. Yan, X. Han, X. Ji, W. Tao, J. Liu, T. Wang, S. Yuan, G. Qu, Z. Chen, Y. Zhang, J. Huang, M. Pan, S. Peng, Y. Hu, H. Li, Y. Huang, H. Zhou, S. Meng, L. Yang, Z. Wang, Y. Yao, Z. Chen, M. Shi, H. Ding, H. Yang, K. Jiang, Y. Li, H. Lei, Y. Shi, H. Weng, and T. Qian, Discovery of a single-band Mott insulator in a van der Waals flat-band compound, Phys. Rev. X 13, 041049 (2023).

[13]	Y. Haraguchi, C. Michioka, M. Ishikawa, Y. Nakano, H. Yamochi, H. Ueda, and K. Yoshimura, Magnetic–nonmagnetic phase transition with interlayer charge disproportionation of Nb3 trimers in the cluster compound Nb3Cl8, Inorg. Chem. 56, 3483 (2017).

[14]	J. Hubbard and B. H. Flowers, Electron correlations in narrow energy bands, Proc. R. Soc. A 276, 238 (1963).

[15]	H. Zhang, Z. Shi, Z. Jiang, M. Yang, J. Zhang, Z. Meng, T. Hu, F. Liu, L. Cheng, Y. Xie, J. Zhuang, H. Feng, W. Hao, D. Shen, and Y. Du, Topological flat bands in 2D breathing-Kagome lattice Nb3TeCl7, Adv. Mater. 35, 2301790 (2023).

[16]	M. J. Wahila, G. Paez, C. N. Singh, A. Regoutz, S. Sallis, M. J. Zuba, J. Rana, M. B. Tellekamp, J. E. Boschker, T. Markurt, J. E. N. Swallow, L. A. H. Jones, T. D. Veal, W. Yang, T.-L. Lee, F. Rodolakis, J. T. Sadowski, D. Prendergast, W.-C. Lee, W. A. Doolittle, and L. F. J. Piper, Evidence of a second-order Peierls-driven metal-insulator transition in crystalline NbO2, Phys. Rev. Mater. 3, 074602 (2019).

[17]	Z. Wang, O. I. Malyi, X. Zhao, and A. Zunger, Mass enhancement in 3d and s-p perovskites from symmetry breaking, Phys. Rev. B 103, 165110 (2021).

[18]	G. Gou, I. Grinberg, A. M. Rappe, and J. M. Rondinelli, Lattice normal modes and electronic properties of the correlated metal LaNiO3, Phys. Rev. B 84, 144101 (2011).

[19]	L. Zhang, Y. Zhou, L. Guo, W. Zhao, A. Barnes, H.-T. Zhang, C. Eaton, Y. Zheng, M. Brahlek, H. F. Haneef, N. J. Podraza, M. H. W. Chan, V. Gopalan, K. M. Rabe, and R. E.-Herbert, Correlated metals as transparent conductors, Nat. Mater. 15, 2 (2016).

[20]	Y. Tomioka, T. Ito, E. Maruyama, S. Kimura, and I. Shindo, Magnetic and electronic properties of single crystals of perovskite nickelate oxide LaNiO3 prepared by the laser diode floating zone method, J. Phys. Soc. Jpn. 90, 034704 (2021).

[21]	B. Li, D. Louca, S. Yano, L. G. Marshall, J. Zhou, and J. B. Goodenough, Insulating pockets in metallic LaNiO3, Adv. Electron. Mater. 2, 1500261 (2016).

[22]	J. Sun, A. Ruzsinszky, and J. P. Perdew, Strongly constrained and appropriately normed semilocal density functional, Phys. Rev. Lett. 115, 036402 (2015).

[23]	S. Grytsiuk, M. I. Katsnelson, E. G. C. P. van Loon, and M. Rösner, Nb3Cl8: A prototypical layered Mott-Hubbard insulator, npj Quantum Mater. 9, 1 (2024).

[24]	M. Ströbele, J. Glaser, A. Lachgar, and H.-J. Meyer, Structure and electrochemical study of Nb3Cl8, Z. Anorg. Allg. Chem. 627, 2002 (2001).

[25]	A. Zunger, S.-H. Wei, L. G. Ferreira, and J. E. Bernard, Special quasirandom structures, Phys. Rev. Lett. 65, 353 (1990).

[26]	A. Baldereschi, Mean-value point in the Brillouin zone, Phys. Rev. B 7, 5212 (1973).

[27]	D. J. Chadi and M. L. Cohen, Special points in the Brillouin zone, Phys. Rev. B 8, 5747 (1973).

[28]	V. Popescu and A. Zunger, Extracting E versus k effective band structure from supercell calculations on alloys and impurities, Phys. Rev. B 85, 085201 (2012).

[29]	J. C. Slater, Magnetic effects and the Hartree-Fock equation, Phys. Rev. 82, 538 (1951).

[30]	G. Catalan, Progress in perovskite nickelate research, Phase Transitions 81, 729 (2008).

[31]	J. Varignon, M. N. Grisolia, J. Íñiguez, A. Barthélémy, and M. Bibes, Complete phase diagram of rare-earth nickelates from first-principles, npj Quantum Mater. 2, 1 (2017).

[32]	C. E. Moore, Atomic Energy Levels as Derived from the Analyses of Optical Spectra: The Spectra of Hydrogen, Deuterium, Tritium, Helium, Lithium, Beryllium, Boron, Carbon, Nitrogen, Oxygen, Fluorine, Neon, Sodium, Magnesium, Aluminum, Silicon, Phosphorus, Sulfur, Chlorine, Argon, Potassium, Calcium, Scandium, Titanium, and Vanadium (U.S. Department of Commerce, National Bureau of Standards, 1949).

[33]	C. E. Moore, A Multiplet Table of Astrophysical Interest (U.S. Department of Commerce, Office of Technical Services, 1959).

[34]	M. Imada, A. Fujimori, and Y. Tokura, Metal-insulator transitions, Rev. Mod. Phys. 70, 1039 (1998).

[35]	A. Fazzio and A. Zunger, Many-electron multiplet effects in the optical spectra of NiO, CoO and MnO, Solid State Commun. 52, 265 (1984).

[36]	A. Zunger, Electronic structure of 3d Transition-Atom Impurities in Semiconductors, in Solid State Phys., edit. F. Seitz, D. Tumbull, and H. Ehrenreich (Acad Press NY) 39, 275-464 (1986).

[37]	J. S. Griffith, The Theory of Transition-Metal Ions (Cambridge University Press, 1971).

[38]	G. M. Lawrence and B. D. Savage, Radiative lifetimes of UV multiplets in boron, carbon, and nitrogen, Phys. Rev. 141, 67 (1966).



 Paper Title: Bound states in doped charge transfer insulators.


	
	
	



Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,
	Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China


Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China


Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China



Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,
	Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China



Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,
	Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
New Cornerstone Science Laboratory, 
	Beijing, 100190, China




Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,
	Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China



§ ABSTRACT 
Understanding the physics of doping a charge transfer insulator is the most important problem in high-temperature superconductivity. In this work, we show that an in-gap bound state emerges from the localized hole of the doped charge transfer insulator. We propose an approximate ground state wavefunction based on one localized Zhang-Rice singlet and the Neel state. By calculating the excitation states with one hole added and removed from this ground state, we successfully identify the existence of bound states inside the charge transfer gap. This feature is further proved by the MPS-based Lanczos study of a system of 4×4 CuO_2 unit cells. How these bound states evolve into metallic states is further discussed. Our findings identify the key component of recent STM results on lightly doped Ca_2CuO_2Cl_2 and provide a new understanding of hole-doped charge transfer insulators.



Bound states in doped charge transfer insulator
    Tao Xiang
    August 5, 2024
===============================================




As a relative of Mott insulator, the concept of charge transfer (CT) insulator plays an important role in the understanding of the insulating behaviors in correlated transition metal insulators, especially in cuprates <cit.>. The key energy scale for charge transfer insulators is the charge-transfer energy between d electrons and their neighbor p electrons Δ_CT, instead of the onsite Hubbard repulsion energy U for the d electrons. Therefore, the doped holes in cuprates reside primarily on oxygen sites and together with the hole in the Cu site. They form the well-known Zhang-Rice (ZR) singlets because the Cu^3+ (3d^8) state has a higher energy <cit.>. These singlets are the dominant charge carriers in the hole-doped cuprates and become superconducting under the influence of super-exchange interaction J𝐒_i𝐒_j between d electrons of Cu <cit.>. Although this effective single-band t-J model and the corresponding Hubbard model are extensively explored <cit.> and the condition for the appearance of superconductivity is studied recently <cit.>, 
a comprehensive understanding of the process of doping the charge transfer insulator and Mott insulator is a central theme for the ultimate understanding of the high-temperature superconductivity <cit.>. 

Recently, a new scanning tunneling microscopy (STM) experiment was carried out on the extremely lightly doped cuprates Ca_2CuO_2Cl_2 (CCOC) <cit.>. The Na^1+ atoms are introduced to replace Ca^2+ atoms resulting in hole doping. Surprisingly, bound states inside the charge transfer gap of CCOC were observed, as illustrated in Fig.<ref>(a) <cit.>. The spatial tunneling spectrum distribution for these bound states shows a similar contour of the Zhang-Rice singlet.
This new observation provides great opportunity to understand the hole-doped charge-transfer insulators and Mott insulators
<cit.>.
In this work, we provide a comprehensive study of the doped charge transfer insulator and point out the bound states are indeed related to the localized Zhang-Rice singlets.

There are two important energy scales for the above observations, the CT gap Δ_CT≃ 2 eV and the bound state energy E_b≃ 0.6-0.7 eV. These two energy scales are much larger than the exchange energy J∼ 0.14 eV and the hopping scale t∼ 0.35 eV <cit.>.

Hence, the kinetic energy and exchange energy can be treated as perturbations for the bound states in a doped CT insulator. This is the central starting point for our following discussions. Another important fact is that the bound states observed in STM experiment <cit.> are localized, or more precisely, the ZR singlets are localized. In the following discussion, we will show that the appearance of in-gap bound states results from the localized ZR singlet state. 




With these two facts, we start from a qualitative perspective and propose the below classical product state wavefunction |GS⟩ as the approximated ground state wavefunction, which provides a qualitative description of the hole-doped CT insulator in the background of antiferromagnetism (AFM):





    |GS⟩= ψ_0A^†d_0A↑|AFM⟩.

Following the conventional notation <cit.>, we define the vacuum |vac⟩ as the fully filled Cu d^10 and O p^6 states from the hole picture. d_iασ^† is the hole creation operator for Cu 3d_x^2-y^2 at unit cell i and AFM sublattice index α=A,B. 
|AFM⟩ is the Neel AFM wavefunction.
The ψ_0A^† is the ZR singlet operator for the bound site at center 0 and sublattice A. The exact form of ψ_0A^† will be discussed below as an extension of the ZR singlet proposed in the original Zhang-Rice paper <cit.>. We also assume there are total N holes for the ground states. More precisely, there is one hole for each magnetic site and two holes for the ZR site.

Indeed, the AFM Neel state should not be the ground state for the Heisenberg model and the half-filled Hubbard model owing to the quantum fluctuations in two dimensions <cit.>. However, the energy difference between the Neel state and the true ground state should be less than order J. Then, we can safely argue the Neel state is enough for the localized bound state problem.
On the other hand, ψ_0A^† may also deviate from the exact state, which is also a controllable perturbation. 

Therefore, |GS⟩ is a faithful approximate ground state.

Before a more detailed discussion, we introduce the three-band Hamiltonian <cit.> below

    H   =   ϵ̃_d ∑_i d_iσ^† d_iσ+ϵ̃_p  ∑_l p_lσ^† p_lσ +U ∑_in̂_d↑n̂_d↓
       +   ∑_⟨ il ⟩ t_pd(d_iσ^† p_lσ+h.c.)
     -∑_⟨ ll' ⟩ t_pp(p_lσ^† p_l'σ+h.c.) 
       +    E_loc∑_δ p_0δσ^† p_0δσ

where  p_lσ^† is the hole creation operator for O (2p_x,2p_y) at site l. t_pd and t_pp are hopping integrals involving p-d and p-p processes respectively. ϵ̃_d/p is the on-site potential for d/p orbitals in the hole picture respectively. More importantly, we add a
E_loc term for the local potential energy at O atoms around the bound site due to dopants, which stabilize a localized ZR solution.
The strong coupling theory of the three-band model without hole doping is the Heisenberg model. 
Moving to the hole doping, the three-band Hamiltonian becomes the standard t-J model as discussed by Zhang and Rice <cit.>.


Since we focus on the tunneling spectrum of STM, the tunneling density of states (DOS) at voltage V at 𝐫 following Fermi's golden rule can be written as

    dI/dV(𝐫,ω) ∝∑_n |⟨ n| ϕ_σ^†(𝐫) |GS⟩|^2 δ(-ω-E_n+E_GS) 
    
        +∑_m |⟨ m| ϕ_σ(𝐫)|GS⟩|^2 δ(-ω-E_GS+E_m)
 
where ϕ_σ^†(𝐫) will be equal to d_iσ^† or  p_lσ^† depending on the position and ω is the energy in the unit of eV. |n⟩ are adding hole excitation eigenstates with hole number n>N and |m⟩ are removing hole excitation eigenstates with hole number m<N. 
Note that we have switched the energy ω in the delta functions to the electron picture for experimental comparison.
Hence, the gap for the insulating behavior is related to 

    Δ_gap=E_N+1+E_N-1-2E_GS


Now, we can apply the above formulas to our case. Firstly, we focus on the spectrum away from the ZR center. The tunneling process is just equivalent to the AFM tunneling spectrum. 
For the |n⟩ excitation with N+1 holes, the problem is reduced to a single hole moving in the background of the quantum antiferromagnet <cit.>. 
As discussed above, t and J are perturbations. We can use the local eigenstates as the approximated states for the spectrum problem.
For example, we use the localized ZR singlet to replace the moving hole as the approximation excited state |n⟩ <cit.>. 

Hence, the spectrum problem reduces to exact diagonalization problems around one Cu cell as illustrated in Fig.<ref> (b) <cit.>. Using the local D_4 symmetry, we can group the four p orbital into one b_1, one a_1 and two degenerate e symmetric basis:

    b_σ   =   (p_xσ-p_yσ-p_-xσ+p_-yσ)/2 
    
        a_σ   =   (p_xσ+p_yσ-p_-xσ-p_-yσ)/2  
    
        e_1σ   =   (p_xσ+p_-xσ)/√(2), e_2σ=(p_yσ+p_-yσ)/√(2)

Their corresponding energies become ϵ_b=ϵ̃_p-2t_pp,ϵ_a=ϵ̃_p+2t_pp and ϵ_e=ϵ̃_p. Due to d_σ only has b_1 symmetry, it only has overlap with b_σ orbital.



Locally, the local ground state |ψ_1h⟩  with one hole filling is the slight mixing between |d_↑⟩ and  |b_↑⟩. Here, we choose one spin-up polarized state without loss of generality. The lowest energy two-hole state |ψ⟩ is the mixing between ZR singlet (|d_↑b_↓⟩-|d_↓b_↑⟩)/√(2), |d_↑d_↓⟩ and |b_↑b_↓⟩.
With these states, we can directly calculate the spectrum as plotted in Fig.<ref>(a). The gap Δ_gap between N → N-1 (ω>0) and N → N+1 (ω<0) is around 2.8 eV owing to the charge transfer process. 

Next, we move to the localized ZR center. We can also calculate the tunneling spectrum as shown in Fig.<ref>(b). Now, the gap Δ_gap^ZR at the ZR center has reduced to 1.2 eV. There is indeed an in-gap bound state inside the charge transfer Δ_gap. The essential reason for this bound state is owing to the fact that the local ground state wavefunction has changed to the localized ZR state as defined in Eq. <ref>. Hence, the removing hole process (N → N-1) is between ZR |ψ⟩ and |ψ_1h⟩. Adding hole process (N → N+1) is between ZR |ψ⟩  and a three-hole excitation state |ψ_3h⟩.
There are two degenerate lowest energy three-hole excitation states |ψ_3h⟩=e_1σ^†|ψ⟩ and |ψ_3h⟩=e_2σ^†|ψ⟩.
Hence, Δ_gap^ZR is reduced to new 1.2 eV bound state. Besides these two states, |ψ_3h⟩ can also be other combinations leading to other peaks at lower energy. 

Clearly, the above bound state result depends on the parameters we choose. Here, we adopt the realistic parameter based on a combined local density functional theory (DFT-LDA) and quantum Monte Carlo work for doped cuprates and CCOC <cit.>. We also test the other parameters used in Ref. <cit.> and <cit.>. Similar bound states show up, although the actual gap sizes vary in different cases. 




Additionally, we also need to consider the motion effect of a single hole or a single electron in the background of quantum antiferromagnetic. 


Historically, this hole motion has been investigated by a self-consistent perturbation theory and other methods <cit.>. The resulting bandwidth W_h is about 2 ∼ 3 J. The bandwidth W_e is also around the same order <cit.>. Hence, the energy level we discussed above should be further broadened to W_h or W_e, as illustrated in Fig.<ref>(c) and Fig.<ref>(d). 






After gaining a qualitative understanding, we want to provide more evidence from many-body computation. We employ the MPS-based Lanczos method for the calculation of the spectrum function <cit.>. The strategy is to represent the Lanczos basis in the matrix-product state (MPS). A special reorthogonalization process is performed after the standard Lanczos method to resolve the loss of orthogonality resulting from the compression of MPS <cit.>. The spectrum functions are then obtained from the eigenstates obtained by Lehmann representation <cit.>. 

The system we studied consists of 4 × 4 CuO_2 unit cells (shown in the inset of Fig. <ref>). Since the ZR singlet state is localized in our setup as discussed above, the finite size effect should be small and the 4 × 4 system is enough for the main feature. The bond dimension for MPS in the calculation is 2000 and the results remain almost unchanged with larger bond dimensions.  
We add the localization potential to the 4 p orbitals surrounding a Cu d orbital (the dashed circle in the inset of Fig. <ref>).
A magnetic pinning field with strength h_m = 0.5 is also applied to the Cu site in the downright corner to show the spin pattern in the system. The distribution of holes and local spin for the one hole-doped system can be found in the inset in Fig. <ref>, where we find the doped hole is indeed pinned to the desired oxygen sites forming a localized ZR with the hole in the center Cu site.

 
 


The local DOS at the same Cu site for the half-filling and one-hole doped systems are plotted in Fig. <ref>.
We distinguish the local DOS of adding electron and hole processes with filled and open markers.
For the blue square lines, the tunneling process is for the half-filling CT insulator. We obtain a charge transfer gap of about 2.15 eV (highlighted by the black line), which is improved closer to the experimental value than 2.8 eV in Fig.<ref>.
More importantly, from the result of the one-hole doped case as shown in the red circle line, we can find a finite density of states inside the charge transfer gap.
This newly emerged bound state matches well with our qualitative discussion, proving the existence of localized ZR. 
The new gap Δ_gap^ZR is about 0.8 eV, which is consistent with the experimental observations <cit.>.



For the understanding of STM measurement, we also need to take the tunneling matrix effect into account <cit.>. STM tip coupling with the CCOC mainly through the top Cl p_z orbitals as shown in Fig.<ref>(a). Cl atoms directly sit above the Cu sites. The overlap between p_z and d_x^2-y^2 directly beneath it is strictly zero by symmetry. However, p_z has overlap with p_x(y) of O atom and through the overlap p_z state hybridizes with the hole state centered on the nearest neighbor (NN) Cu atom. Hence, the operator ϕ_σ^†(𝐫) in Eq.<ref> should be modified to

    ϕ_σ^†(𝐫) =  ∑_τ (-1)^P_τ⟨𝐫|i,Cl⟩ p^†_i,τ,σ + ∑_τ (-1)^D_τ⟨𝐫|i,Cl⟩ d^†_i+τ,σ
 
where ⟨𝐫|i,Cl⟩ is the overlap between STM tip and Cl p_z state at site i. The first term describes the tunneling to O p_x(y) around the Cu at site i in which (-1)^P_τ=-1 for τ=-x,-y and (-1)^P_τ=1 for τ=x,y according to symmetry. The second term describes the tunneling to NN Cu d_x^2-y^2 through O in which (-1)^D_τ=-1 for τ=± x and (-1)^D_τ=1 for τ=± y. 



Fig.<ref>(b) shows the DOS map if STM scans the sample at the energy of in-gap bound state. When the STM tip is placed at the position of the ZR center, it couples with a_1 state of the four O atoms around it due to the factor in the first term of Eq.<ref>. However, the in-gap state spectrum has no contribution of O a_1 state and NN Cu in the previous calculation. Thus the DOS signal vanishes at the ZR center. Whereas at the position of NN Cu, STM tip can couple to one O p_x(y) and Cu d_x^2-y^2 of the ZR cell, which has spectral weight at in-gap state energy. Thus, STM maps the DOS at NN Cu of the ZR center, which is qualitatively consistent with the experiment observed ZR-like contour <cit.>.


Finally, a natural question arising from the in-gap bound states is how these bound states evolve into metallic states.
A straightforward calculation is challenging owing to the complicated spin pattern and random doping. 
There is one way to determine the critical hole doping density x_c by comparing E_b and the coherent hole motion bandwidth. We use the renormalized mean field of t-J model based on Gutzwiller approximation to estimate bandwidth W_GW <cit.>. The hopping renormalization factor is g_t=2x/1+x while the spin exchange renormalization factor is g_s=4/(1+x)^2. x is the average hole doping density 1-⟨n̂⟩. By decoupling the J𝐒_i𝐒_j into the nearest neighbor hopping bond, we find the W_GW of renormalized t-J model is about 0.65 eV at x=0.046. Hence, the x_c is estimated around 5%.
This estimation is consistent with experimental findings in Na-doped CCOC <cit.>, where superconductivity only emerges after x>5%.
Hence, if the charge-transfer insulator is undoped, a charge-transfer gap Δ_gap dominates the charge dynamics of the antiferromagnetic insulating ground state, as illustrated in Fig.<ref>(c).
Then, keeping light-doping the system, a bound state with the tunneling gap Δ_gap^ZR emerges from the charge-transfer gap as illustrated in Fig.<ref>(e), where ZR singlets are randomly localized. 
If the kinetic energy overcomes the binding energy, an itinerant ZR band overlaps with the valence band showing a metallic behavior as illustrated in Fig.<ref>(f). 







In summary, we successfully identify the origin of the in-gap bound states emerged from the lightly doped charge transfer insulator
using an approximate ground-state wavefunction based on one localized Zhang-Rice singlet and the Neel state.
By calculating the tunneling spectrum, the tunneling gap Δ_gap^ZR at the ZR center is around half of the charge-transfer gap Δ_gap. Moreover,
we carry out MPS-based Lanczos calculations on system consisted of 4×4 CuO_2 unit cells with local potential to pin the Zhang-Rice singlet. The results show a bound state with 0.8 eV emerges within the charge transfer gap. Taking the STM tunneling matrix effect into account, the tunneling contour at the bounding energy around the ZR center is weak while the four neighbors are strong.
How these bound states evolve into metallic states is further discussed.
All these findings are consistent with the recent experiment observations in lightly hole-doped CCOC <cit.> and provide a comprehensive understanding of doped charge-transfer insulators. 








Acknowledgement:

We thank Yayu Wang and Shusen Ye for helpful discussion of their experimental results. 
We acknowledge the support by the Ministry of Science and Technology  (Grant No. 2022YFA1403900), the National Natural Science Foundation of China (Grant No. NSFC-11888101, No. NSFC-12174428, No. NSFC-11920101005 and No. NSFC-12274290), the New Cornerstone Investigator Program, and the Chinese Academy of Sciences Project for Young Scientists in Basic Research (2022YSBR-048).










Supplemental Material: Bound states in doped charge transfer insulator






























































 §.§ LDOS for broader energy range for the 4 × 4 system

In Fig. <ref>, we show the LDOS result for the same system as in Fig. <ref> in the main text (consisted of 4 × 4 CuO_2 unit cells) from MPS-based Lanczos method. In Fig. <ref> in the main text, results between -3 eV and 3 eV are shown, while in Fig. <ref>, the LDOS for broader energy range is plotted, where we can identify the lower (around -9 eV) and upper (around 4 eV) Hubbard bands.  





 Paper Title: Quantum Order by Disorder: A Key to Understanding the Magnetic Phases of BaCo$_2$(AsO$_4$)$_2$.








Present Address: Department of Physics and National High Magnetic Field Laboratory, University of Florida, Florida 32611, USA.
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.


National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.


Los Alamos National Laboratory, Los Alamos, NM 87545, USA.


Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA.


Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA.


National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310-3706,  USA.


[Corresponding authors:]vzapf@lanl.gov
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.


[Corresponding authors:]winters@wfu.edu
Department of Physics and Center for Functional Materials, Wake Forest University, Winston-Salem, NC 27109, USA.


[Corresponding authors:]ml10k@lanl.gov
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.




§ ABSTRACT 
 (BCAO), a honeycomb cobaltate, is considered a promising candidate for materials displaying the Kitaev quantum spin liquid state. This assumption is based on the distinctive characteristics of  Co^2+ ions (3d^7) within an octahedral crystal environment, resulting in spin-orbit-coupled J_ eff = 1/2 doublet states. However, recent experimental observations and theoretical analyses have raised questions regarding this hypothesis. Despite these uncertainties, reports of continuum excitations reminiscent of spinon excitations have prompted further investigations. In this study, we explore the magnetic phases of BCAO under both in-plane and out-of-plane magnetic fields, employing dc and ac magnetic susceptibilities, capacitance, and torque magnetometry measurement. Our results affirm the existence of multiple field-induced magnetic phases, with strong anisotropy of the phase boundaries between in-plane and out-of-plane fields. To elucidate the nature of these phases, we develop a minimal anisotropic exchange model. This model, supported by combined first principles calculations and theoretical modeling, quantitatively reproduces our experimental data. In BCAO, the combination of strong bond-independent XXZ anisotropy and geometric frustration leads to significant quantum order by disorder effects that stabilize colinear phases under both zero and finite magnetic fields. 


Quantum Order by Disorder: A Key to Understanding the Magnetic Phases of 
    Minseong Lee
    
=========================================================================









§ INTRODUCTION
 



The interplay between crystal field symmetry, spin-orbit coupling, and the electron configuration of cobalt ions (Co^2+, 3d^7) in a nearly ideal octahedral oxygen environment stabilizes the J_ eff = 1/2 state, a fundamental building block of numerous intriguing physical phenomena in various configurations. One of the intriguing theoretical proposals is that cobaltates with a J_ eff = 1/2 state can potentially exhibit bond-dependent exchange interactions, including what is lately called the Kitaev exchange interaction <cit.>. It is well known that when the Kitaev exchange interaction dominates the spin Hamiltonian, the ground state can become a Kitaev quantum spin liquid (KQSL) due to strong magnetic frustration <cit.>. The KQSL is highly sought after among frustrated magnets because it has the potential to host exotic emergent phenomena such as Majorana Fermions and topological order, with significant applications in quantum computing <cit.>. Therefore, several honeycomb cobalt-based magnets, including Na_2Co_2TeO_6 <cit.>, and Na_3Co_2SbO_6  <cit.> have been investigated intensively. 

Recently, another cobalt-based honeycomb-lattice material  (BCAO) has been proposed as an ideal platform for realizing a KQSL <cit.> (Figure  <ref>a). BCAO shows a series of magnetic phase transitions under applied magnetic field (H). The zero-field order is identified as either an incommensurate spiral or double-stripe (DS), which is suppressed in favor of a collinear up-up-down (UUD) phase under field <cit.>. These spin structures are illustrated in Figure <ref>b. For magnetic fields in the crystallographic ab-plane, the latter phase is subsequently suppressed by relatively weak fields (0.5 T), above which a KQSL phase has been suggested <cit.>. For field in the out-of-plane direction, which is theoretically more conducive to a field-induced KQSL phase <cit.>, a broad magnetic continuum was observed <cit.>. However, conflicting reports exist regarding whether BCAO is best described by an extended Kitaev model with strongly bond-dependent magnetic couplings or as an XXZ-J_1-J_3 model with weakly bond-dependent couplings; recent inelastic neutron scattering measurements, as well as ab initio calculations <cit.>, align more with the XXZ-J_1-J_3 model <cit.>. It has also been pointed out that strong distortions of oxygen octahedra, whose the trigonal term in crystal-field splitting matrix is larger than one fifth of spin-orbit coupling, lead to Ising- and XY-like spin behavior <cit.>. Additionally, the weak ligand-assisted exchange interactions weaken bond-dependent exchange interactions <cit.>. Hence, further investigation is still required to characterize the phases of BCAO comprehensively. Investigating the angular dependence of phase boundaries is an effective method to understand the origin of magnetic anisotropy <cit.>. Recently, Safari et al. explored the phase diagram of BCAO using magnetotropic susceptibility measurements and theoretical models <cit.>. Their findings indicate that while classical models qualitatively describe the system's behavior, quantum corrections are essential for precise critical field predictions <cit.>.






In this study, we explored the details of magnetic field-induced ground states in BCAO using ac and dc magnetic susceptibility, capacitance measurement, and torque magnetometry measurements at various temperatures (T). With the aim of establishing the origin of the magnetic anisotropy, we study the angular dependent T-H phase boundaries by tuning the magnetic field orientation. With the application of a magnetic field, multiple-phase transitions were identified for H along the in-plane direction, consistent with previous studies <cit.>. In comparison, for H close to the c-axis, the critical field for suppression of ordering is drastically increased, and intermediate phase transitions disappear or are washed out at the lowest temperatures.

To understand the full details of the phase diagram, we employ first-principles-based calculations to provide a starting point for analyzing the magnetic couplings. The resulting model is refined by fitting the experimental phase diagram employing second-order energy corrections <cit.> to account for quantum fluctuation effects on the stability of various magnetic phases. This combined approach allows for the precise reproduction of the experimental observations. Consistent with recent assertions <cit.>, we conclude that quantum order-by-disorder (QOBD) effects dominate in BCAO, influencing both the critical field values and specific progression of field-induced phases.







§ RESULTS
 


 §.§ Magnetic properties and phase diagrams for in-plane fields




Figure <ref>a illustrates the magnetization measured at various temperatures as a function of the field applied along the a-axis. Hysteresis loops emerge between 0.1 T (H_1) and 0.2 T (H_2), which become smaller with increasing temperature and are almost invisible above 4.0 K. Beyond 0.5 T, there is a steep increase in magnetization before reaching saturation around 0.6 T (H_3). The saturation magnetization is about 2.3 μ_B/Co^2+, from which we extracted g_ab∼ 4.6. The magnetization between the saturation field and ∼0.15 T constitutes a one-third magnetization plateau. These findings are consistent with prior studies <cit.>. When the temperature exceeds T_N∼ 5.5 K, all features disappear, and a monotonic increase of the magnetization with field is observed, consistent with a thermal paramagnetic state. Figure <ref>b shows the ac magnetic susceptibility as a function field. The temperature dependent ac magnetic susceptibility is in Supplementary Figure 1. Three peaks are observed: The first two peaks at ∼H_1 and ∼H_2 are consistent with the jump in magnetization in the hysteresis loop and the end of the hysteresis loop. The third peak above ∼H_3 corresponds to the saturation field. ac susceptibility at 6 K does not show any anomalies. The magnetic phase diagram with field along the a-axis deduced from these measurements is presented in Figure <ref>c. We observed three thermodynamic phases (I, III, and IV) and one intermediate state (II) due to the hysteresis loop <cit.>. Neutron scattering experiments identified phase I as the DS phase or closely related spiral phase with in-plane wavevector close to Q = [1/4,0] <cit.>. Phase III is the UUD phase with 1/3 saturation magnetization and Q = [1/3,0] <cit.>. Given the differing wavevectors and the discontinuous change in wavevector through this phase transition, it is not surprising that these phases are separated by a first order phase transition. Phase II corresponds to a coexistence region of Phase I and III. Lastly, phase IV is the asymptotically polarized paramagnetic phase that is smoothly connected to high temperature paramagnetic phase.

Figures <ref>d-<ref>f show analogous results for field along b^*-axis. For this field direction, we observe the same progression of phase transitions. The hysteresis loops are wider but less pronounced, and H_3 is slightly lower in comparison with H||a. Therefore, the width of 1/3 plateau (phase III) is slightly narrower with magnetic field along b^*-axis. 



 §.§ Magnetic properties and phase diagrams for out-of-plane field


Figure <ref>a shows the temperature dependent dc magnetic susceptibility (M_c/H (T)) measured at various constant external magnetic fields. At 1 T, a clear phase transition from the paramagnetic to DS state is observed and marked with an empty circle around 5.3 K. Raising the field to 14 T only suppresses T_N to 4 K. The phase transition is therefore very robust against out-of-plane magnetic field, which is in sharp contrast with the behavior for in-plane fields. 

Figure <ref>b shows the magnetization versus out-of-plane field up to 14 T. At 2 K, the magnetization linearly increases without showing any anomaly, demonstrating a gradual spin canting towards the field direction without phase transitions. At 4 K, we observe a weak slope change around 14 T, denoted as a black arrow, which is consistent with the thermal phase transition between the DS and thermal paramagnet observed in Figure <ref>a. With increasing temperature to 4.5 K, the slope change shifts to slightly lower field of about 11.5 T. Suppression of the phase boundary between AFM and paramagnetic phase with increasing temperature is a typical behavior of antiferromagnets. 



Since 14 T is not enough to observe the magnetic saturation, we measured the magnetization with field along c-axis up to 50 T at Pulsed Field Facility in Los Alamos National Laboratory. 
Results from pulsed field magnetization experiments at various temperature are shown in Figure <ref>c and  <ref>d. The data are admittedly noisy, but the important features are clearly discernible. In the paramagnetic state at 10 K, the magnetization curve resembles the Brillouin function. 
Below T_N, the magnetization increases linearly and shows a slope change in the vicinity of 25 T, which we assign as the out-of-plane saturation field. Therefore dM_c/dH is a constant below ∼20 T and above ∼35 T. Due to the small crystal field excitation gap of Co^2+, it is common to observe a linearly increasing magnetization above the saturation field, due to Van Vleck paramagnetism <cit.>. By extrapolating the Van Vleck contribution to the zero field, we found a saturation moment of about 0.71 μ_B/Co^2+, from which we extract g_c∼ 1.5. We note that the saturation field with out-of-plane configuration in this study is higher than the field claimed by Zhang et al. <cit.> and Safari et al. <cit.>. As we demonstrated in Supplementary Figure 4 and 5, a tiny misalignment of the crystal can lead to a drastically suppressed saturation field. For example, only 6^∘ degrees off from the c-axis suppresses the saturation field 27 T to 4.5 T. 

The 25 T phase transition is smoothly connected to the zero field AFM to paramagnetic phase transition, additionally confirming that it is the saturation field. Thus, the out-of-plane magnetization measurements demonstrate that BCAO is highly anisotropic between in-plane and out-of-plane fields, with a saturation field ratio H_sat^c/H_sat^a of about 50. 




 §.§ Angular dependence of magnetic phase transitions


To investigate how the phase boundaries evolve with the field direction, we measured the magnetization, capacitance, and magnetic torque below T_N while varying the magnetic field orientation across the c-b^* plane. In Figure <ref>a, we first show the evolution of the magnetization. For an out-of-plane field (θ = 0^∘), the magnetization does not show any features up to 14 T. However, with a small rotation of the field direction towards the b^* direction, the transitions to the intermediate UUD and polarized phases appear. At θ = 6.3^∘, the phase transitions to the UUD and polarized phases occur at 1 T and 4 T, respectively. These move towards ∼H_1 and ∼H_2 as the field direction approaches b^*-axis, consistent with Figure <ref>d. The magnetic moment also monotonically increases as the field is rotated towards b^* direction, consistent with the anisotropy of the g-values.



Capacitance and magnetostriction measurements can be sensitive probes of magnetic phase transitions as the change in spin structure accompanies the change in electric susceptibility and lattice parameter of magnetic insulators. We found this to be the case for . Figure <ref>b shows the change in capacitance as a function of magnetic field at various angles. We applied the electric field at 16 kHz along b^*-axis. At 27^∘, the capacitance change shows clear two phase transitions corresponding to the phase transition from DS to UUD, and UUD to polarized phase. They evolve to higher fields as the field becomes close to c-axis. The anomaly from the phase transition from DS to UUD phase becomes weaker while that from UUD to polarized phase is more or less similar at all angles. The capacitance change and magnetostriction data with in-plane magnetic field can be found in Supplementary Figure 2 and 3.

Figure <ref>c and <ref>d show magnetic torque (τ) and its differentiation with magnetic field (dτ/dH) up to 30 T as a function of magnetic field at different fixed angles. Both τ and dτ/dH display a clear anomaly at phase transitions. When the field is only at 0.36^∘ from c-axis, a sharp increase in τ and a peak dτ/dH appear at about 23 T. This is reasonably well consistent with the saturation field observed in pulsed field magnetization shown in Figure <ref>c. The anomaly shifts quickly to lower field as the angle becomes larger than 1^∘ and ends up being only 2 T at 19.7^∘. 

In Figure <ref>a, we show the magnetic torque measured for continuous field-angle sweeps between the c and b^* axes at T = 0.5 K at various fixed magnitudes of H. All curves exhibit 180^∘ periodicity and a saw-tooth shape due to the strong magnetic anisotropy. The sharp change of sign of the torque near 0^∘ indicates the c-axis is the hard axis. Additional step-like anomalies are observed near θ = 0^∘ marking the first order transitions H_1 and H_3. These two features are only clearly observable up to 10 T and they merge into one above 10 T. 

In general, the magnetic torque may be expanded as <cit.>:

    τ/H = ∑_n = 1^∞ c_2nsin(2nθ) = c_2sin(2θ) + c_4sin(4θ) + ⋯

The experimental field-dependence of |c_2| and |c_4| is plotted in Figure <ref>c. Both coefficients increase steeply up to 4 T and then slowly decrease above 4 T up to the maximum field. The ratio of c_2 to c_4 remains more or less constant at ∼0.4 across the entire field range due to the dominant saw-tooth curve shapes.  In the limit of large field, such that the majority of the field angles correspond to the polarized phase, the c_2n coefficients have two contributions. The first contribution is field-independent, and depends exclusively on the g-anisotropy. For the implied range of g-values (g_ab∼ 4.6, g_c = 1.5 - 2.5), the limiting ratio for this contribution leads to |c_4/c_2| = 0.17 - 0.27. The next order contribution to the c_2n scales as 1/H, and is related to both the XXZ exchange anisotropy and  <cit.>.  The findings that c_2 and c_4 decay with increasing field, and that their ratio exceeds the g-only value both demonstrate significant exchange anisotropy is necessary to explain the observed torque. Below, we develop an exchange model for BCAO, and use these observations to verify the model.

From the phase transitions observed in the angular dependence of magnetic properties in Figure <ref> and <ref>a, an angular dependence of the magnetic phase diagram is constructed, as shown in Figure <ref>e. The red empty symbols represent the phase transition from the DS to the UUD phase (H_1), and the black empty symbols indicate the saturation field (H_3). As the field direction moves from the b^*-axis to the c-axis, both phase boundaries sharply increase below 10^∘. The saturation field reaches above 23 T, whereas the phase boundary between the DS and the UUD merges with the saturation field within 5^∘ above about 10 T.



 §.§ Microscopic magnetic model

To first define a realistic range of magnetic couplings, we employ the des Cloizeaux effective Hamiltonian (dCEH) approach <cit.>, based on exact diagonalization of the electronic d-orbital Hamiltonian for clusters include one and two Co sites [see Methods]. 

We first estimate the g-tensor by diagonalizing the electronic Hamiltonian on one Co site, and projecting the Zeeman operator into the low-energy space of j_1/2 states. This yields g_ab = 5.1 and g_c = 2.4, which further confirms accurate reproduction of the trigonal crystal field splitting in BCAO. To estimate the magnetic couplings, we exactly diagonalize the electronic Hamiltonian on pairs of sites and project the resulting low-energy Hamiltonian onto j_1/2 states. The relatively low symmetry crystalline environment allows for a large number of independent exchange constants. In particular, in the R3̅ space group, each first and third neighbor bond is constrained only by inversion symmetry, allowing six independent exchange parameters per bond:


    ℋ =    ∑_ij J_ij^xy (S_i^x S_j^x + S_i^y S_j^y) + J_ij^z S_i^zS_j^z  
        + 2J_ij^±±(S_i^xS_j^x - S_i^y S_j^y) + Γ_ij^yz (S_i^y S_j^z + S_i^z S_j^y) 
        + Γ_ij^xz(S_i^x S_j^z + S_i^z S_j^x) + Γ_ij^xy(S_i^y S_j^x + S_i^x S_j^y)

where the bond-dependent local x-axis is oriented parallel to the bond, and the z-axis is parallel to the crystallographic c-axis. Second neighbor couplings tend to be weaker and are ignored in first approximation. By varying the strengths of the Coulomb interactions within a realistic range [see Methods], we establish reasonable ranges for the first and third neighbor couplings: 
J_1^xy∼ -8 to -12 meV, 
J_1^z/J_1^xy∼ 0.5 to 0.6, 
J_1^±±∼ -0.6 to -0.8 meV, 
Γ_1^yz∼ 0.35 to 0.5 meV,
J_3^xy/J_1^xy∼ -0.25 to -0.4, and 
J_3^z/J_3^xy∼ 0.2. The remaining couplings are all found to have magnitude smaller than 0.1 meV.


There are several aspects of the computed couplings that bear mentioning. In BCAO, the local trigonal distortion of the CoO_6 octahedra has a significant impact on the magnetic Hamiltonian, inducing both the large g-anisotropy, and bond-independent XXZ anisotropy of the intersite exchange. Microscopically, the third neighbor couplings are dominated by a single exchange path involving the e_g electrons, which leads to antiferromagnetic interactions that are rendered anisotropic only by the crystal field effects. As such, the off-diagonal anisotropies should be nearly vanishing and the third neighbor interactions should be parameterized by a pure XXZ anisotropy, with J_3^xy being the same sign as J_3^z. For the first neighbor couplings, the dominant anisotropy is also of the bond-independent XXZ type. While Co^2+ materials can, in principle, host strongly bond-dependent couplings <cit.>, in BCAO we find that these are suppressed by a combination of factors. First, the large trigonal distortion from a pure octahedral environment significantly modifies the composition of the J_ eff = 1/2 moments, and imposes significant XXZ anisotropy on all couplings instead of bond-dependent exchange interactions. Second, the largest contribution to the first neighbor exchange comes from ferromagnetic “charge-transfer” contributions arising from hybridization of the Co e_g orbitals with the oxygen p-orbitals. Unlike the couplings between t_2g electrons, these interactions do not display large bond-dependent anisotropies. As such, the bond-dependent anisotropies (J_1^±± and Γ_1^yz) are subleading. Although some of these features are found in previously proposed models for BCAO <cit.>, not all are reproduced. This raises the question of whether the phase diagram can be understood by employing a model that follows more closely these microscopic considerations. 






To address this question, we made a comprehensive search of the parameter space around the predicted range, with a focus on fitting the experimental phase diagram. BCAO necessarily lies in a parameter region with multiple competing phases, making it necessary to consider quantum corrections to the classical state energies. We employ the method outlined in Ref. jackeli2015quantum [see Methods], which considers states defined by classical spin vectors, but accounts for local fluctuation effects on the state energies at second order in perturbation theory. This naturally includes the leading quantum order-by-disorder (QOBD) effects. From this search, we propose a simplified, but extremely finely tuned, six-parameter model:

J_1^xy = -11.75, J_1^z = -5.89, J_1^±± = -0.33, Γ_1^yz = -0.15, J_3^xy = +3.37, and J_3^z = +0.79 meV. The parameters are more consistent with microscopic considerations but otherwise broadly compatible with previously proposed models for BCAO with dominant XXZ anisotropy <cit.>.

In Figure <ref>, we show the computed evolution of the state energies and magnetization per site of the above model for in-plane fields. We evaluate m = |∑_i g·𝐒_i| from the classical spin vectors for which the second order energies correspond. The zero-field ground state is found to be a colinear DS phase, with spins oriented in the plane, as shown in Figure <ref>b. Under applied field, there are two first order transitions. The first is to the magnetization plateau UUD state, and the second to the polarized state. We estimate for H  ||  a: H_1 = 0.16 T, H_3= 0.55 T, and for H  ||  b^*: H_1 = 0.18 T, H_3 = 0.52 T. Thus, the model reproduces the slight anisotropy in the in-field critical fields observed in experiment. The predicted magnetization as a function of field is shown in Figure <ref>c and <ref>d, and reproduces well the experimental magnetization curves in Figure <ref>.


In Figure <ref>e and  <ref>f, we show a comparison of the experimental and theoretical phase diagram for fields oriented in the b^*c-plane. For strictly out-of-plane fields H||c, the model reproduces the direct transition from the DS state to the polarized phase. The critical field depends very strongly on the field angle; for θ = 0^∘, we compute H_3 = 36 T, while for θ = 1^∘, we compute H_3 = 18 T. This range is comparable to the broadened saturation observed in the pulsed-field measurements in Figure <ref>.
The sharp increase of the critical fields in the vicinity of the c-axis is mainly due to the large bond-independent XXZ anisotropy, which leads to an easy ab-plane. Overall, the model reproduces the phase boundaries well, confirming that the specific progression of field-induced phases can be understood within a microscopically compatible model. 

Finally, to further check that the model reproduces the characteristic magnetic anisotropy of BCAO, we simulated the magnetic torque data from the second-order corrected energies, where τ = ∂ E/ ∂θ. Results are shown in Figure <ref>b alongside the experimental data. The simulations reproduce very well both the positions and magnitudes of the first order jumps in τ corresponding to the various phase boundaries, as well as the rounding of τ(θ) at high field. A precise comparison can be made on the basis of the coefficients c_2n. In Figure <ref>d, we show the theoretical evolution of these coefficients with field. The finding that c_2 and c_4 are well reproduced by the model implies the relative magnitudes of both the g-anisotropy and XXZ exchange anisotropy are well captured. 

Overall, we conclude that the microscopically motivated model provides a very good reproduction of the experimental phase diagram of BCAO and therefore may serve as a starting point for future refinement. The present results confirm the validity of recently proposed models with primarily bond-independent XXZ anisotropy <cit.>, but show that the complex phase diagram of BCAO illuminated by various recent studies is compatible with the microscopic expectations for this material. 



§ DISCUSSION

As a consequence of the exchange anisotropy and frustration between first and third neighbor couplings, the phase diagram of BCAO is uniquely dominated by QOBD effects. Over the region of couplings we considered for BCAO, we observe several trends when comparing classical state energies with the second order corrected energies. First, primarily due to the strong XXZ anisotropy, both antiferromagnetic and polarized (ferromagnetic) states have finite energy corrections. Similar to the DMRG studies in Ref. <cit.>, for the physically relevant magnitudes of the XXZ anisotropy, we find that fluctuations tend to stabilize the polarized phase over the other phases. 
Overall, we find that quantum fluctuations tend to reduce the critical fields by favoring the polarized state, consistent with the conclusions of <cit.>. Second, we emphasize that the collinear DS and UUD phases only appear as ground states due to QOBD effects. We did not find any realistic parameters for which these are classical ground states for any magnetic field. Conversely, the spiral phase is never found to be stable once fluctuation effects are included. These findings are consistent with <cit.>. Finally, we emphasize that the present model for BCAO is extremely finely tuned. As can be anticipated from the (necessarily) small differences of the state energies in Figure <ref>a and  <ref>b deviations of any coupling on the order of ± 0.01 meV may be sufficient to ruin the agreement with experiment. This sensitivity likely also leads to variations in the results of different numerical approaches. In this work, we opted to use the inexpensive approximation of second-order energy corrections to facilitate a fine parameter search. Future refinement of the model could employ more sophisticated numerical methods, provided those approaches adequately treat the QOBD effects. 


In conclusion, this work is a comprehensive study of the magnetic field-induced ground states of BCAO, employing various techniques such as dc magnetization, capacitance, ac magnetic susceptibility at low temperatures. We mapped out the full T-H phase diagram for different magnetic field orientations, including H along a, b^*, c, and c'. To understand the specific progression of field-induced phases, we employed first-principles-based calculations to derive a model that reproduces all essential features of the low-temperature phase diagram. The stabilization of two colinear phases (I and III) in both zero and finite fields is attributed to the significant influence of QOBD effects. BCAO presents an intriguing material where quantum fluctuation effects dominate the phase diagram at both zero and finite field. This investigation highlights new avenues for understanding the complex behavior of BCAO under external magnetic fields.




§ METHODS
 


 §.§ Experimental details


High-quality single crystals of BaCo_2(AsO_4)_2 synthesized through a flux method at Oak Ridge National Laboratory in Oak Ridge, for which the detailed methods are described elsewhere  <cit.>. At low temperature, dc magnetization (M) is measured up to 60 T through millisecond timescales in pulsed magnet using custom-built ^3He pumping (400 mK–3 K), and ac magnetic susceptibility (χ_ac) and capacitance (C) measurements are shown here performed in PPMS for 2 K–30 K under the magnetic fields up to 14 T at Pulsed Field Facility in Los Alamos. For dc magnetization and ac magnetic susceptibility measurements, the sample with 1 mm^2 was aligned along a, b^*, c, and c' and mounted on the sample holder inside the compensation and pick-up coil, where c'-axis is 6^∘ from c-axis on the b^*-c plane.
The electrical contacts were made by painting silver epoxy on the two parallel opposite surfaces for the capacitance measurement and Andeen-Hagerling AH-2700A capacitance bridge was employed to measure the capacitance.

Magnetic torque was measured with the 31 T resistive magnet at the National High Magnetic Field Laboratory in Tallahassee, Florida using a home-made torque magnetometer. The torque magnetometer was formed from two parallel BeCu plates. A tiny aligned crystal with the mass of 0.13 mg was attached using epoxy on the cantilever plates. The capacitance change was monitored while the field or angle was swept using an AH-2700A capacitance bridge. More details can be found in https://nationalmaglab.org/user-facilities/dc-field/measurement-techniques/torque-magnetometry-dc/. 

We then converted the measured capacitance to the deflected angle (α) using the following equation

    C(α) = ϵ_0ϵ_rw/tanαlnltanα-h_0/h_0,

where α is the defected angle that is proportional to the magnetic torque, ϵ_0 and ϵ_r are vacuum and relative permittivity, respectively, and w, l, and h_0 are width, length of the BeCu plate and the gap distance between two BeCu plates at zero magnetic field. We extracted the coefficients c_2n using the following equation

    c_2n = 2/π∫_θ_1^θ_i+πτ/Hsin(2nθ) dθ,

where θ_i is the initial angle, τ is the measured magnetic torque signal.




 §.§ Theoretical phase diagrams


In order to account for quantum fluctuation effects, we employ the method of second-order energy corrections outlined in Ref. jackeli2015quantum. At each site i, the local quantization axis (denoted ẑ_i^') is along the local spin vector, such that ⟨ S_i^z'⟩ = +1/2 for all sites. Fluctuations are then driven by terms acting like J_ij^–S_i^-'S_j^-'. The second-order energy correction due to such fluctuations is:


    Δ E^(2) = ∑_ bonds   ijJ_ij^++ J_ij^–|⟨ m| S_i^-'S_j^-'|gs⟩|^2/E_gs, classical - E_m,  classical

where E_gs, classical is the classical energy of the spin configuration, and E_m, classical is the classical energy of the state with the two spins along the bond ij flipped. For the example of a Heisenberg honeycomb antiferromagnet, the classical ground state energy is E_gs, classical = -0.375  J per site, whereas the present approach provides E_gs, classical + Δ E^(2) = -0.563  J per site. The latter estimate (which includes local singlet fluctuations) is in much better agreement with E_gs = -0.544  J per site from series expansion <cit.> and quantum Monte Carlo <cit.>. 

In order to capture the energetically competitive magnetic orders depicted in Figure <ref>b, we employ a combination of 4-, 6-, and 8-site clusters of spins with appropriate boundary conditions, and numerically minimize E_ classical + Δ E^(2) with respect to the spin orientations. 
We consider the commensurate spiral state with Q = [1/4, 0] as a proxy for the proposed incommensurate spiral with Q = [0.27,0] <cit.>. We find the Q = [1/4, 0] spiral to be the classical ground state for a range of parameters, but it is completely replaced by the colinear ferromagnetic and Q = [1/4, 0] DS phases once Δ E^(2) is included. 



 §.§ First principles calculations




To estimate the magnetic couplings, we consider an electronic model for the Co 3d-orbitals, which is a sum of, respectively, one- and two-particle terms: ℋ_ el = ℋ_1p+ℋ_2p. The one-particle terms include intersite hopping, intrasite crystal field, and spin-orbit coupling, ℋ_1p = ℋ_hop+ℋ_ CF + ℋ_ SO:


    ℋ_hop =    ∑_ijαβσt_ij^αβ c_i,α,σ^† c_j,β,σ
    ℋ_ CF =     ∑_iαβσd_i^αβc_i,α,σ^† c_i,β,σ
    ℋ_ SO =     ∑_iαβσσ^'λ_ Mn⟨ϕ_i^α(σ)|𝐋·𝐒|ϕ_i^β(σ^')⟩ c_i,α,σ^† c_i,β,σ^'

where c_i,α,σ^† creates an electron at Co site i, in d-orbital α, with spin σ. To parameterize the one-particle terms, we performed fully relativistic density functional theory (DFT) calculations at the GGA (PBE <cit.>) level using FPLO <cit.>, a 12×12×12 k-grid, and the structure from Ref. djordevic2008baco2. The resulting Kohn-Sham Bloch functions were projected onto Co d-orbitals to yield single-particle terms from the Kohn-Sham Fock matrix written in the Wannier basis <cit.>. For BCAO, this approach yields accurate crystal-field excitation energies <cit.>, confirming the validity of the computed single-particle parameters. 

For the two-particle terms, we consider a combination of on-site and nearest neighbor intersite contributions: ℋ_2p = ℋ_U + ℋ_J^ nn, respectively. The on-site contributions are given by:


    ℋ_U = ∑_iαβδγ∑_σσ^'U_αβγδ c_i,α,σ^† c_i,β,σ^'^† c_i,γ,σ^' c_i,δ,σ


In the spherically symmetric approximation <cit.>, U_αβγδ are parameterized by the Slater parameters F_0^dd, F_2^dd, F_4^dd. For the (screened) on-site metal Coulomb interactions, we use F_2^dd = 11.55 eV, and F_4^dd = 7.68 eV, following the fitted parameters for LiCoO_2 from Ref. van1991electronic. We consider U_t2g= F_0^dd + (4/49)(F_2^dd+F_4^dd) within the realistic range of 5.0 to 5.5 eV.

The nearest neighbor intersite Hund's coupling is given by:


    ℋ_J^ nn = ∑_ijαβσσ^' J_H,ij^αβ c_i,α,σ^† c_j,β,σ^'^† c_i,α,σ^'c_j,β,σ


A significant contribution to the intersite magnetic couplings comes from ligand “charge transfer” exchange terms <cit.>, which account for perturbative processes where two holes meet on a single oxygen ligand. When downfolded into the d-orbital Wannier basis, these appear as intersite Hund's coupling terms. They give an overall ferromagnetic contribution to the intersite magnetic couplings between the low-energy j_1/2 doublets, which may be rendered anisotropic (bond-independent XXZ anisotropy) by the crystal-field splitting of the t_2g orbitals. In a previous study of BCAO <cit.>, some of the authors accounted for this term via an estimated correction added to the computed couplings. Here, we employ the improved approach utilized in <cit.>  that captures these effects at the level of the electronic Hamiltonian. The d-orbital Wannier functions can be written:


    |d̃_i,α,σ⟩ = ϕ_i,α^0|d_i,α,σ⟩ + ∑_n,βϕ_i,α^n,a|p_n,a,σ⟩


where i labels a given Co metal site, α is a d-orbital, n labels an oxygen ligand site, and a∈{x,y,z} is a p-orbital. The tilde on the left side of the equation indicates a Wannier function, whereas the absence of a tilde indicates a bare atomic orbital. Rotating the p-orbital Coulomb terms into the d̃ Wannier basis results in residual intersite Hund's coupling given by:


    J_H,ij^αβ =  ∑_n ∑_αβ    
    (U^p-J_H^p) ϕ_i,α^n,aϕ_i,α^n,bϕ_j,β^n,aϕ_j,β^n,b
        
    + J_H^p  ϕ_i,α^n,aϕ_i,α^n,aϕ_j,β^n,bϕ_j,β^n,b


where U^p and J_H^p are the atomic ligand Hubbard and Hund's couplings, respectively. The computed magnetic couplings depend most strongly on the overall magnitude of J_ij^αβ, rather than the specific orbital dependence, which provides for some inconsequential flexibility in the choice of U^p and J_H^p.  
We take U^p = 2J_H^p, and 
consider J_H^p within the range of 0.4 to 0.6 eV. This choice corresponds to roughly 60% to 80% of the atomic value for oxygen <cit.>. The sum over oxygen sites in Eq.  (<ref>) is then evaluated from the ab-initio Wannier function coefficients for the bridging oxygen atoms for each bond. 




§ ACKNOWLEDGEMENTS
 
The authors would like to thank A. L. Chernyshev, P. A. Maksimov, Y. B. Kim, F. Desrochers, E. Z. Zhang, K. A. Modic, and S. A. Crooker for insightful discussions. 
The experimental work, analysis and manuscript writing  (S.L., S.Z., S.M.T., L.P., C.A.B., V.S.Z., and M.L.) was funded by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers. The theoretical work, analysis and manuscript writing of S. M. W. was funded by the Oak Ridge Associated Universities
(ORAU) through the Ralph E. Powe Junior Faculty Enhancement Award. Computations were performed using the Wake Forest University (WFU) High-Performance Computing
Facility <cit.>, a centrally managed computational resource available to WFU researchers including
faculty, staff, students, and collaborators. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-2128556, the State of Florida, and the U.S. Department of Energy.



§ AUTHOR CONTRIBUTIONS
 
V.S.Z. and M.L. conceived and designed the project. S.L., S.Z., S.M.T., E.S.C. and M.L. performed measurements. S.M.W. performed the theoretical calculations.
L.P. and C.A.B. prepared high quality samples. S.L., V.S.Z., S.M.W., and M.L. wrote the manuscript, with contributions from all authors.



§ COMPETING INTERESTS
 
The authors declare no competing interests.




Correspondence and requests for materials should be addressed to
Vivien S. Zapf, Stephen M. Winter or Minseong Lee.







§ SUPPLEMENTARY MATERIAL ­– 
 “QUANTUM ORDER BY DISORDER: A KEY TO UNDERSTANDING THE MAGNETIC PHASES OF ”






















 Paper Title: Ge-based Clinopyroxene series: first principles and experimental local probe study.







IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.


IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.
High Performance Computing Chair, University of Évora, Largo dos Colegiais 2, 7004-516 Évora, Portugal


IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.


IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.


CNR-SPIN c/o Università degli Studi dell’Aquila, Via Vetoio 10, 67010 Coppito, L’Aquila, Italy.



Université Grenoble Alpes, CNRS, Institut Néel, 38000, Grenoble, France


Université Grenoble Alpes, CNRS, Institut Néel, 38000, Grenoble, France


C2TN, DECN, Instituto Superior Técnico, Universidade de Lisboa, Bobadela, Portugal.



Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970, São Paulo-SP, Brazil


Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970, São Paulo-SP, Brazil


IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.


jearaujo@fc.up.pt
IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.




§ ABSTRACT 
The structural and electronic properties of the CaMnGe2O6 and SrMnGe2O6 clinopyroxene systems have been investigated by means of perturbed angular correlation (PAC) measurements, performed at ISOLDE, 
combined with  ab initio electronic structure calculations within the density functional theory (DFT) framework. The partial density of states (PDOS) 
of the CaMnGe2O6 and SrMnGe2O6 stable compounds 
has been 
determined, and it has been observed that the requirement of including an on-site Hubbard-U potential was necessary  
in order to describe the highly correlated 
Mn 3d-states. By considering U_eff=4 eV, we obtained  
a band gap width  
of 1.82 eV and 1.70 eV, for the CaMnGe2O6 and SrMnGe2O6, respectively. 
Combining electric field gradient (EFG) first principles calculations, using a supercell scheme,  with experimental PAC results,  we were able to infer that the Cd probe can replace either the A (Ca, Sr) or the Mn sites in the crystalline structures. We also showed that Cd substitution is expected to lead to a reduction in the width of the band gap in these systems, evidencing opportunities for potential band-gap engineering.







Ge-based Clinopyroxene series: first principles and  experimental local probe study
    João P. Araújo
    August 5, 2024
===================================================================================







§ INTRODUCTION




Pyroxenes are a class of materials with general formula AMX2O6,  where A is a monovalent or a divalent cation, M is respectively a trivalent or a divalent cation, and X is typically either Si or Ge. They are well known in mineralogy and geology as they are one of the main rock-forming minerals of the Earth's crust <cit.>. Within this rich family of compounds, monoclinic pyroxenes (clinopyroxenes), with a 3d transition metal at the M-site, have recently been the subject of interest in the field of condensed matter physics due to the diversity of their magnetic properties, 
namely: multiferroicity in SrMnGe2O6 <cit.> and NaFe(Si/Ge)2O6 <cit.>; magnetoelectric effect in CaMnGe2O6 <cit.> and Li(Cr/Fe)Si2O6 <cit.>; and also ferrotoroidal ordering in LiFeSi2O6. 

In spite of their natural prevalence, pyroxenes have not been the subject of much computational or theoretical investigations. It is noteworthy of mentioning, that recent density functional theory (DFT) studies have shown that substitution of Al3+ by Tl^3+ in NaAlSi2O6  results in a reduction in the band gap width from 5.32 eV to 2.05 eV. A further reduction is observed when replacing Na by the organic cation CH3SH2, showing the high tunability of the band gap in these materials<cit.>. Other works have shown their potential usefulness in batteries, demonstrating that pyroxenes CaFe/Mn/Si2O6 could display high theoretical energy densities in Ca-based batteries<cit.>, whereas LiFeSi2O6 has been shown to undergo a reversible electrochemical reaction against Li, thus exhibiting potential for use as an electrode material in Li-ion batteries<cit.>. Lastly, the magnetic properties of CaMnGe2O6 were studied by Temnikov et al. <cit.> who computed the exchange parameters 
and showed that the magnetic frustration in the system is weak, which could explain the frequent occurrence of commensurate collinear antiferromagnetic structures in Ca2+-bearing pyroxenes.


The ground state structure for the clinopyroxenes considered in this study is a monoclinic Bravais lattice with space group C2/c (#15), where the M-cations are in octahedral sites (Wyckoff 4e atomic positions), the A-cations are in 8-coordinated sites (Wyckoff 4e atomic positions) and the X-cations are in tetrahedral sites (Wyckoff 8f atomic positions). Their structure is characterised by quasi-one-dimensional chains with edge-sharing MO6 octahedra running along the c-axis, and connected by corner-sharing XO4 tetrahedra, along the c-axis, as schematically represented in Fig.<ref>.



When the M-site is occupied by a magnetic cation, such an arrangement leads to low-dimensional magnetic properties and to magnetic frustration, due to the competition between interchain and intrachain interactions <cit.>. It is the existence and the possible interplay between the low dimensionality and frustration that is thought to give rise to the aforementioned diversity of magnetic properties in these materials <cit.>. Moreover, it has been suggested that the magnetic frustration can lead to spiral spin structures, which may be favourable towards magnetically driven ferroelectricity, making these compounds good candidates for multiferroic behaviour <cit.>. Since these properties are thought to arise from local features that are often not accurately described by long-range order based techniques, such as X-ray diffraction and Raman spectroscopy, it is important to utilise techniques that are more sensitive to local order to aid in the characterisation of these intriguing systems. Several works have, in the past, studied pyroxene systems by means of nuclear techniques, such as nuclear magnetic resonance spectroscopy <cit.> and Mössbauer spectroscopy <cit.>; however, and to the best of the authors' knowledge, this is the first work being reported where the pyroxene systems are studied via perturbed angular correlation (PAC) spectroscopy. In particular, none of the  
nuclear techniques above mentioned have been used to characterise the systems here under study.

In the present work, the AMnGe2O6 germanate clinopyroxene series were explored,  where the A-site is occupied by periodic table group-II elements, i.e.,  Be, Mg, Ca, and Sr. The former two compounds have so far not been experimentally realised, and they were thus studied solely through ab initio calculations. (Ca, Sr)1-xCdxMnGe2O6 and (Ca, Sr)Mn1-xCdxGe2O6 were explored by means of ab initio computational simulations, from which we were able to estimate the band gaps, as well as the effective masses of the electrons and holes of the systems; and by local Cd probe experiments using PAC spectroscopy, allowing one to closely follow the temperature evolution of the electric field gradient (EFG) tensor.



§ METHODS





 §.§ Experimental Methods


For the purpose of performing the PAC measurements, pelletised samples of CaMnGe2O6 and SrMnGe2O6 materials were implanted at ISOLDE-CERN by a 30 keV beam of 111mCd probes [111mCd→111Cd, t_1/2 = 48.6 minutes], which have a nuclear spin I=5/2 and an electric quadrupole moment of Q = 0.83(13) b  in the intermediate state of decay <cit.>. Small doses of the order of 1011 atoms/cm2 were implanted and subsequently an annealing was performed at a temperature of ∼1123 K for 20 minutes in air to recover from implantation damage. Confirmation for the recovery of point defects and for the incorporation of Cd into the lattice was given by the PAC measurements themselves, which yielded well-defined experimental R(t) anisotropy functions. The measurements were performed on 6-BaF2 detectors PAC spectrometers<cit.>, equipped with either a closed-cycle refrigerator or a special high-temperature furnace for temperature control. Due to the relatively short lifetime of the parent probe, each temperature point required the implantation of a new sample for satisfactory activity to be maintained during measurement.
The experimental R(t) function was fitted with exact numerical methods that build the expected observable by solving the hyperfine interaction Hamiltonian's characteristic equations<cit.>. 
The EFG, that characterizes the surrounding charge density that interacts with a
probe nucleus, can be represented as a rank-2 symmetric and traceless tensor. 
As such it is always possible to diagonalise it by rotating the axis system into the EFG eigenbasis,  that is referred to as the principal reference frame. The experimentally observable EFG is conventionally described by two parameters, the principal component V_zz and the axial asymmetry parameter:

    η=V_xx-V_yy/V_zz,

with V_zz, V_yy and V_xx being the eigenvalues of the EFG tensor, defined such that |V_zz|≥|V_yy|≥|V_xx|<cit.>.


The signature for the interaction between the probe atoms and their local environment is the perturbation function G_kk(t). In the case of static electric quadrupole interactions, the interaction can be written as<cit.>:

    G_kk(t)=∑_n s_k_ncos(ω_nt)e^-δω_nt/2,

where s_k_n and ω_n are functions of η and the fundamental frequency ω_Q. The exponential term arises from randomly distributed vacancies, defects, and lattice strains that result in an attenuation of the experimental R(t) function. In the case of the 111mCd probes, the intermediate level in the decay has a nuclear spin of I=5/2, which leads to a split of the intermediate level into three sub-levels due to the quadrupole interaction and three transition frequencies, ω_1, ω_2, and ω_3=ω_1+ω_2. The fundamental frequency is usually defined as <cit.>:

    ω_Q=eQV_zz/4I(2I-1)ħ,

 where Q is the nuclear quadrupole moment, e is the electron charge, and ħ is the reduced Planck constant.



 §.§ Computational Methodology


DFT calculations were performed by using the Quantum Espresso (QE) <cit.> suite with Projector-Augmented Wave (PAW) datasets<cit.>. The atomic valence configurations that were considered for the calculations were: Be[2s2], Ca[3s23p64s2], Cd[4d9.55s25p0.5], Ge[3d104s24p2], Mg[2s22p63s2], Mn[3s23p63d54s2], O[2s22p4], and Sr­[4s24p65s2]. 
The kinetic energy cutoffs were converged to the values of 70 and 540 Ry for the wavefunctions and the charge density, respectively. However, in order to converge the EFGs to the experimental data, and for the calculations that included the Cd substitutional atom, these values were raised to 113 and 900 Ry, for the wavefunctions and the charge density, respectively. Regarding the exchange-correlation (XC) functional, the generalised gradient approximation (GGA) with Perdew-Burke-Ernzerhof<cit.> (PBE) parametrization was employed. In order to treat the strongly correlated 3d-states of Mn, an effective Hubbard-U potential was considered by defining U_eff = 4 eV <cit.>, by employing the simplified version of Cococcioni and de Gironcoli <cit.>, with PBE (PBE+U). 


In order to obtain the ground-state properties, total energy minimization with respect to the changes in the atomic positions and cell parameters were taken into account, until the total force did not exceed 0.05 eV/Å, and constraining the system to zero internal pressure, with the variable cell shape relaxation, considering the damped Beeman ionic dynamics and the Wentzcovitch extended Lagrangian for the cell dynamics <cit.>.
For the four compounds, the enthalpies of formation were determined by comparing respective total energies to that of the respective DFT total energy of the AGeO3 and MnGeO3 stable constituent oxides (Eq. <ref> in Appendix  <ref>).
With respect to the CaMnGe2O6 and SrMnGe2O6 compounds, the EFGs were evaluated for the pristine unit-cell, as well as for the unit-cell and the 1 × 1 × 2 supercell by considering Cd substitution of either A or Mn cation sites  (Fig. <ref>), in order to compare with previously obtained PAC measurements. The computation of the EFGs was performed by using the Gauge Including Projector Augmented Waves (GIPAW)<cit.> package routine implemented in QE. 

For the supercell calculations, only the atomic positions were relaxed, by using the lattice parameters obtained from the pristine unit-cell structural minimization(by employing the variable cell shape relaxations). The Brillouin-zones (BZ) 
were sampled with a (9 × 9 × 18) Monkhorst-Pack k-point grids for unit-cell calculations, and a (9 × 9 × 9) for calculations on the 1 × 1 × 2 supercells; whereas for the calculation of EFGs and PDOS, k-point grids of (11 × 11 × 22) and (11 × 11 × 11) were used, for the unit-cells and supercells, respectively.



§ RESULTS AND DISCUSSION




 §.§ Perturbed Angular Correlation

The PAC measurements on the SrMnGe2O6 (SMGO) system were performed at selected temperatures in the 13-902 K range. 
Figure <ref> shows a few representative experimental γ - γ R(t) perturbation functions obtained for the SrMnGe2O6 compound along with their corresponding Fourier transforms. Several models were tested in order to numerically obtain the best fit to the experimental R(t) spectra before arriving at the presented model, wherein two distinct fractions (two distinct local environments) are evident across the whole temperature range, indicating the probes interacted with two distinct EFG distributions, henceforth referred to as EFGSMGO1 and EFGSMGO2, and represented by the green and blue lines in figure <ref>, respectively. 







The temperature behaviour of the experimental R(t) fit parameters of SMGO  is displayed in figure <ref>. One can observe, in figure <ref> a), green circles, that the V_zz value for the EFGSMGO1 fraction has a noticeable increase at low temperatures, with increasing temperature, reaching a maximum; whereupon it starts to decrease. On the other hand, the axial asymmetry η(1) parameter increases as the temperature increases, as displayed in figure <ref> b) by the green circles.  The EFG aforementioned  
behaviour is typical for a phonon contribution to V_zz and, accordingly, a temperature dependence least-squares fit by the Bayer-Kushida<cit.> model, with a single phonon frequency, was done, according to

    V_zz(T)=V_zz(0){ 1-3/8π^2hA/ν(1/2+1/e^hν/kT-1)}  ,

 where h is the Planck constant, ν is the phonon frequency, k is the Boltzmann constant and A is a quantity that in general can only be said to have the dimensions and order of magnitude of the inverse of the moment of inertia associated to the molecular vibration, except in specific cases where it is indeed the inverse of the moment of inertia associated with a normal mode of molecular vibration<cit.>. 
 From this fitting, we have obtained ν = 8.01(7) × 10^12 Hz, which is within the range of frequencies that are typical for intra-molecular vibrations and is comparable to the value of 6.60(85) × 10^12 Hz reported for Ca2MnO4<cit.>, suggesting that the 
 V_zz temperature behaviour for the EFGSMGO1 local environment 
 is indeed due to a phonon contribution. Regarding the axial asymmetry parameter η(1)  at the  EFGSMGO1 site, it displays an intermediate asymmetry, which increases as temperature raises. 
 
For the EFGSMGO2 fraction, depicted by the blue triagles 
 in figure <ref> a),  the V_zz shows an atypical behaviour of increasing as temperature also increases, though such behaviour has nevertheless been reported in other compounds. Specifically, analogous behaviour was experimentally observed   in the A21am phase of Ca3Mn2O7 for which DFT simulations indicated that such behaviour is actually due to structural effects, that prevail over the phonon contributions <cit.>. 
  The asymmetry parameter η(2) for the EFGSMGO2 fraction, as displayed in figure <ref> b) by the blue triangles, 
 on the other hand, remains approximately constant, up to 474 K, whereupon an anomalous increase occurs. It is unclear if such behaviour could be related to structural phenomena, as there are no measurements reported for temperatures above 350 K. It is worth noting that the probes distribution between both fraction remained fairly constant across the whole temperature range, suggesting that they are
related to the probes occupying two distinct lattice sites.
 



With respect to the CaMnGe2O6 (CMGO) system, the PAC measurements were likewise performed at selected temperatures, in the 11-898 K range. Representative fits to the experimental R(t) perturbation functions for CMGO are presented in figure <ref>, along with the corresponding Fourier transforms. Similar to how the data from the previous compound was treated,  several models were also tested in order to numerically obtain the best fit to the experimental R(t) spectra. Distinctly from SrMnGe2O6, wherein two noticeable fractions are present across the whole temperature range, for the CaMnGe2O6 compound there is only one fraction present from low temperatures, up to 351 K, hereby referred to as EFGCMGO1 and represented by the green lines in figure <ref>. It can be noticed in this figure that a second fraction, labeled EFGCMGO2, appears for temperatures above 351 K, represented by the blue lines. However, it is likely that this second fraction is likewise present at lower temperatures, but  owing to the very small observable frequency associated with it, that is not well defined in the time window available in the PAC experiments, it is not possible to fit this fraction at lower temperatures. Even at high temperatures the EFGCMGO2 fitting parameters have nonetheless an associated  high uncertainty.  



The temperature behaviour of the fit parameters relative to the CaMnGe2O6 system are shown in figure <ref>. One can observe, in figure <ref> a), green circles,  that the EFGCMGO1 distribution has a V_zz value of similar magnitude to EFGSMGO1 (figure <ref> a)), with an analogous decrease as temperature raises. 


Also η(1), displayed in figure <ref> b), green circles,  similarly  increases with increasing temperature, when compared to SMGO η(1) variation (figure <ref> b)). 
As was done in the case of the SrMnGe2O6 compound, a least-squares fits was likewise performed to V_zz, according to equation <ref>. From the fit, a frequency of ν = 10.0(2) × 10^12 Hz was obtained, once more indicating that this behaviour is likely attributable to a phonon contribution to V_zz. Similarly to what was observed for SrMnGe2O6, we note that the magnetic hyperfine interactions were not apparent in the fits to the experimental R(t) functions.






 §.§ Density Functional Theory Calculations




  §.§.§ Structural Parameters


We begin this section by summarizing in Table <ref> the calculated lattice parameters for the SrMnGe2O6 and CaMnGe2O6 systems, computed by applying both the PBE and PBE+U methodologies, as well as the reported experimental values,<cit.> for comparison purposes. 




Regarding the SrMnGe2O6 system, we can observe that 
the lattice parameters computed with the PBE functional are overestimated, with respect to the experimental values, corresponding to a relative increase of ∼1.7%, ∼0.8%, and ∼1.6%, respectively,  for a, b, and c parameters, which is in line with the well-known tendency of the PBE functional.<cit.> The inclusion of the on-site effective Hubbard-U potential results in a further relative increase of these parameters with ∼2.1%, ∼1.7%, and ∼2.1%, when compared to the experimental values. Though the β monoclinic angle is overestimated for both PBE and PBE+U approximations, when comparing to experimental data, the difference is in fact smaller when considering the PBE+U calculations, resulting in a value closer to the experimental one.

With respect to CaMnGe2O6, the a and b theoretical lattice parameters obtained from PBE calculations likewise are overestimated, relative to experimental data, by ∼1.2% and ∼0.5%, respectively. On the other hand we observed that the c parameter is underestimated  by ∼0.9%. Analogously to what was observed for SrMnGe2O6, the a and b lattice parameters that resulted from the PBE+U approach display an increase with respect to the PBE values, presenting an overestimation of ∼1.7% and ∼1.6%, respectively,  with respect to experimental results; whereas c parameter is still underestimated, though smaller than the one obtained by the PBE functional, of  ∼0.4%. Also, and distinctly from what was observed for SrMnGe2O6, the theoretical monoclinic angle is underestimated  relative to the experimental angle, though again, results from the PBE+U approximation is closer to the experimental value.




  §.§.§ Enthalpies of Formation


The enthalpy of formation of the constituent oxides of the AMnGe2O6 series has also been examined through DFT calculations and we have determined that only the CaMnGe2O6 and SrMnGe2O6 clinopyroxene compounds are thermodynamically stable. This is evidenced by their negative enthalpies of formation of  -0.19 eV/f.u. and -0.11 eV/f.u., respectively (calculated by considering equation <ref> of Appendix <ref>). On the other hand, the BeMnGe2O6 and MgMnGe2O6 systems both evidenced positive enthalpies of formation of 0.12 eV/f.u., supporting that these are not stable against dissociation into the respective stable constituent oxides BeGeO3 or MgGeO3 and MnGeO3. This observation is consistent with the fact that only the former two compounds have been experimentally synthesised, whereas the synthesis of the latter two has not yet been reported. Such observation implies that different reaction pathways towards synthesis will have to be explored if these systems were to be experimentally realised. Details on how the enthalpies of formation were obtained are presented in the Appendix <ref>.




  §.§.§ Partial Density of States


The PDOS for the CaMnGe2O6 and SrMnGe2O6 compounds, obtained considering the PBE calculation, without the Hubbard-U correction (Ueff = 0 eV), were computed and are displayed in figure <ref>, in which the valence band maximum (VBM) is aligned at the Fermi energy and is set to zero.
Two narrow bands form the VBM, and these are mainly composed by the Mn occupied 3d-states, hybridized with a smaller contribution from the O p-states. Energetically below these narrow bands (< -2.5 eV), a broader band is observed and it is mostly dominated by the O p-states, with a small mixture of 3d-states from Mn. The conduction band minimum (CBM), on the other hand, presents low density of states and hybridization between the p-states of O and Ge. For increasing energies, the DOS significantly increases forming a narrow resonant band, between 2.5 and 3.0 eV, that is mainly composed by the Mn 3d-states unoccupied states. 

For both studied systems the PDOS are qualitatively very similar, although at the conduction band the observed resonant peaks of Mn for SrMnGe2O6 are much broader when compared to those of CaMnGe2O6.

 






The electronic band gaps are also noted to be fairly narrow, being 0.46 eV and 0.51 eV, respectively, for CaMnGe2O6 and SrMnGe2O6, which is narrower than what has been reported for similar Si-based compounds<cit.>.

Since the band gap widths are likely underestimated in the absence of the Hubbard-U correction, we have also considered the PDOS by including the Ueff = 4 eV, applied to the Mn 3d-states. These are shown in figure <ref> along with the respective electronic band structures in the high-symmetry directions of the primitive cell first Brillouin zone
 (figure <ref>). 

 





When including the Ueff = 4 eV, and comparing to the PDOS results when Ueff = 0 eV, differences are mainly evidenced at the VBM, with the Mn 3d-states. The two narrow bands observed in figure <ref>, now overlap and form one narrow band for CaMnGe2O6 (figure <ref> a)). As for the SrMnGe2O6 system, the narrow bands are energetically very close, almost forming also one narrow band (figure <ref> b)). Moreover, these narrow bands are now dominated by a stronger hybridization between Mn d-states with the O p-states.




Since the inclusion of the U parameter localises and positions the unoccupied narrow 3d-states of Mn higher in energy (between 5.0 and 5.5 eV), the CBM shows slightly increased p hybridized states of O and Ge, than when U is not considered. The PBE+U theoretical band gaps are now wider, when compared to the previous case, with 1.82 eV and 1.70 eV for the CaMnGe2O6 and SrMnGe2O6 compounds, respectively. Despite this, the gap widths are still below the range of other similar Si-based clinopyroxenes<cit.>.




  §.§.§ Electronic Bandstructures





The electronic bandstructures for the pristine CaMnGe2O6 and SrMnGe2O6 systems, as well as the respective Cd-containing supercells were also computed through DFT calculations, in a path along the high-symmetry points depicted in figure <ref>, generated with the SeeK-path tool.<cit.>



The bandstructures for the pristine systems are presented in figure <ref>, whilst those for the supercells can be found in Appendix <ref>.

Differences between both pristine systems is evidenced by the character of the band gap: while SrMnGe2O6 exhibits a direct band gap, CaMnGe2O6 has an indirect gap. We must stress that subtleties persist with respect to the analysis of the VBM, since the valence bands are quite flat, and thus the electronic bands along the different high symmetry segments/points are energetically very close.   


From figure <ref>, one can observe that the top of the valence band is much flatter than the bottom of the conduction band, which shows narrow parabolas around the zone-centre, indicating that the effective mass of the holes, m^*_h,  is much larger than the effective mass of electrons, m^*_e. Such an observation is an indication of a higher mobility of the electrons comparatively to the holes of the system. Since the effective mass of the charge carriers is an essential parameter for the performance of photoelectric and photo-catalytic activities<cit.>, and also for semiconductor radiation detectors<cit.>; we have considered further analysing respective properties (table <ref>). Therefore and to quantify the effective masses of holes and electrons, we employ the well-known definition of the effective mass as

    m^*=ħ^2/∂^2 E/∂ k^2,


with the second derivative of the electronic dispersion relation being obtained from a quadratic fit around the energy range where the band edges are located.



In the case of CaMnGe2O6, we obtain a m^*_e,  around the Γ-point, very close to that of the free electrons, with m^*_e=1.028 m_0 (where m_0 is the mass of the free electron with 9.11×10^-31 kg). While for the hole carriers, and as expected by considering the flatness-shape of the VBM, the mass of holes proved to be substantially larger (in absolute terms) with m^*_h=26.125 m_0, and calculated along the high-symmetry segment of AΓ→ A (table <ref>). Regarding the Cd-containing supercells of CaMnGe2O6, when Cd substitution was considered at the Ca site, we found that the effective masses of the electrons only yielded mild variation when compared to the pristine system. For the electrons, the m^*_e showed very little to no increase
with  m^*_e=0.996 m_0 and m^*_e=1.028 m_0, for the Ca3/4Cd1/4MnGe2O6 and Ca7/8Cd1/8MnGe2O6 systems, respectively. In the case of the holes, we observe that for the Ca3/4Cd1/4MnGe2O6 cell, and by comparing to the pristine system, m^*_h slightly increases along the AΓ→Γ with m^*_h=45.259 m_0, although showing similar values along AΓ→ A with m^*_h=27.875 m_0.

With respect to the larger supercell (CaMn7/8Cd1/8Ge2O6), we observe a very high increase with respect to the pristine system, with m^*_h=67.562 m_0, along the same high-symmetry segment. 

On the other hand, when the Cd atom was placed at the Mn site, we found that the dopant affected the valence band edge even further, leading to a further flattening of the VBM; however, and once again, the m^*_e remained very similar to the pure unit-cell. A significant increase of the hole effective mass was found for the CaMn3/4Cd1/4Ge2O6 
structure (V_2 →Γ direction), with  m^*_h=118.325 m_0. 
Moreover, for the more diluted system we observed that along 
AΓ→ A the effective mass decreases slightly (m^*_h=28.65 m_0), showing an increase of m^*_h in the direction of AΓ→ A (m^*_h=68.128 m_0).



Regarding the pristine SrMnGe2O6 structure, we found a similar value for the electron effective mass with m^*_e=1.023 m_0. However, a significantly more parabolic dispersion relation at the VBM when compared to CaMnGe2O6, leading to a hole effective mass of m^*_h=9.380 m_0, when considering the Γ→ D segment. We must however state that since the valence band dispersion is quite asymmetric, the interpolation with a quadratic function did not yield a very accurate fit to the dispersion relation; therefore the value of the m^*_h should be understood more as an order of magnitude rather than a quantitative value. 
Once again, we observe that with the Cd substitution, there is only a mild variation of m^*_e, leading to a slightly increased value of m^*_e=1.028 m_0 for Sr7/8Cd1/8MnGe2O6, and a more pronounced increase for the 
Sr3/4Cd1/4MnGe2O6 compound with m^*_e=1.106 m_0. As for the SrMn3/4Cd1/4Ge2O6 and SrMn7/8Cd1/8Ge2O6 systems, a slight increase of the effective masses is observed to m^*_e=1.068 m_0 and m^*_e=1.057 m_0, respectively. 
With respect to the m^*_h, we again note that due to the flat-shape of the bands, a parabolic interpolation does not provide a dependable quantitative value, so we can only state with confidence that this value should remain of the same order as that of the pristine SrMnGe2O6 compound. We must state that such observed differences of the masses between the charge carriers is quite interesting because of the increased possibility to enhance the separation of the electron–hole pairs, and even of evidencing large electron and
hole drift lengths<cit.>, thus affecting the opto-electronic properties of the studied clinopyroxenes <cit.>.
The band gaps were also determined for all systems from the bandstructure calculations, and are summarised in Table <ref>. As already observed for the PDOS, the band gaps for all systems lie between 1.70 eV and 1.82 eV for the pristine systems, SrMnGe2O6 and CaMnGe2O6, respectively; whereas the doped structures show narrower band gaps. All the doped structures show indirect band gap character, even for the Sr-based strucutures, where the pristine counterpart evidences a direct one (figure <ref> and figures Sr2Cd2Mn2Ge4O12_bands_DOS,SrCdMnGe2O6_bands_DOS,Sr2Mn2Cd2Ge4O12_bands_DOS,SrMnCdGe2O6_bands_DOS,Ca2Cd2Mn2Ge4O12,CaCdMnGe2O6,Ca2Mn2Cd2Ge4O12,Ca2Mn2Ge4O12AFM_bands,CaMnCdGe2O6 in Appendix <ref>).





  §.§.§ Ab Initio Electric Field Gradients


The DFT  EFGs calculations 
were performed for the pristine unit-cells, along with the EFGs for the unit-cells and 1 × 1 × 2 supercells with Cd substitution, either at the A- or the Mn-sites. Results were compared with those obtained for the experimental EFG fractions and are summarised in Table <ref>, whilst a more comprehensive list of computed values can be found in appendix <ref>, Table <ref>. It should be noted that in the cases where Cd was placed at a Mn site, the systems were treated as ferromagnetic, for the sake of computational simplicity. An antiferromagnetic calculation for the CaMn7/8Cd1/8Ge2O6 system was nevertheless performed to check how much of an effect the magnetic ordering might have on the computed EFG. Very similar results were obtained in both the ferromagnetic and antiferromagnetic calculations, with V_zz=12 V/Å^2 and η=0.21, in the former case, 
and V_zz=13 V/Å^2 and η=0.37, in the latter, 
thus showing that this approximation is reliable. 





Comparing the EFG parameters for the experimental EFGSMGO1 fraction of SrMnGe2O6  with those computed for the supercells with Cd substitution at the Sr-site, where the 1/4 and 1/8 Cd dilution was considered (unit-cell and 1 × 1 × 2 supercell, respectively), we may observe a very good agreement between V_zz and η. From the PAC measurements we obtain for EFGSMGO1the values of V_zz=120 V/Å^2 and η=0.28; for the Cd probe nucleus for Sr3/4Cd1/4MnGe2O6 they are V_zz=103 V/Å^2  and η=0.14 and for Sr7/8Cd1/8MnGe2O6 these result as V_zz=108 V/Å^2 and η=0.14. Likewise, good agreement is also observed for the EFGSMGO2 fraction of SrMnGe2O6 for the case of the Cd substitution at the Mn-site. We obtain for the EFGSMGO2 fraction the values of V_zz=20 V/Å^2 and η=0.81. From the DFT results we get V_zz=22 V/Å^2 and η=0.67, and V_zz=21 V/Å^2 and η=0.72, for the SrMn3/4Cd1/4Ge2O6 and SrMn7/8Cd1/8Ge2O6 cells, respectively. Based on the obtained theoretical results, we may conclude that the supercells are the systems that simulate quite well the experimental data when the Cd probe is considered, with the EFGs being in good agreement to the EFGSMGO1 and EFGSMGO2 experimental fractions, which correspond to the Cd probe replacing the Sr-sites and the Mn-sites, respectively, in the crystalline structure. 
 Similarly, this same correspondence seems to hold true for the CaMnGe2O6 compound. 
For the EFGCMGO1 experimental fraction, we observe the values of V_zz=130 V/Å^2 and η=0.15, and for the DFT data we obtain values of V_zz=108 (109)  V/Å^2 and η=0.01 (0.02) for  Cd concentration of 1/4 (1/8)  in the Ca atom position.  
The obtained theoretical EFGs are slightly lower than those of the experimental fraction; such is mostly evident for the asymmetry parameter, η. As for the EFGCMGO2 fraction from the PAC experiments, the EFG  result in V_zz=10 V/Å^2 and an indeterminate η, whereas for the DFT diluted supercell of CaMn7/8Cd1/8Ge2O6 we have V_zz=12 V/Å^2 and η=0.21, similar to those obtained for CaMn3/4Cd1/4Ge2O6. These results indicate that first-principles calculations combined with PAC spectroscopy provide a valuable tool to explain the observed two distinct local environments, indicating that Cd probes are distributed between the Sr- or Ca-sites and the Mn-sites in the  SrMnGe2O6 and CaMnGe2O6 basic compounds, respectively. 





§ CONCLUSIONS

We have undertaken DFT calculations to obtain the structural and electronic properties of the AMnGe2O6 clinopyroxene systems, with A=Be, Mg, Ca, Sr. We found, through enthalpy of formation calculations, that the Be- and Mg-based systems
are not thermodynamically stable against dissociation into the stable constituent oxides BeGeO3 or MgGeO3 and MnGeO3. On the other hand, the Ca- and Sr-based systems are thermodynamically stable against dissociation and, accordingly,  have been realized experimentally. From the PDOS, we found that the strongly correlated Mn 3d-states form the narrow band close to the VBM together with the O p-states.  The CBM is composed mainly of low density hybridized p-states of O and Ge and, for higher energies, the density of states significantly increases
forming a narrow resonant band  that is mainly composed of the anti-bound unnocupied Mn 3d-states. The theoretical band gaps of CaMnGe2O6 and SrMnGe2O6 compounds, computed using the PBE+U approximation, 
are 1.82 eV and 1.70 eV, respectively. The calculated effective masses of the charge carriers of the two studied pristine compounds show considerable differences, which implies that the separation of the electron–hole pairs can be enhanced, thus improving the photoelectric efficiency. By considering the Cd dopants, we do not observe much variation on the value of the charge carrier masses, whereas a decrease of the band-gap widths is observed.

The temperature behavior of the experimental EFG V_zz fractions, labeled  EFGSMGO1 and EFGCMGO1, respectively related to the Cd probe on the SrMnGe2O6 and CaMnGe2O6 compounds,   fits precisely, according to the Bayer-Kushida model <cit.>,  with a single phonon frequency, producing frequencies of 8.01(7) × 10^12 and 10.0(2) × 10^12 respectively, suggesting that their behavior can be attributed to a phonon contribution to V_zz.  


The hyperfine parameters were computed to obtain the EFGs and these were compared to PAC measurement results. Data between both sets are consistent with the two obtained experimental fractions, EFGSMGO1 and EFGSMGO2, when considering a supercell approach with a Cd ion replacing  Sr and Mn sites, respectively. 
We also find that this same correspondence seems to be true for the  CaMnGe2O6 compound, where the  EFGCMGO1 and EFGCMGO2 experimental fractions can be atributed, respectively,  to Cd replacing  Ca and  Mn ions.   
Furthermore, through the combination of theoretical ab-initio calculations and PAC results, we could infer that the Cd probe in the AMnGe2O6 clinopyroxene compounds, with A=Ca, Sr, can replace 
the ion located at the A-site as well as the Mn ion, offering new perspectives for new systems synthesis with possibly improved funcionalities. 



The authors acknowledge the support of the technical teams at ISOLDE for their exceptional work in delivering high-quality beams for the presented PAC measurements. The authors also acknowledge project NECL under NORTE-01-0145-FEDER-022096, and FCT projects UIDP/04968/2020 (doi.org/10.54499/UIDP/04968/2020), UIDB/04968/2020 (doi. org/10.54499/UIDB/04968/2020), LaP-MET LA/P/0095/2020, POCI-01-0145-FEDER-029454, POCI-01-0145-FEDER-032527 and CERN/FIS-TEC/0003/2021. Also, BMBF through grants 05K16PGA and 05K22PGA, EU Horizon Europe Framework research and innovation programme under grant agreement no. 101057511 (EURO-LABS) for supporting IS679 ISOLDE-CERN experiment, and PRACE project with reference 2021240118, with access to the Irene Skylake computer. RPM acknowledges support from the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the EC Research Innovation Action under the H2020 Programme, with reference HPC171030W; in particular, the support of Dr. A. Stroppa and the hospitality provided by CNR-SPIN at the Department of Physical and Chemical Science of the University of L'Aquila (Italy) during the visit from 15/10/2021 
to 20/12/2021 as well as computer resources and technical support provided by CINECA. RPM also acknowledges computer resources and technical support from Minho Advanced Computing Center through project CPCA/A1/460622/2021, financed by FCT. LVCA (Project 314884/2021-1) and HMP (Project 308438/2022-1) acknowledge funding from CNPq and support from FAPESP (Project 2022/10095-8). AMLL acknowledge the FCT 2021.04084.CEECIND (doi.org/­10.54499/­2021.04084.CEECIND/­CP1655/­CT0002) and ELdS   the 2022.00082.CEECIND (doi.org/­10.54499/­2022.00082.CEECIND/­CP1719/­CT0001) grants. ELdS further acknowledges the High Performance Computing Chair - a R&D infrastructure (based at the University of Évora; PI: M. Avillez), endorsed by Hewlett Packard Enterprise, and involving a consortium of higher education institutions, research centers, enterprises, and public/private organizations. 











 
   






§ CALCULATION OF FORMATION ENERGIES





The formation energies for the studied oxides of the  AMnGe2O6 (A = Be, Mg, Ca, Sr) clinopyroxene series,  were calculated according to the following equation:

    E_form = E_AMnGe2O6 - E_AGeO3 - E_MnGeO3,

where E_AMnGe2O6, E_AGeO3, E_MnGeO3 are the total electronic energies obtained from DFT+U calculations, per formula unit (f.u.), which are summarised in  table <ref>.





§ ELECTRIC FIELD GRADIENT PARAMETERS




 
The calculated EFG parameters are presented in table <ref>. From these results we can observe that the EFGs at the Ge nuclei  were only slightly affected by the Cd substitution, as evidenced by the fact that the symmetry of the Ge sites remained quite similar to SrMnGe2O6 and CaMnGe2O6 basic compounds after Cd substitution. The same cannot be said for the Sr/Ca and Mn sites, which in many cases deviated substantially.




§ ELECTRONIC BAND DISPERSION OF THE CD DOPED SYSTEMS





The bandstructure and the density of states for the AMnGe2O6 (A = Ca, Sr) clinopyroxene compounds with Cd substitution in  A- and Mn-sites are  presented in figures Sr2Cd2Mn2Ge4O12_bands_DOS,SrCdMnGe2O6_bands_DOS,Sr2Mn2Cd2Ge4O12_bands_DOS,SrMnCdGe2O6_bands_DOS,Ca2Cd2Mn2Ge4O12,CaCdMnGe2O6,Ca2Mn2Cd2Ge4O12,Ca2Mn2Ge4O12AFM_bands,CaMnCdGe2O6.



















48
natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL 


[Streltsov and Khomskii(2008)]Streltsov2008
authorS. V. Streltsov
  and authorD. I.
  Khomskii, journalPhys. Rev. B
  volume77, pages064405
  (year2008).

[Ding et al.(2016a)Ding,
  Colin, Darie, and Bordet]Ding_JMCC2016
authorL. Ding,
  authorC. V. Colin,
  authorC. Darie, and
  authorP. Bordet,
  journalJ. Mater. Chem. C volume4,
  pages4236 (year2016a).

[Colin et al.(2020)Colin, Ding,
  Ressouche, Robert, Terada, Gay, Lejay, Simonet, Darie, Bordet
  et al.]PhysRevB.101.235109
authorC. V. Colin,
  authorL. Ding,
  authorE. Ressouche,
  authorJ. Robert,
  authorN. Terada,
  authorF. Gay,
  authorP. Lejay,
  authorV. Simonet,
  authorC. Darie,
  authorP. Bordet,
  et al., journalPhys. Rev. B
  volume101, pages235109
  (year2020).

[Jodlauk et al.(2007)Jodlauk, Becker,
  Mydosh, Khomskii, Lorenz, Streltsov, Hezel, and Bohatý]Jodlauk2007
authorS. Jodlauk,
  authorP. Becker,
  authorJ. A. Mydosh,
  authorD. I. Khomskii,
  authorT. Lorenz,
  authorS. V. Streltsov,
  authorD. C. Hezel, and
  authorL. Bohatý,
  journalJ. Phys. Condens. Matter. volume19,
  pages432201 (year2007).

[Kim et al.(2012)Kim, Jeon, Patil, Patil,
  Nénert, and Kim]Kim2012
authorI. Kim,
  authorB.-G. Jeon,
  authorD. Patil,
  authorS. Patil,
  authorG. Nénert,
  and authorK. H. Kim,
  journalJ. Phys. Condens. Matter. volume24,
  pages306001 (year2012).

[Ding et al.(2016b)Ding,
  Colin, Darie, Robert, Gay, and Bordet]Ding_PRB2016
authorL. Ding,
  authorC. V. Colin,
  authorC. Darie,
  authorJ. Robert,
  authorF. Gay, and
  authorP. Bordet,
  journalPhys. Rev. B volume93,
  pages064423 (year2016b).

[Legesse et al.(2018)Legesse, Park,
  El Mellouhi, Rashkeev, Kais, and
  Alharbi]https://doi.org/10.1002/cphc.201701155
authorM. Legesse,
  authorH. Park,
  authorF. El Mellouhi,
  authorS. N. Rashkeev,
  authorS. Kais, and
  authorF. H. Alharbi,
  journalChem. Phys. Chem. volume19,
  pages943 (year2018).

[Torres et al.(2019)Torres, Luque,
  Tortajada, and Arroyo-de Dompablo]Torres2019
authorA. Torres,
  authorF. J. Luque,
  authorJ. Tortajada,
  and authorM. E.
  Arroyo-de Dompablo, journalSci. Rep.
  volume9, pages9644 (year2019).

[Zhou et al.(2014)Zhou, King, Scanlon,
  Sougrati, and Melot]Zhou_2014
authorS. Zhou,
  authorG. King,
  authorD. O. Scanlon,
  authorM. T. Sougrati,
  and authorB. C. Melot,
  journalJ. Electrochem. Soc. volume161,
  pagesA1642 (year2014).

[Temnikov et al.(2019)Temnikov, Komleva,
  Pchelkina, and Streltsov]Temnikov2019
authorF. V. Temnikov,
  authorE. V. Komleva,
  authorZ. V. Pchelkina,
  and authorS. V.
  Streltsov, journalJETP Lett.
  volume110, pages595 (year2019).

[Ashbrook et al.(2002)Ashbrook, Berry,
  and Wimperis]NMRpyro1
authorS. E. Ashbrook,
  authorA. J. Berry, and
  authorS. Wimperis,
  journalJ. Phys. Chem. B volume106,
  pages773 (year2002).

[Flemming and Luth(2002)]NMRpyro2
authorR. L. Flemming and
  authorR. W. Luth,
  journalAm. Mineral. volume87,
  pages25 (year2002).

[Begaudeau et al.(2012)Begaudeau,
  Morizet, Florian, Paris, and Mercier]NMRpyro3
authorK. Begaudeau,
  authorY. Morizet,
  authorP. Florian,
  authorM. Paris, and
  authorJ.-C. Mercier,
  journalEur. J. Mineral. volume24,
  pages535 (year2012).

[Dowty and Lindsley(1973)]Mosspyro1
authorE. Dowty and
  authorD. H. Lindsley,
  journalAm. Mineral. volume58,
  pages850 (year1973).

[Dyar et al.(2013)Dyar, Klima, Fleagle,
  and Peel]MossPyro2
authorM. D. Dyar,
  authorR. L. Klima,
  authorA. Fleagle, and
  authorS. E. Peel,
  journalAm. Mineral. volume98,
  pages1172 (year2013).

[Oshtrakh et al.(2007)Oshtrakh, Petrova,
  Grokhovsky, and Semionkin]MossPyro3
authorM. I. Oshtrakh,
  authorE. V. Petrova,
  authorV. I. Grokhovsky,
  and authorV. A.
  Semionkin, journalHyperfine Interact.
  volume177, pages65 (year2007).

[Raghavan(1989)]Raghavan1989
authorP. Raghavan,
  journalAt. Data Nucl. Data Tables volume42,
  pages189 (year1989).

[Butz(1989)]Butz1989
authorT. Butz,
  journalHyperfine Interact. volume52,
  pages189 (year1989).

[Schatz and Weidinger(1996)]Schatz1996
authorG. Schatz and
  authorA. Weidinger,
  titleNuclear condensed matter physics : nuclear methods and
  applications (publisherJohn Wiley, year1996), ISBN
  isbn9780471954798.

[Lopes et al.(2006)Lopes, Araújo,
  Ramasco, Amaral, Suryanarayanan, and Correia]PhysRevB.73.100408
authorA. M. L. Lopes,
  authorJ. P. Araújo,
  authorJ. J. Ramasco,
  authorV. S. Amaral,
  authorR. Suryanarayanan,
  and authorJ. G.
  Correia, journalPhys. Rev. B
  volume73, pages100408
  (year2006).

[Barradas et al.(1993)Barradas, Rots,
  Melo, and Soares]PhysRevB.47.8763
authorN. P. Barradas,
  authorM. Rots,
  authorA. A. Melo, and
  authorJ. C. Soares,
  journalPhys. Rev. B volume47,
  pages8763 (year1993).

[Correia(1992)]NNFITBarr
authorJ. G. Correia,
  titleNnfit the pac manual (year1992).

[Correia(2018)]NNFIT
authorJ. G. Correia,
  titleNnfit and fft upgrades 2018 (year2018).

[Rasera and
  Catchen(1993)]doi:10.1080/00150199308008701
authorR. L. Rasera and
  authorG. L. Catchen,
  journalFerroelectr. volume150,
  pages151 (year1993).

[Giannozzi et al.(2009)Giannozzi, Baroni,
  Bonini, Calandra, Car, Cavazzoni, Ceresoli, Chiarotti, Cococcioni, Dabo
  et al.]QE-2009
authorP. Giannozzi,
  authorS. Baroni,
  authorN. Bonini,
  authorM. Calandra,
  authorR. Car,
  authorC. Cavazzoni,
  authorD. Ceresoli,
  authorG. L. Chiarotti,
  authorM. Cococcioni,
  authorI. Dabo, et al.,
  journalJ. Phys. Condens. Matter volume21,
  pages395502 (year2009).

[Giannozzi et al.(2017)Giannozzi,
  Andreussi, Brumme, Bunau, Nardelli, Calandra, Car, Cavazzoni, Ceresoli,
  Cococcioni et al.]QE-2017
authorP. Giannozzi,
  authorO. Andreussi,
  authorT. Brumme,
  authorO. Bunau,
  authorM. B. Nardelli,
  authorM. Calandra,
  authorR. Car,
  authorC. Cavazzoni,
  authorD. Ceresoli,
  authorM. Cococcioni,
  et al., journalJ. Phys. Condens. Matter
  volume29, pages465901
  (year2017).

[Blöchl(1994)]PhysRevB.50.17953
authorP. E. Blöchl,
  journalPhys. Rev. B volume50,
  pages17953 (year1994).

[Dal Corso(2014)]DALCORSO2014337
authorA. Dal Corso,
  journalComput. Mater. Sci. volume95,
  pages337 (year2014).

[Perdew et al.(1996)Perdew, Burke, and
  Ernzerhof]Perdew1996
authorJ. P. Perdew,
  authorK. Burke, and
  authorM. Ernzerhof,
  journalPhys. Rev. Lett. volume77,
  pages3865 (year1996).

[MP()]MP
noteData retrieved from the Materials Project for SrMn(GeO_3)_2
  (mp-1208680) from database version v2022.10.28..

[Jain et al.(2013)Jain, Ong, Hautier,
  Chen, Richards, Dacek, Cholia, Gunter, Skinner, Ceder
  et al.]doi:10.1063/1.4812323
authorA. Jain,
  authorS. P. Ong,
  authorG. Hautier,
  authorW. Chen,
  authorW. D. Richards,
  authorS. Dacek,
  authorS. Cholia,
  authorD. Gunter,
  authorD. Skinner,
  authorG. Ceder,
  et al., journalAPL Mater.
  volume1, pages011002 (year2013).

[Cococcioni and
  de Gironcoli(2005)]PhysRevB.71.035105
authorM. Cococcioni and
  authorS. de Gironcoli,
  journalPhys. Rev. B volume71,
  pages035105 (year2005).

[Wentzcovitch(1991)]Wentzcovitch-PhysRevB-44-2358-1991
authorR. M. Wentzcovitch,
  journalPhys. Rev. B volume44,
  pages2358 (year1991).

[Kim et al.(1991)Kim, Levin,
  Wentzcovitch, and Auerbach]PhysRevB.44.5148
authorJ. H. Kim,
  authorK. Levin,
  authorR. Wentzcovitch,
  and authorA. Auerbach,
  journalPhys. Rev. B volume44,
  pages5148 (year1991).

[Wentzcovitch et al.(1993)Wentzcovitch,
  Martins, and Price]PhysRevLett.70.3947
authorR. M. Wentzcovitch,
  authorJ. L. Martins,
  and authorG. D. Price,
  journalPhys. Rev. Lett. volume70,
  pages3947 (year1993).

[Charpentier(2011)]CHARPENTIER20111
authorT. Charpentier,
  journalSolid State Nucl. Magn. Reson.
  volume40, pages1 (year2011).

[Bayer(1951)]Bayer1951
authorH. Bayer, journalZ.
  Phys. volume130, pages227
  (year1951).

[Kushida(1955)]kushida1955influence
authorT. Kushida, journalJ.
  Sci. Hiroshima Univ., Ser. A volume19,
  pages327 (year1955).

[Kushida et al.(1956)Kushida, Benedek,
  and Bloembergen]kushida1956dependence
authorT. Kushida,
  authorG. Benedek, and
  authorN. Bloembergen,
  journalPhys. Rev. volume104,
  pages1364 (year1956).

[Rocha-Rodrigues
  et al.(2020a)Rocha-Rodrigues, Santos, Oliveira,
  Leal, Miranda, dos Santos, Correia, Assali, Petrilli, Araújo
  et al.]PhysRevB.102.104115
authorP. Rocha-Rodrigues,
  authorS. S. M. Santos,
  authorG. m. c. N. P. Oliveira,
  authorT. Leal,
  authorI. P. Miranda,
  authorA. M. dos Santos,
  authorJ. a. G. Correia,
  authorL. V. C. Assali,
  authorH. M. Petrilli,
  authorJ. a. P. Araújo,
  et al., journalPhys. Rev. B
  volume102, pages104115
  (year2020a).

[Rocha-Rodrigues
  et al.(2020b)Rocha-Rodrigues, Santos, Miranda,
  Oliveira, Correia, Assali, Petrilli, Araújo, and
  Lopes]PhysRevB.101.064103
authorP. Rocha-Rodrigues,
  authorS. S. M. Santos,
  authorI. P. Miranda,
  authorG. N. P. Oliveira,
  authorJ. G. Correia,
  authorL. V. C. Assali,
  authorH. M. Petrilli,
  authorJ. P. Araújo,
  and authorA. M. L.
  Lopes, journalPhys. Rev. B
  volume101, pages064103
  (year2020b).

[Haas et al.(2009)Haas, Tran, and
  Blaha]PhysRevB.79.085104
authorP. Haas,
  authorF. Tran, and
  authorP. Blaha,
  journalPhys. Rev. B volume79,
  pages085104 (year2009).

[Streltsov et al.(2010)Streltsov, McLeod,
  Moewes, Redhammer, and Kurmaev]PhysRevB.81.045118
authorS. V. Streltsov,
  authorJ. McLeod,
  authorA. Moewes,
  authorG. J. Redhammer,
  and authorE. Z.
  Kurmaev, journalPhys. Rev. B
  volume81, pages045118
  (year2010).

[Hinuma et al.(2017)Hinuma, Pizzi,
  Kumagai, Oba, and Tanaka]HINUMA2017140
authorY. Hinuma,
  authorG. Pizzi,
  authorY. Kumagai,
  authorF. Oba, and
  authorI. Tanaka,
  journalComputational Materials Science
  volume128, pages140 (year2017),
  ISSN issn0927-0256.

[Togo et al.(2024)Togo, Shinohara, and
  Tanaka]togo2024textttspglibsoftwarelibrarycrystal
authorA. Togo,
  authorK. Shinohara,
  and authorI. Tanaka,
  title: a software library for crystal
  symmetry search (year2024), 1808.01590.

[LIU(2016)]LIU20161
journalComput. Mater. Sci. volume123,
  pages1 (year2016).

[Lee et al.(1999)Lee, James, Olsen, and
  Hermon]Lee_JElectronMater_28_766
authorE. Y. Lee,
  authorR. B. James,
  authorR. Olsen, and
  authorH. Hermon,
  journalJ. Electron. Mater. volume28,
  pages766 (year1999).

[Zhang et al.(2019)Zhang, Deng, Gao,
  Chen, Au, Li, Yin, and Cai]C9CY00997C
authorJ.-R. Zhang,
  authorX.-Z. Deng,
  authorB. Gao,
  authorL. Chen,
  authorC.-T. Au,
  authorK. Li,
  authorS.-F. Yin, and
  authorM.-Q. Cai,
  journalCatal. Sci. Technol. volume9,
  pages4659 (year2019).





 Paper Title: Minimal Quantum Circuits for Simulating Fibonacci Anyons.



















































 

graphs






compat=1.18


    |ττ̅⟩1|τ1⟩t|1τ̅⟩|11⟩⟩1/ϕ1/√(ϕ)Department of Physics, University of Illinois at Urbana-Champaign,
    1110 West Green Street, Urbana, IL 61801, USAInstitute for Quantum Information,
    RWTH Aachen University, D-52056 Aachen, GermanyCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USARudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford, OX1 3PU, UKhormozi@bnl.govComputational Science Initiative, Brookhaven National Laboratory, Upton, NY 11973, USA§ ABSTRACT 
    The Fibonacci topological order is the prime candidate for the realization of universal topological quantum computation. We devise minimal quantum circuits to demonstrate the non-Abelian nature of the doubled Fibonacci topological order, as realized in the Levin-Wen string net model.  Our circuits effectively initialize the ground state, create excitations, twist and braid them, all in the smallest lattices possible. We further design methods to determine the fusion amplitudes and braiding phases of multiple excitations by carrying out a single qubit measurement.  We show that the fusion channels of the doubled Fibonacci model can be detected using only three qubits, twisting phases can be measured using five, and braiding can be demonstrated using nine qubits. These designs provide the simplest possible settings for demonstrating the properties of Fibonacci anyons and can be used as realistic blueprints for implementation on many modern quantum architectures.
    
    Minimal Quantum Circuits for Simulating Fibonacci Anyons
        Layla Hormozi
        August 5, 2024
    ========================================================
    
    
    
    § INTRODUCTION
    
    The emergence of topological order in interacting quantum systems is one of richest phenomena in modern condensed matter physics <cit.>. The possibility of creating long-range entangled states <cit.> and quasi-particle excitations with non-Abelian statistics has opened the door to new vistas in fault-tolerant quantum information processing, both in the form of (passively) error-tolerant topological quantum processors, and (active) topological quantum error correcting codes (QECCs) <cit.>. For almost three decades, this two-way relationship has been a wellspring of creative work, from quantum algorithms with super-polynomial speedups <cit.> to topological codes such as the surface code, which are among the most promising QECCs <cit.>. 
    
    In recent years, rapid progress in simulating topological phases has been achieved, and observation of Abelian statistics has been reported in a variety of platforms. This includes minimal implementations using optical qubits <cit.> to intermediate-scale experiments in superconducting qubits <cit.>. More recently, progress has been reported on quantum simulation of certain non-Abelian phases, including dislocation-induced non-Abelian excitations in Abelian states <cit.>, Majorana states <cit.>, as well as non-Abelian states, such as those with D_4 and S_3 topological order <cit.>.
    
    Despite this progress, some of the most interesting topological phases, particularly those which can be used for universal topological quantum computation and quantum error correction, have remained out of reach. Fortunately, rapid advances in the science and engineering of controllable arrays of synthetic qubits, in various platforms, are increasingly enabling more advanced digital quantum simulation. Given efficient circuits which prepare and manipulate the ground states and quasiparticle excitations of topological phases of matter, it is now possible to directly probe their fascinating properties on a variety of nascent quantum hardwares.
    
    
    
    
    
    In this article, we describe minimal quantum circuits which demonstrate the non-Abelian properties of Doubled Fibonacci (DFib) anyons, arising as excitations of Levin and Wen's string-net model <cit.>. Fibonacci topological order <cit.>, is particularly interesting, since it is the simplest candidate for universal topological quantum computing, having braiding operations which are universal and can be implemented fault tolerantly <cit.>. As suggested above, our approach is based on digital quantum simulation: applying a series of quantum gates which effectively project into the desired eigenstate of the string-net Hamiltonian. <cit.>. 
    
    In constructing these circuits, we identify the minimal settings in which the intrinsically “topological” properties  of Fibonacci anyons (i.e., the building blocks for topological quantum computation) can be verified. In particular, each property is realized and measured using the fewest number of qubits possible, and with circuits that are sufficiently shallow to be accessible on virtually any modern quantum hardware. Though our constructions generalize to larger system sizes, asymptotic scaling is not the focus of this paper. Rather, we aim to build a specific set of primitive subroutines to enable further progress in the experimental understanding of topological phases of matter on programmable quantum devices.
    
    
    This paper is organized as follows. In Sec. <ref>, we review the essential features of both the chiral Fibonacci and the achiral DFib models and establish the notation. In Sec. <ref>, we describe our approach to initialization and measurement of DFib anyons. In Sec. <ref>, we discuss the details of constructing minimal circuits for demonstrating fusion, twisting and braiding of DFib anyons in different settings. Our circuits use, three, five and nine qubits, respectively — the smallest number possible in each case. Section <ref> concludes and provides an outlook.
    
    
    
    
    
    
    § THE FIBONACCI TOPOLOGICAL ORDER
    
    
    In this section, we first briefly review the (chiral) Fibonacci anyon model (Fib), then describe the basics of the Levin-Wen string-net model <cit.> that realizes the (achiral) DFib topological order.  
    
    
    
     §.§ The Chiral Fibonacci Model
    
    The Fibonacci anyon model <cit.> can be described by a set of anyon types, here represented by 1 and τ, where the former represents the trivial (vacuum) state and the latter the only non-trivial anyon type, carrying a quantum number known as topological charge. Similar to other quantum numbers (e.g., spin or electric charge), there are rules for combining topological charge. In the case of Fibonacci anyons, these rules are: 
    
    τ× 1 = τ,
    
    1 × 1 = 1,
    τ×τ = 1 + τ.
    
    
    The first two equations mean that combining (fusing) any object of charge 1 or τ with the vacuum state 1 will not affect the charge of that object. The third equation means that combining  the topological charges of two objects with charge τ will result in either an object with trivial charge 1 or an object with charge τ.
    
    An important consequence of the Fibonacci fusion rule is that the Hilbert space of n Fibonacci anyons, each of charge τ, has a dimension which grows according to the Fibonacci sequence (hence the name). In the asymptotic limit, the dimensionality of the Hilbert space grows as ϕ^n, where
    
    
        ϕ = √(5) + 1/2
     
    
    is the golden ratio. Since ϕ sets the growth rate of Hilbert space in this model, it is referred to as the quantum dimension of Fibonacci anyons, d_τ = ϕ. The quantum dimension of the vacuum particle is unity d_1 = 1.
    
    The order of anyon fusion defines a choice of basis for the Hilbert space. For example, given three anyons in a row, we can choose to fuse the first two, then add the third one. Equivalently, we can choose to first fuse the last two, then add the first one. These two choices define two different bases for the three-dimensional Hilbert space of three anyons. These two bases can be mapped to each other by a unitary operation known as F operator, or F-move <cit.>, defined pictorially in Fig. <ref>(a). In its simplest non-trivial form, the F  operator has the following matrix representation:
    
    
    F ≡ F^ττ m_ττ m' = [    1/ϕ 1/√(ϕ); 1/√(ϕ)   -1/ϕ;        ],
    
    
    where m,m' ∈{1, τ}.
    
    
    In the quantum computing community, Fibonacci anyons are best known for the universality of their braiding: their braid operators are generators of SU(2), and hence can be used to approximate all single-qubit gates with arbitrary accuracy <cit.>. In general, the braid operator R^ij_m describes the phases resulting from a counterclockwise exchange of two anyons with topological charges i and j, such that the total charge of the system is m, as depicted in Fig. <ref>(b). The matrix representation of the non-trivial  braiding of two Fibonacci anyons with total charge m is as follows:
    
    
    R 
    ≡ R^ττ_m 
    =  [ e^-4π i/5         0;         0 e^+3π i/5;           ].
    
    The diagonal nature of this matrix indicates that braiding two anyons does not change their total charge. The first and second diagonal elements correspond to the phase acquired by braiding two Fibonacci anyons with total charges 1 and τ, respectively. These are the braiding phases for so-called “right-handed” Fibonacci anyons; we can define a separate left-handed Fibonacci model whose braiding phases are the complex conjugate of those above. 
    
    
    
    
    
    Closely related to braiding is the concept of twisting, or a counterclockwise rotation of  an anyon around itself by 2π, as shown in  Fig. <ref>(c). The action of twisting is encoded in the  topological twist operator, or topological spin, θ_i, which is related to the braid operator as follows, 
    
    θ_i = (R^i i_1)^-1.
    
    
    It follows that braiding and twisting are related to each other as,
    
    
    (R_m^ij)^2 = θ_mθ_i^-1θ_j^-1,
    
    
    provided that 
    i and j   fuse to m.
    Here  i,j,m∈{1, τ}, θ_1 = 1 and θ_τ = e^+4π i/5. A visual depiction of this relation is shown in Fig. <ref>(d).
    
    
    Another element of the Fibonacci anyon model that will be relevant to us is the modular S-matrix, which has the form,
    
    
    S 
    = 1/D[  1  ϕ;  ϕ -1;    ].
    
    
    Here D = √(1 + ϕ^2) is the so-called total quantum dimension of the Fibonacci model. This operator describes the process in which two pairs of τ particles are created out of vacuum, then two anyons from different pairs exchange twice (one making a full wrap around the other), and finally each re-annihilates with its respective partner, forming the so-called “Hopf link.’’ A visual description of this operator is also given in Fig. <ref>(e).
    
    Finally, braiding and fusion rules can be used to derive relations to resolve string crossings. A version of this relation, which will be used extensively in the next section, is given in Fig. <ref>(f).
    
    
    
    
    
    
    There is no known exactly solvable model for the realization of the chiral Fibonacci model. However, Levin and Wen's string-net construction provides a framework for the doubled (achiral) realization of all types of topological order, including DFib. This model can be thought of as two copies of the Fibonacci model with opposite chiralities, where fusion and 
    braiding rules apply independently to the right-handed τ and the left-handed τ̅ excitations. The resulting DFib topological order, Fib⊗Fib,  has four distinct anyon types: {1,τ}⊗{1, τ̅} = { 11, τ1, 1τ̅, ττ̅}. Our realization of the DFib topological order is based on Levin and Wen's original <cit.> and extended string-net models <cit.>. In what follows we will briefly describe the properties of this model. 
    
    
    
     §.§ String-net Realization of the DFib Model
    
    
    
    
    The Levin-Wen (LW) string-net model <cit.> is defined by a set of string types on the edges of a 2D trivalent lattice and a set of self-consistent rules for combining (fusing) different string types. These fusion rules  define the string-net Hilbert space. A family of exactly solvable Hamiltonians can be introduced in this Hilbert space, which realize all achiral topological phases, and can be thought of as a generalization of Kitaev's toric code [Technically,  a string net model can be defined for any “unitary fusion category”C, resulting in an  anyon theory which is the so-called Drinfeld center Z( C).]. These Hamiltonians have the following form, 
    
    
    H = -∑_v Q_v -∑_p B_p,
    
    
    where Q_v and B_p are mutually commuting vertex and plaquette projection operators. The former acts on the three edges surrounding the vertex v, while the latter acts on the edges surrounding the plaquette p. Fig. <ref>(a) shows these operators for a hexagonal lattice. The system is in its ground state when both the vertex and plaquette operators aresatisfied with eigenvalue +1.   We say an operator is  violated if it has eigenvalue 0 rather than +1, and this corresponds to an excited state of the system.
    
    
    The operators Q_v and B_p act on the space of “string types,” which can be encoded as multi-level spins, or qudits, in general. In this picture, the vertex operator enforces certain branching rules, consistent with the fusion rules of the strings. In particular, the branching rules decree that Q_v is always violated if a single non-trivial string ends at a vertex (an open string) [This means a single anyon cannot be created out of vacuum.]. Thus, non-trivial string types that satisfy Q_v always form closed configurations (loops or nets). We will refer to the space of states for which all Q_v's  are satisfied as the “string-net space.”
    
    
     
    The plaquette operator B_p effectively measures the topological charge of plaquette p and is satisfied when the plaquette contains trivial topological charge. This operator, which also controls the dynamics of string-nets, imposes further restrictions on the string-net space. Together with Q_v these operators favor a certain superposition of loops and nets, which defines the ground state of the string-net model. In the original Levin-Wen model, various excitations result from different combinations of Q_v and B_p violations.
    
    
    The DFib topological order <cit.> is a specific instance of the LW string-net model, with two string types, corresponding to {1,τ} in the chiral Fibonacci phase. Here we identify the string types 1 and τ with qubit states |0 and |1, respectively. We will sometimes refer to an edge with string type τ, or equivalently qubit state |1, as an activated edge. The vertex operators, Q_v, essentially enforce branching rules  that are consistent with the fusion rules of the Fibonacci topological order Eq. (<ref>). As a result, Q_v is only violated when a single incoming edge at vertex v is activated. 
    
    
    The plaquette operator in this case can be written as,
    
     
    B_p = 1/D^2(1 + ϕ B_p^τ),
    
    
    where the operator B_p^τ effectively adds a loop of string type τ to the plaquette p. In fact, B_p can be thought as of an operator that adds what is known as a vacuum loop, a weighted sum of vacuum and a τ loop, to the plaquette, which can then be absorbed into the surrounding edges using a series of F-moves [For a detailed description of the process of absorbing strings through lattice edges with a series of F-moves see the original work in <cit.>]. Fig. <ref> depicts  the vacuum loop and the effect of the B_p operator on plaquette p. 
    
    
    As was mentioned above, the DFib model  has four  anyon types, { 11, τ1, 1τ̅, ττ̅}. Here 11 indicates the ground state, corresponding to no topological charge, which results from satisfying all the Q_v and B_p operators. In the original string-net model, the excitations ττ̅, τ1 and 1τ̅ result from combinations of Q_v and B_p violations. 
    
     The ττ̅ excitation consists of four components: ττ̅_11, ττ̅_1τ, ττ̅_τ1 and ττ̅_ττ, all of which have the same topological charge <cit.>. Out of these, the ττ̅_11 component can be realized just by violating B_p, thus it can be realized in the original LW model without leaving the string-net space. This excitation is essentially achiral since the phases resulting from braiding τ  and τ̅ generally cancel each other. However, due to its multi-channel fusion properties, a restricted set of braiding phases can be observed for these anyons. We will discuss this in more detail in Sec. <ref>.
     
    
    
    To demonstrate the full braiding phases of the DFib model, we need to consider the chiral excitations τ 1 or 1 τ̅. These excitations possess braiding statistics that are essentially equivalent to that of chiral Fibonacci anyons (right- and left-handed, respectively) and can therefore realize universal quantum computation <cit.>.
    
    
    To realize these excitations without violating Q_v, we use the extended LW model, which is defined on a tailed lattice with two extra edges added to each plaquette <cit.>, as shown in Fig. <ref>(e-h). The addition of these tails will allow us to correct Q_v violations while preserving the excitations, thus effectively permitting the creation of chiral excitations within the string-net space.
    
    
    
    § INITIALIZATION AND CHARGE MEASUREMENT
    
    
    
    The string-net space is defined by satisfying the vertex operator Q_v on every vertex v.  We can project to this space by measuring this operator at every vertex and then correcting where an error is detected. 
    Relatively simple circuits for measuring the vertex operator and correcting the affected vertices have been introduced <cit.>. Here, we do not explicitly show the vertex measurement/correction circuits, but we assume all Q_v operators are satisfied before applying our circuits. 
    
    
    Projection to the ground state requires further satisfying B_p operators at every plaquette, thus selecting a particular superposition of string-nets, which is a highly entangled state. Since the plaquette operators are projectors, we can ensure they are satisfied by a projective measurement, though the measurement itself is very nontrivial, as we shall now see.
    
    We follow the method introduced in <cit.>, which is inspired by the idea of entanglement renormalization <cit.>. This method is based on the observation that the F-move acting on a given edge e, which effectively redraws the lattice at that edge, and the operator B_p, which effectively measures the  charge of plaquette p,  “commute" with each other in the following sense: 
    
    
    F_e B_p = B_p' F_e.
    
    
    This means that if we measure the charge of an n-sided plaquette p with operator B_p, then apply an F-move to an adjacent edge e to reduce the size of the plaquette, the result is the same as applying the F-move first, followed by B_p' applied to the reduced-size (n-1)-sided plaquette p' (see Fig. <ref>). An important consequence of this observation is that F-moves can be used to locally modify the lattice while preserving the ground state. 
    
    Following <cit.>, we use this fact to both initialize the plaquettes and also to measure their charge. The basic idea is that by repeatedly applying F-moves to the edges of an n-sided plaquette, we can reduce it to a single tadpole: a two-edge graph defined by a a closed edge forming the “head”  and a connecting edge forming the “tail”: 
    
    
        111.
    
    
    We can use the above rule to reduce any lattice to a series of tadpoles, which can be easily initialized to the 11 (B_p = 1) state using single qubit gates , then reverse the F-moves to return to the initial lattice, now in its ground state <cit.>. Similarly, to measure the charge of a plaquette, we can reduce it to a tadpole with the same procedure, then carry out single qubit measurements of the head and tail edges to determine the charge of the original plaquette. In what follows, we discuss how this notion can be  generalized to initialize and measure the charges of other states with anyon types τ1,  1τ̅ and ττ̅. We also generalize this notion to devise a procedure for measuring the total charges of multiple plaquettes. 
    
    
    
     §.§ Excitations
      Not only do F-moves preserve plaquettes satisfying B_p = 1, but the statement that F and B_p operators commute (Eq. (<ref>)) guarantees that this holds for all eigenstates of the LW Hamiltonian. We can use this fact to both create and measure the entire DFib spectrum. 
    
    As noted earlier, in the original string-net model with no tails, the only other anyon type that can be initialized with a tadpole, without leaving the string-net basis, is ττ̅. To realize the chiral excitations τ1 and 1τ̅ without violating the vertex operator, we use the tailed lattice <cit.> in which each plaquette is equipped with two extra edges realizing a tail (see Fig. <ref>).  Similar to the original tail-less plaquette, we can reduce the tailed plaquette using a series of F-moves. The result is a generalized tadpole where an  extra inward tail is added inside the head of the tadpole:
    
    1111,
    
    
    (see also Fig. <ref>(f)). This generalized tadpole is minimally represented by four qubits and  can be used to initialize all four anyon types of the DFib model, including the four components of the ττ̅ anyon. 
    
    
    
    
    Just as the ground state can be created by initializing tadpoles with a vacuum loop, excitations can be created by superimposing τ strings to the vacuum loops <cit.>. This process is shown in Fig. <ref>. If a string passes over a vacuum loop, it results in  a generalized tadpole which generates the τ 1 excitation. The crossings between the τ strings and the vacuum loops can be resolved by using the relation given in Fig. <ref>(f). 
    
    Likewise, if a string passes under a vacuum loop, it leads to a 1τ̅ generating (generalized) tadpole. If two strings pass over and under the vacuum loop, the result is a ττ̅ generating tadpole. Note that the end points of two strings passing over and under the vacuum loop can fuse in four possible ways, resulting in the four components of the ττ̅ anyon. In the tailed lattice, these τ strings must end on the added tails to avoid Q_v violations. These are the same tails that show up in the generalized tadpoles that generate the DFib spectrum, a summary of which is also given below,
    
     =  1/D0000 +  ϕ/D1100
    1 =  1/D0111 +  ξ^2/D1011 +   √(ϕ) ξ/D1111
    t =  1/D0111 +  ξ^-2/D1011 +   √(ϕ) ξ^-1/D1111
    _11 =  ϕ/D0000 -  1/D1100
    _1τ =  1110
    _τ1 =  1101
    _ττ =  √(ϕ)/D0111 +  √(ϕ)/D1011 +   1/D ϕ1111.
    
    
    
    Here the red (shaded) lines indicate  strings of type τ (or qubit state |1), while dotted  lines depict the type 1 strings (qubit state |0), and ξ = e^3π i/5. 
    
    
    
    
    
    
    
    Thus, to create a pair of excitations of any kind, we first reduce the lattice to a series of generalized tadpoles using  F-moves. Then, depending on the desired excitation, we initialize them according the rules given in Eqs. (<ref>)-(<ref>). Finally we return to the original lattice by reversing the F-moves.  The circuits for both initialization and measurement of these generalized tadpoles are shown in Fig. <ref>. The S gate used in the figure is the modular S-matrix (Eq. (<ref>)) while the U gate is defined as,
    
    
    U = [ ξ^-1/√(ϕ)     ξ^2/ϕ;   -ξ^-2/ϕ    ξ/√(ϕ);           ].
    
    
    
    Finally, note that given the spherical boundary conditions, the excitations are always created in pairs — all of the quantum numbers of all the excitations in the entire system must fuse to the identity.   Thus we cannot create a single plaquette excitation from the ground state without also creating its partner.   In Fig. <ref> (and also in Eqs. (<ref>)-(<ref>)) for simplicity, we show only half of the process, where a string is superimposed on a single vacuum loop inside the head of a (generalized) tadpole. In general, we need to form the tadpoles such that plaquettes that share a pair of excitations will be transformed to tadpoles that share tails, which can then be initialized together. We will discuss this in more detail the following section. 
    
    
    
    
    
    
    
     §.§ Multiple Plaquettes
     
    
    In the extended LW model defined on lattices with tails where excitations are created in the absence of Q_v violations, they are essentially created on plaquettes. Thus, to determine the fusion amplitudes resulting from combing several excitations, we need a method to measure the total charge of multiple plaquettes. We discuss this problem from two different perspectives, first by building projective operators that determine the charge of two (or more) plaquettes, then designing quantum circuits which allow us to measure the total charge of multiple plaquettes with a computational basis measurement.
    
    As was noted in Sec. <ref>, the action of the plaquette operator B_p is to determine the charge content of a plaquette (whether it is trivial or not) by inserting a vacuum loop inside the plaquette, and absorbing it through the edges by a sequence of F-moves (see Fig. <ref>). This idea  can be generalized to an operator that determines the charge content of two neighboring plaquettes by inserting  a vacuum loop inside the two plaquettes. However, in this case, there is a choice of whether to insert the loop over or under the dividing edge. It turns out that these two options lead to two different operators, which we call B^o_p_i p_j and B^u_p_i p_j, where the superscripts o and u refer to over and under respectively. A visualization of these operators is depicted in Fig. <ref>. 
    
    
    Satisfying the operator B^o_p_i p_j indicates that the total charge of the right-handed anyons 1 in the two plaquettes p_i and p_j is trivial. Likewise, satisfying  B^u_p_i p_j implies that the total charge of left-handed anyons t on the corresponding plaquettes is trivial. Satisfying both of these the operators B^o_p_i p_j and B^u_p_i p_j  indicates that the total charge content of the two plaquettes is trivial. Note that this can be true even if individual plaquette operators B_p_i and B_p_j are not satisfied. This is analogous to the case where the collective charge of two τ anyons can be trivial. 
    
    
    Similar to single plaquette operators B_p, the double plaquette operators defined above are also projection operators, hence  cannot be implemented directly as a  unitary quantum circuit but instead must be implemented as a (rather complicated) projective measurement. Here we design quantum circuits to directly measure the charge of multiple plaquettes. Our approach is to generalize the single plaquette strategy — initialization and charge measurement by first reducing to a  tadpole — to a charge measurement by reducing multiple plaquettes to a series of concentric circles with each circle tied to the next by a single edge, resembling a series of concentric generalized tadpoles. Examples of this process for small lattices are shown in Fig. <ref>. 
    
    The key idea is that, through a series of F-moves, we reduce one of the plaquettes in the original lattice to the central tadpole in the concentric circles basis. The next-outermost (generalized) tadpole then contains the total charges of the original plaquette and another plaquette, which is reduced to an annulus surrounding the central tadpole. Similarly, with more plaquettes, the outer tadpoles hold the charge of all the original plaquettes which are subsequently enclosed by those tadpoles. Since the system should be thought of as living on the surface of a sphere, the outermost ring can be viewed as a tadpole as much as the innermost ring. 
    
    
    By choosing different series of F-moves, we can select different concentric tadpole bases that correspond to different combinations of plaquettes and use it as a basis for both initialization and measurement of combined charges. As an example, Fig. <ref>(b) and (c) depict two different choices of basis, suitable for measuring the charges of plaquettes p_1 ∪ p_2 or p_3 ∪ p_2. [Note that, in general, different sequences of F-moves can be used to transform a lattice into different concentric tadpoles. The  number of these choices increases in the case of tailed lattices and sometimes there are multiple sequences of F-moves that lead to seemingly similar concentric tadpoles. However, in general different choices can lead to additional twists and sometimes braids. Thus, caution needs to be exercised when selecting the F-move sequence to avoid unwanted phases. ]
    
    The equivalence of this approach to the projective operators above can be understood by recalling the action of the two-plaquette operator B_p_i p_j, which inserts a vacuum loop inside the two plaquettes. By keeping track of the F-moves which modify the lattice into concentric tadpoles, we can see that this vacuum loop effectively covers the area around two concentric tadpoles. 
    
    
    
    
    
    
    
    
    
    
    
    § SIGNATURES OF NON-ABELIAN ANYONS IN MINIMAL LATTICES
    
    
    In this section, we provide an overview of the signatures of the DFib model that we investigate in various settings, postponing the details to the following sub-sections. The basic properties that we are interested in are fusion, braiding and twisting of various excitations within the DFib model using the minimum number of qubits. Fig. <ref> depicts these operations for (chiral) Fibonacci anyons. 
    
    We restrict our study to lattices with spherical boundary conditions. The simplest lattices that can be defined on a sphere are shown in Fig. <ref>. The smallest possible lattice  consists of  two plaquettes each forming a hemisphere with a single common edge forming the meridian (see Fig. <ref>(b)). This two-plaquette lattice can be initialized into ground state or host a pair of ττ̅ = ττ̅_11 excitations, with total charge 11. The former can be realized by by applying an S gate to the single qubit on the edge initialized at |0, while the latter by applying an S gate to the qubit on the edge initialized at |1, as can be seen in Fig. <ref>.
    
    The smallest non-trivial lattice for our purposes consists of three plaquettes, separated by three edges that meet at the two poles of the sphere. A 2D depiction of this lattice, which resembles the Greek letter Θ, is shown in Fig. <ref>(c). We will use this lattice to demonstrate the fusion of two  excitations (Fig. <ref>). 
    
    The next smallest lattice with spherical boundary conditions consists of six edges and four plaquettes and is equivalent to a tetrahedron, as shown in Fig. <ref>(d).  We will use this lattice to realize a different implementation of the  fusion, which reveals additional details about the  structure of fusion channels, and discuss the difference between the two cases. 
    
    To realize the braiding properties of the DFib model (see Fig. <ref>(b)) we need to create the chiral excitations 1 or t without violating the vertex operators Q_v. This can be done using the extended tailed lattice. The simplest tailed  lattices are shown in Fig. <ref>(e-h). These lattices have additional qubits to account for the addition of a tail to each plaquette. The simplest lattice on which the creation and braiding of two pairs of chiral anyons can be demonstrated is the tailed Θ lattice, which consists of nine qubits (see Fig. <ref>(g)).
    
    We will also briefly discuss a simple procedure for revealing the topological twist phase of a single 1 anyon, which is created as part of a pair on the simplest two-plaquette lattice, or a generalized tadpole, consisting of four qubits (Fig. <ref>(c)). A simple generalization of this procedure can then be used to verify the relation between braiding and twisting.
    
    
    A note on notation: For the remainder of the paper, we will use two different notations for our states, depending on the context. The first is the qubit basis, in which we simply refer to the states of the qubits (|0 or |1) that form a particular lattice. The second notation is used when considering the concentric tadpole basis (see Fig. <ref>), where  we use a tensor product of kets that represent the sectors of the DFib model corresponding to each tadpole in this basis. 
    
    For example, in the case of the Θ lattice where the concentric tadpole basis consists of two tadpoles (see Fig. <ref>(a)), we use the notation |inn.|out. to refer to the states of the innermost and outermost tadpoles. Likewise for the tetrahedron lattice, where the concentric tadpole basis consists of three tadpoles (See Fig. <ref>(b) and (c)), we use the notation |inn.|mid.|out., which again refer to specific anyon types of the DFib model represented by each of the innermost, middle and outermost tadpoles in this basis.    
    
    In what follows, we explain the details of how various excitations can be created, braided, twisted and measured in specific lattices, and 
    present the corresponding quantum circuits. The circuit representations of the various F-moves used in this section are given in Fig. <ref>.
    
    
    
    
    
     §.§ Fusion with Three Qubits: The Θ Lattice
    
    
    The simplest setting for detecting the fusion properties of the DFib model is to realize two pairs of  excitations on a minimal lattice consisting of three plaquettes on a sphere, or the Θ lattice. 
    The process of initialization and detection of fusion channels is shown in Fig. <ref>.
    
    
    Initialization:  We start by initializing two pairs of _11 excitations on plaquette pairs p_1, p_3 and p_2, p_3. This can be done by starting from a basis in which each of the plaquettes p_1 and p_2 are reduced to single tadpole, as shown by the trivial state  shown as |ψ_00 in Fig. <ref>. Here  qubits q_1 and q_3 form the heads of the two tadpoles and qubit q_2 serves as the common tail.   We then apply single-qubit gates to the two heads, resulting in the state |ψ_0, which in qubit basis has the form,
    
    
    |ψ_0   =    |q_1q_2q_3
       =   (ϕ|0 - |1/D)|0(ϕ|0 - |1/D).
    
    
    We then apply an F-move to qubit q_2  to restore the original Θ lattice basis, resulting in the state |ψ_1.  At this point, we have created two pairs of  excitations on plaquette pairs p_1, p_3 and p_2, p_3 in the Θ lattice, such that the total charges of plaquettes p_1∪ p_3 and p_2∪ p_3 correspond to the  sector of the DFib model. In other words, we have effectively pulled two  pairs of _11 excitations out of vacuum. 
    
    
    Measurement:  We expect the total charge of plaquettes p_1 ∪ p_2, (or equivalently the charge of plaquette p_3, due to the spherical boundary conditions) to be in a superposition state consistent with the fusion rules of the Fibonacci model. To read this total charge, we need to change the basis into concentric tadpoles (see also Fig. <ref>(a)) such that, either plaquette p_1 or plaquette p_2 will be enclosed by a tadpole, which is nested inside an annulus, containing the charge of the other plaquette. This can be done by applying an F-move, acting on qubit q_3, in |ψ_2 as shown in Fig. <ref> [This could also be done with an F_1 move resulting in a shape where p_2 is enclosed by the central tadpole.]. 
    
    As was discussed in Sec. <ref>, the outer tadpole, which encloses plaquettes p_1 and p_2, can now be measured to ascertain the total charge of p_1 ∪ p_2. Note that in this case this outer boundary is also equivalent to the boundary of the single tadpole that encloses plaquette p_3 (the remainder of the sphere) and contains the same charge as p_1 ∪ p_2. Note also that while we use the term fusion in referring to combining topological charge, we don't actually fuse excitations in plaquettes p_1 and p_2. We merely change the basis such that we can measure the superposition state that contains information on fusion amplitudes, should an actual fusion occur. 
    
    
    
    Using the notation |ψ_2  = |inn.|out., this state can be written as,
    
    
    |ψ_2   =   _11
       +   (_11_11 
    - _1τ_τ1),
    
    
    which represents the anyon types contained by the inner and outer tadpoles, as shown in Fig. <ref>. We see that the inner tadpole, which contains the charge of plaquette p_1, corresponds to a single  anyon in all three terms. This is consistent with our initialization, in which plaquettes p_1 and p_2 each contained a single  excitation. 
    
    Focusing on the outer tadpole, which contains the total charge of plaquettes p_1 ∪ p_2, the first term in |ψ_2 indicates trivial total charge with coefficient 1/ϕ, resulting in the state , while the second and third terms indicate that the total charge corresponds to the charge of a single  anyon with coefficient 1/√(ϕ). Thus, the total charge of of p_1∪ p_2, signifies the following cross fusion relation,
    
    × =  + ,
    
    
    i.e., only two fusion channels appear. We will return to this point at the end of this section.
    
    Note that while we started by initializing the _11 component of the  excitation, cross fusion results in additional components of the  excitation. This can be seen in the third term of |ψ_2 (see Fig. <ref> and Eq. (<ref>)) where the excitation _1τ1 appears in the inner tadpole and _τ1 in the outer one. Since the components of  contain the same charge, this is consistent with Eq. (<ref>), where the second term can contain different combinations of the components of .  These terms can also be obtained by resolving the loops on the right hand side of the equation shown in Fig. <ref>(a) followed by the F-move.
    
    To simplify the measurement of the fusion channels of p_1 ∪ p_2 in a quantum circuit, we apply a series of two-qubit gates to |ψ_2, which effectively maps it to a product state. These two-qubit gates are two controlled-S gates acting on qubits q_2 and q_3 and a controlled-L gate, acting on qubit q_3 where,
    
    
    L = [  1/√(ϕ)    -1/ϕ;    -1/ϕ -1/√(ϕ);         ],
    
    
    resulting in the state
    
    
    |ψ_3   =    |q_1q_2q_3
       =   (|0 + |1)|10.
    
    
    Thus, the information on the fusion channels of  p_1 ∪ p_2 is effectively transferred to qubit q_1 in this state. 
    
    To detect the fusion coefficients, we can perform statistical measurements on q_1 at this stage and observe the corresponding probability distribution. Alternatively, we can apply an F gate (Eq. (<ref>)) to q_1, then use the result as the control for a CNOT gate acting on a target ancilla qubit. This F gate acts as a “filter,” so that if q_1 is in the correct superposition, the F gate maps it to |0, which does not affect the ancilla qubit in the subsequent CNOT. Thus, with this protocol we can verify the coefficient of fusion by carrying out a single measurement of the ancilla qubit. 
    
    Alternatively, we can determine the state of  |ψ_2⟩ by first measuring the state of q_3.  If q_3 is found to be in the |1⟩ state, so should q_1 and q_2 (the last term in the expression for |ψ_2⟩ in the Fig  <ref>.c).  If q_3 is found to be in the |0⟩ state, then q_1 and q_2 can be tomographed, and we should find q_2 to be in the  |ττ⟩_11 state and q_1 to be in the |1 1 ⟩ + |ττ⟩_11 state, which directly reflects the fusion in Eq. <ref>. In cases below we will always favor read-out schemes that minimize the number of measurements, although other read-out schemes are certainly possible.
    
    
    To conclude this section, we return to the observation in  Eq. (<ref>), namely the fact that in the Θ lattice, the result of cross fusing two  excitations is two fusion channels, namely  and . This might look unexpected at first sight, since the two sets of chiral excitations in the DFib model (the right-handed 1 and the left-handed t) must fuse independently from each other, thus in principle we expect the combination of two  excitations to result in four fusion channels: , 1, t and . However, recall that 1 and t are created by passing a τ string over or under the lattice, respectively. On the Θ lattice, two excitations must share the same plaquette and for a single plaquette there is no way to distinguish between “over” and “under.” Thus only the achiral excitations can exist on this plaquette. Consequently,  the combined charge of p_1∪ p_2 lead to only two  fusion  channels:  and .
    
    
    
    
    
    
    
    
    
    
    
    
    
    To explore this idea further, we create two pairs of _11 excitations on the four plaquettes of a  tetrahedral lattice, consisting of six qubits, and measure the cross-fusion channels of opposite plaquettes. We expect to see four fusion channels in this case, including the chiral excitations. We will discuss this in the next section. 
    
    
    
     §.§ Fusion with Six Qubits: Tetrahedron Lattice
    
    
    The process of initialization and measurement of the fusion channels of two pairs of excitations on the four plaquettes of a tetrahedral lattice is shown in Fig. <ref>. 
    
    Initialization:  We start by initializing two pairs of  excitations out of vacuum on pairs of plaquettes p_1, p_2 and p_3, p_4 such that the total charge of each pair corresponds to . This can be done by starting from a basis in which plaquettes p_1 and p_2 form concentric tadpoles, enclosed by a third tadpole containing p_3. In Fig. <ref>, one such starting configuration is depicted as the basis for the state |ψ_00. We then create pairs of _11 excitations on plaquettes p_1, p_2 and p_3, p_4 by initializing the corresponding generalized tadpoles. 
    
    The innermost tadpole, which represents plaquette p_1, consists of qubits q_2 and q_5, forming its head and tail, respectively. This tadpole is initialized as, 
    
    
    |q_2 q_5 = ϕ|0 - |1/D|0.
    
    
    
    The middle (generalized) tadpole, which contains the information about total charges of p_1 ∪ p_2, consists of qubits q_1 and q_6,  together forming its head, and q_5 and q_3, which form its two tails. This tadpole is initialized as,
    
    
    |q_1 q_6 q_5 q_3 = |00 + ϕ |11/D|00
    
    Finally, the outermost tadpole, containing the charge of p_1 ∪ p_2 ∪ p_3, consists of qubit q_4 as its head and qubit q_3 as its inward tail. These qubits are initialized as,
    
    |q_4 q_3 = ϕ|0 - |1/D|0.
    
    
    The resulting state in the qubit basis is the following,
    
    
    |ψ_0   =    |q_2 q_4 |q_1 q_6 |q_5 q_3
       =   (ϕ|0 - |1/D)^⊗2(ϕ|00 +ϕ |11/D)|00.
    
    
    
    
    We can also re-write the state |ψ_0 in the concentric tadpole basis, where each ket represents the topological charge of the corresponding generalized tadpole,
    
     
    |ψ_0   = |inn. |mid. |out.
    
    where,
    
    
    |inn.   = _11
    
    |mid.   = 
    
    |out.   = _11.
    
    
    
    Here the first equality means that the plaquette represented by the innermost tadpole (p_1) and the plaquette represented by the annulus surrounding the innermost tadpole (p_2) each have a charge of _11. The second equality indicates that the total charge of plaquettes p_1 ∪ p_2, represented by the middle tadpole, is ; in other words, the pair of excitations on p_1 and p_2 are effectively pulled out of vacuum. Finally, the third equality implies that the plaquette represented by the annulus surrounding the middle tadpole (p_3) and the outside plaquette representing the remainder of the sphere (p_4) each also have a charge of _11. Due to spherical boundary conditions, the total charge of these two plaquettes is also , thus the excitations on these plaquettes are also pulled out of vacuum. 
    
    To return to the original lattice basis (tetrahedron) we apply three F-moves to qubits q_3, q_5 and q_1, as shown in Fig. <ref>(b), resulting in the state |ψ_1. The circuit definitions of the F-moves can be found in Fig. <ref>. At this point, we have created two pairs of  excitations on plaquettes p_1, p_2 and p_3, p_4 such that each pair fuse to vacuum. 
    
    
    
    Measurement: To measure the fusion coefficients resulting from cross-fusing excitations from opposite pairs, e.g., p_2 ∪ p_3 and p_1 ∪ p_4, we need to switch to a concentric tadpole basis in which two plaquettes from opposite pairs are enclosed by the middle tadpole. One such basis is shown in Fig. <ref>(c), in which the middle tadpole encloses the charge of p_2∪ p_3. In this basis, the innermost tadpole contains the charge of plaquette p_1, and the outermost tadpole corresponds to the total charge of plaquettes p_1 ∪ p_2 ∪ p_3 or, equivalently, plaquette p_4. This basis change can be achieved by applying three additional F-moves acting on qubits q_4, q_3 and q_5,  resulting in the state |ψ_2 in Fig. <ref>. In the concentric tadpole notation, this state has the form,
    
    
    |ψ_2   = 1/ϕ^2_11_11
       + 1/ϕ√(ϕ)_1τ(1 + t) _τ1
       +1/ϕ[
    1/ϕ_11_11_11 
    -1/ϕ√(ϕ)( _τ1_1τ_11 + _11_1τ_τ1) 
    + 1/ϕ^2_τ1_ττ_1τ].
    
    
    
    As was noted above, the information on the fusion channels of the total charge of p_2∪ p_3 is now represented by the state of the middle tadpole in |ψ_2. To identify the components of the fusion channels, we can use Eq. (<ref>)-(<ref>) to re-write |mid. = × as follows,
    
    
    |mid.   =   1/ϕ^2
       +   1/ϕ√(ϕ)(1  + t)  
       +   ,
    
    
    where,
    
       =   _11
       -   1/ϕ√(ϕ)( _1τ + _τ 1)
       +   1/ϕ^2_ττ.
    
    
    This is consistent with our expectation that in general the two chiral sectors of the  excitation fuse independently, resulting in four fusion channels, as given in Eq. (<ref>).
    
    The charges of individual plaquettes p_3 (represented by the innermost tadpole |inn.) and p_4 (represented by the outermost tadpole |out.) remain at . However, as we see in Eq. (<ref>), for these states different components of the  sector appear in different terms. 
    
    Similar to the previous case, it is possible to design a quantum circuit to simplify the state |ψ_1 such that the expected configuration will be mapped to an un-entangled state which can then be measured for verification.
    
    
    
    
    
     §.§ Fusion and Braiding with Nine Qubits: Tailed Θ Lattice
    
    
    To detect non-Abelian braiding phases in the DFib model, we need to create the chiral excitations 1 or t. To realize these excitations without violating the vertex operators Q_v, we employ the extended tailed lattice <cit.>. The simplest nontrivial tailed lattice for this purpose would be a tailed Θ lattice, consisting of nine qubits, as shown in Fig. <ref>(g). In what follows we describe circuits to create these excitations, change the basis to measure fusion channels, then braid the excitations and finally detect the braiding phases. 
    
    
    
    
    
    
      §.§.§ Fusion
    
    
    Here we focus on creating two pairs of 1 excitations on three plaquettes, aiming at detecting the following fusion relation,
    
    1 ×1 =  + 1,
    
    
    which results from fusing excitations from opposite pairs, created out of vacuum. The general procedure resembles the case of creating  excitations in the Θ lattice described above, but the initialization and measurement processes involve details that are specific to the chiral excitations, which we describe bellow. The full circuit is depicted in Fig. <ref>.
    
    Initialization: We initialize the tailed Θ lattice in a basis in which plaquettes p_1 and p_2 are enclosed by separate (generalized) tadpoles, resulting in |ψ_0, as shown in Fig. <ref>(c). The general form of initialization circuit for  1 anyons is given in Fig. <ref>. Here we apply these circuits independently to the two tadpoles, but initialize the tail of plaquette p_3, with an F-gate. This can be understood by first imagining two tails for plaquette p_3 by using two extra qubits q_10 and q_11, then initialize each tadpole independently according to Fig. <ref>. We then combine these tails by applying an F-move to q_9 and through away the extra qubits  q_10 and q_11, leaving us with |q_9=|0 + |1. Since everything about this process is deterministic, we choose to do away with the extra qubits and directly initialize the tail qubit q_9 with an F gate. 
    
    We then apply three F-moves to |ψ_0 to return the state to the Θ lattice basis, resulting in |ψ_1. This process is depicted in Fig. <ref>(c) where the effects of the F-moves on the lattice are shown. Here the colored solid lines are merely guides to the eye, to keep track of the locations of pairs of excitations. At this point we have created two pairs of 1 excitations on plaquettes p_1, p_3 and p_2, p_3, such that each of these pairs would fuse to the vacuum state .
    
    Measurement: To measure the  amplitudes of the superposition state |ψ_1, resulting from cross-fusing the charges of plaquettes p_1∪ p_2, we change into a concentric tadpole basis where plaquettes p_1 and  p_2 are enclosed by concentric tadpoles. This can be done with the aid of three F-moves, resulting in the state |ψ_2 as shown in Fig. <ref>(d). 
    
    In this state, the outer tadpole, consisting of qubits  q_5 and q_6, which form the head, and qubits q_2 and q_9, forming its tails, contains the information on the total charge content of plaquettes p_1 ∪ p_2, or equivalently, p_3. Likewise, qubits q_1, q_3, q_4 and q_7, which form the components of the inner tadpole, contain information on the charge content of plaquette p_1. Written in the concentric tadpole basis, |ψ_2 = |inn.|out., we have, 
     
    |ψ_2 = 
    1(1/ϕ +
    1/√(ϕ)t ).
    
    
    The components of the  and 1 sectors for the inner and outer tadpoles are shown in Fig. <ref>. In terms of these components, the inner tadpole consists of three terms (Fig. <ref>(c)) and the outer tadpole consists of two terms for the  sector (Fig. <ref>(a)) and three terms for the 1 sector (Fig. <ref>(b)), leading to fifteen terms for |ψ_2.
    
    To detect the fusion amplitudes, we  first simplify |ψ_2 using the  circuits given in Fig. <ref> applied independently to the inner and outer tadpoles. Switching to the qubit basis, these circuits first map the two tadpoles to the following:
    
    
    |q_4 q_8 q_7 q_1 q_3   =    |11110
    
    |q_9 q_2 q_6 q_5   =   |0000 +  |1110.
    
    
    To further simplify, we apply CNOT gates to q_2, q_6 and q_9 to un-entangle them, resulting in 
    
    
    |ψ_3   =    |q_1 q_2 q_3 q_4 q_5 q_6 q_7 q_8 q_9
       =     |1(|0 +  |1)|0100110.
    
    
    Thus, the information on the fusion channels of the combined plaquettes p_1 ∪ p_2 is now stored in the state of a single qubit, namely q_2 [By choosing different combinations of CNOTs we could choose any of q_2, q_6 or q_9 as our flag qubit.]. 
    
    To detect these amplitudes, as before, we either carry out statistical measurements of q_2 and measure the probability distribution of each fusion channel, or apply an F gate to q_2 as a “filter,” mapping it to |0, then use the result to control a CNOT acting on ancilla qubit A. If the fusion channels are as expected, the ancilla qubit should not change. 
    
    
    
    
      §.§.§ Braiding
    
    
    Staying within the tailed Θ lattice, here we discuss how to create and braid two 1 excitations and measure the resulting phases. This process is shown in Fig. <ref>.
    
    Initialization: We repeat the same initialization as the one we used in the previous section and create two pairs of 1 excitations, on plaquettes p_1, p_3 and p_2, p_3, resulting in the state  |ψ_1 in the Θ lattice basis. 
    
    
    
    
    Braiding: To braid the charge contents of plaquettes p_1 and p_2, we simply exchange the corresponding plaquettes in a counterclockwise manner using a series of F-moves <cit.> as shown in Fig. <ref>(d), resulting in the state |ψ̃_1. Note that, while we end up in the same lattice basis as the starting point, braiding effectively changes the arrangement of qubits on the lattice. We need to keep this in mind when preparing the state for read-out. 
    
    Measurement: To measure the braid phases, we again need to change the basis into one where plaquettes p_1 and p_2 form concentric tadpoles, so that the total charges of combined plaquettes p_1 ∪ p_2 will be enclosed by the outer tadpole. This process is the same as the one in the fusion case but with different qubits, as shown in Fig. <ref>(e). Using the concentric tadpole basis,   |ψ̃_2 = |inn.|out., we get,
    
    
    |ψ̃_2 = 1 
    ( e^-4π i/5/ϕ + e^+3π i/5/√(ϕ)1
    ).
     
    
    Thus the two states of the outer tadpole have acquired the phases corresponding to a counterclockwise exchange of two τ1 anyons. 
    
    
    To verify these phases with a quantum circuit, we begin by simplifying |ψ̃_2, using the detection circuits in Fig. <ref>. This simplification effectively maps the states of the two tadpoles to the following, 
    
    
    |q_1 q_7 q_8 q_5 q_4    =    |11110
    
    | q_3 q_9 q_2 q_6   =   e^-4π i/5/ϕ|0000 +e^+3π i/5/√(ϕ)|1110.
    
    
    To further simplify this state, we un-entangle qubits q_2, q_3 and q_9 by applying two CNOT gates, resulting in 
    
    
    |ψ̃_3   =    |q_1 q_2 q_3 q_4 q_5 q_6 q_7 q_8 q_9
       =    
    |1(e^-4π i/5/ϕ|0 + e^+3π i/5/√(ϕ) |1) |1010110.
    
    
    Thus, all the information on braiding phases of the two fusion channels is now stored in q_2. For the final detection, similar to the previous cases, we design another filter, which has the form,
    
    F̃ = [     ξ^2/ϕ    ξ/√(ϕ); ξ^-1/√(ϕ)  -ξ^-2/ϕ.;           ]
    
    
    
    We believe this is the simplest setting in which to demonstrate the braiding properties of the chiral excitations of the DFib model. One may also perform these exercises on a tailed tetrahedron (Fig. <ref>(h)) with fourteen qubits, such that each of the four plaquettes contains a single 1 excitation. The results are identical to the three plaquette lattice for both fusion and braiding. 
    
    Since this article is focused on minimal models, we do not provide that circuit here. 
    
    
    
    
    
    
     §.§ Twisting with Five Qubits: Tailed Tadpole
    
    
    A chiral excitation can be twisted within a plaquette and the resulting phase can be detected using additional tricks (see, e.g., <cit.>). The simplest lattice for this purpose consists of only four qubits, forming a single generalized tadpole, which can be thought as representing two plaquettes on a sphere, one shown and the other being the outside region representing the remainder of the sphere (see Fig. <ref>(f)). The circuit for the initialization, twisting and detection of the phase is shown in Fig. <ref> (with two additional qubits to simplify readout). 
    
    
    We start by initializing  the 1 state on the four qubits forming the generalized tadpole, using the initialization circuits in Fig. <ref>. We then apply a special case of an F-move, which effectively exchanges the two tails. This F-move is defined in Fig. <ref>. Written in the original basis, this is equivalent to twisting the 1 excitation in the front plaquette in a counterclockwise manner, resulting in,
    
    θ̂1 = e^+4π i/51.
     
    
    
    To detect the overall phase, we initialize an ancilla qubit |q_0 with a Hadamard gate. Re-writing the 1 state in qubit basis as, 
    
    1    =    |q_1 q_2 q_3 q_4
       =   1/D|00(|10 + ξ^2|01 +√(ϕ)ξ|11).
    
    
    results in the initial state, 
    
    
    |ψ_0   =    |q_0|q_1q_2q_3q_4
       =   1/√(2) (|0 + |1)1.
     
    
    We then then use |q_0 as a control qubit for the application of a controlled-F-move acting on q_3, resulting in
    
    
    |ψ_1   =    |q_0|q_1q_2q_3q_4
       =   1/√(2) (|0 + e^4π i/5|1)1.
    
    
    Thus, we use the phase kickback from twisting the 1 excitation in the tadpole consisting of  qubits q_1-q_4 to impart a phase shift on the ancilla qubit |q_0. We can then measure this phase shift by using the appropriate filter, which maps the state of q_0 to |0. The filter in this case is the following gate,
    
    M = [         1  e^4π i/5; e^-4π i/5        -1;           ].
    
    Finally we can use q_0 as a control for a CNOT acting on another ancilla qubit to verify the expected phase, as in the previous cases. Since the state of the ancilla qubit q_0 is un-entangled from the qubits forming the tadpole, we do not need to apply a simplification circuit to these qubits. However for ease of checking the results, simplifying gates can be adopted from Fig. <ref> to map the 1 state to |q_1q_2q_3q_4 = |1110. 
    
    
    We end by mentioning that this procedure can easily be generalized to larger plaquettes and be used to verify the relation between braiding and twisting, Eq. (<ref>), in the nine-qubit lattice. 
    
    
    
    
    § CONCLUSIONS AND OUTLOOK
    
    
    This work was inspired by a theoretical question: what are the smallest possible lattices in which the string-net rendition of the Doubled Fibonacci model can be realized and its non-Abelian nature revealed? We answered this question by designing quantum circuits that simulate the spectrum of this model and demonstrated their fusion, twisting and braiding properties, all on minimal lattices with the fewest number of qubits possible. 
    
    We argued that the ground state as well as a  pair of achiral  excitations can be realized on a lattice consisting of only one qubit. We then showed that with only three qubits, we can create two pairs of  excitations in a minimal three-plaquette lattice and demonstrate the amplitudes of their fusion channels. We further showed that by moving to a lattice with six qubits, additional fusion structure resulting from combining  excitations can be revealed. 
    
    We then created two pairs of chiral 1 excitations in a three-plaquette tailed lattice, consisting of nine qubits, braided them and measured the relative braiding phases. Finally, we showed that a total of four qubits is enough to create a pair of chiral excitations (1 or t) in two plaquettes, twist one of them and verify the resulting phase with the aid of a fifth (ancilla) qubit. 
    
    The Fibonacci anyon model exhibits one the simplest, yet most exotic, forms of topological order, with possible applications in active and passive fault-tolerant quantum computation. Our goal here was to find the minimal resources required to demonstrate the non-Abelian nature of Fibonacci anyons and we believe our circuits require sufficiently small quantum volume that they can be run on any modern qubit hardware with reasonable coherence times and gate fidelity, thus making this model more accessible to the wider physics community. 
    
    The methods discussed here can  be generalized to larger lattices, where more anyons can be created and quantum gates can be carried out topologically. An important future direction would be to quantify and measure the stability of these states in the presence of noise. Another interesting direction would be the design of minimal circuits that simulate other instances of the  string-net model, resulting in other forms of topological order, especially those that require multi-level qudits.
    
    
    
    
    § ACKNOWLEDGMENT
     
    
    We thank R. Koenig, M. Levin and A. Schotte for clarifying discussions at an earlier stage of this work, and F. Burnell for discussions and comments on an earlier draft of the manuscript. We also thank the Aspen Center for Physics
    for its hospitality during the completion of part of this
    work. This project was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. LH and JP acknowledge partial support from BNL LDRD 20-022. SB and RRA were supported by the U.S. Department of Energy, Office of Science’s SULI program.  SHS is supported by EPSRC Grants
    EP/S020527/1 and EP/X030881/1.
    
    
    Note Added: During the preparation of this work, we became aware of two other independently carried-out works on a related but different simulation of DFib anyons on specialized hardware <cit.> and <cit.>. Besides using different methods, the present work is distinct in that it focuses on designing minimal circuits suitable for realization on any quantum platform.


 Paper Title: Emergent dipole field theory in atomic ladders.



SPstyle




Emergent dipole field theory in atomic ladders







Hernan B. Xavier1,2⋆,
Poetri Sonya Tarabunga1,2,3,
Marcello Dalmonte1, and 
Rodrigo G. Pereira4








1 The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy


2 International School for Advanced Studies (SISSA), Trieste, Italy


3 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy


4 International Institute of Physics and Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal-RN, Brazil








⋆ mailto:hxavier@ictp.ithxavier@ictp.it







§ ABSTRACT


We study the dynamics of hard-core bosons on ladders, in the presence of strong kinetic constrains akin to those of the Bariev model. 

We use a combination of analytical methods and numerical simulations to establish the phase diagram of the model. The model displays a paired Tomonaga-Luttinger liquid phase  featuring an emergent dipole symmetry, which encodes the local pairing constraint into a global, non-local quantity.

We scrutinize the effect of such emergent low-energy symmetry during quench dynamics including single particle defects. We observe that, despite being approximate, the dipole symmetry still leads to very slow relaxation dynamics, which we model via an effective field theory. 

The model we discuss is amenable to realization in both cold atoms in optical lattices and Rydberg atom arrays with dynamics taking place solely in the Rydberg manifold. We present a blueprint protocol to observe the effect of emergent dipole symmetry in such experimental platforms, combining adiabatic state preparation with quench dynamics.

















1pt
1pt






§ INTRODUCTION


Sparked by a series of remarkable atomic physics experiments <cit.>, constrained dynamics in quantum many-body systems have attracted a great deal of attention in recent years <cit.>. From a fundamental viewpoint, these systems offer a rich playground for studying complex non-equilibrium properties, where the interplay of correlations and dynamical frustration can result in a variety of elusive phenomena, such as Hilbert space fragmentation <cit.>, slow relaxation dynamics <cit.>, as well as intriguing links to lattice gauge theories <cit.> and fracton models <cit.>. Such phenomena are inherently related to strong correlations and lack a counterpart in the context of non-interacting particles. 

The imposition of higher, multipole conservation laws in the many-body theory <cit.> represents a new opportunity to generate nontrivial dynamics. This has been recently demonstrated in experiments with ultracold atom gases <cit.>, where a tilted optical lattice is used to enforce a dipole (center-of-mass) preserving dynamics. In view of that, a number of works have  not only established the ground state phase diagram of dipole-conserving lattice models of fermions or bosons <cit.>, but have also considered the nonequilibrium dynamics of such models <cit.>. The core idea about this research line is to enforce as well as possible the dipole symmetry as a Hamiltonian property (regardless of energy scale).

In this work we pursue a different approach, where the symmetry constraint emerges as a low-energy property of the ground state. We draw inspiration from a Bariev-like model <cit.>, whose phase diagram features a Tomonaga-Luttinger liquid (TLL) state formed by bound pairs. We use field theory arguments and exact diagonalization (ED) to link this ground state to an emergent dipole-type symmetry, which constrains the local dynamics of single-particle excitations that need to find partners in order to move. Having in mind atomic physics realizations (Fig. <ref>), we propose a quasi-adiabatic protocol <cit.> to prepare the dipole TLL state from biased optical ladders, as well as Rydberg atom arrays. We benchmark the state preparation by using a time-dependent variational principle (TDVP) algorithm to dynamically evolve an initial product state. Finally we consider the quench dynamics of isolated defects placed on top of the dipole TLL state. We use a combination of density matrix renormalization group (DMRG) and time-evolving block-decimation (TEBD) algorithms to prepare and evolve the single-defect states, contrasting our numerical results to field theory predictions. Figure  <ref> gives a general perspective onto a protocol that prepares and observes the time evolution of defects. 




The rest of the paper is organized as follows. We set our notation and introduce the model Hamiltonian as a hardcore boson ladder in Sec. <ref>. We inspect analytically specific parameter regimes, and then focus on the regime where the model exhibits strictly confined excitations. We utilize a duality map to reveal a connection to a PXP-type model featuring Hilbert space fragmentation, providing an alternative viewpoint on dipole symmetry at one of the exactly solvable points the model features. We discuss experimental implementations of the microscopic dynamics in Sec. <ref>: we derive the effective Hamiltonian from physically sensible Rydberg- and cold-atom models, discussing pertinent perturbations to each platform. A low-energy field theory description for the lattice model is presented in Sec. <ref>, which we use to assess the stability of the dipole TLL state and devise the mobile impurity model. This analysis is complemented by numerical simulations in Sec. <ref>. We give particular focus to benchmark the state preparation protocol and test the nontrivial dynamics of isolated defects. We offer a summary and point out open perspectives in Sec. <ref>.




§ EFFECTIVE HARD-CORE BOSON MODEL



We consider hard-core bosons on a two-leg zigzag ladder (shown in Fig. <ref>), where the number of particles is preserved separately in each sub-lattice. 
The system dynamics is described by a Bariev-like <cit.> Hamiltonian:

    H=-J∑_i(σ_i^+σ_i+2^-+)-W∑_i(σ_i^+σ_i+1^zσ_i+2^-+)+V∑_iσ_i^zσ_i+1^z,

where i=1,2,…,L are sites in the zigzag geometry, and σ_i^+ is the hard-core boson creation operator. The hard-core boson occupation number n_i=σ^+_iσ^-_i is related to the Pauli matrix σ_i^z by n_i=(1+σ_i^z)/2. The lattice model depends on three coupling parameters. The intra-leg hopping amplitude J, the correlated hopping term W, and the Ising-like interaction V. Note that single-particle tunneling among different legs is not allowed as it violates the number conservations.

The two global U(1) symmetries of the model   can be expressed as the conservation of the total number of excitations and the so called inter-leg magnetization:

    N=∑_in_i,    
    M=1/2∑_i(-1)^in_i.

We call N and M the charge and spin quantum numbers, given the similarity to electronic systems. We use these two conserved charges  to fix a subspace of interest. Hereafter, we concentrate on the scenario where both legs are half-filled, meaning we shall consider chains whose thermodynamic limit is defined by N/L→1/2, with vanishing magnetization M/L→0. We use J=1 to set the energy scale, and study the model as a function of W and V. We only consider positive values of W, given that we can flip its sign via a global particle-hole transformation, σ_i^z→-σ_i^z.






 §.§ Phase diagram overview


Bariev-like models and their corresponding phase diagrams have been discussed in the literature <cit.>. Below, we exploit some of these earlier results to clarify the phase diagram of the hard-core boson ladder, referring the reader to Refs. <cit.> for more details. We will complement those with a field theory approach we describe below.

The phase diagram is schematically depicted in Fig. <ref>. At V=W=0, we find the 2TLL phase, a critical state with power-law decaying correlation functions described by two independent TLL theories (as for decoupled chains). The TLLs govern the low-energy spectrum of collective excitations of charge and spin that are generated from the hybridization of the original excitations in the legs. This phase arises from the competition among the correlated hopping W and the antiferromagnetic V interaction that prevents excitations from pairing up.

If we keep V small and move towards dominant W coupling, spin excitations are gapped out and we enter the dipole TLL phase. The dipole TLL phase can be viewed as a quantum liquid of molecular dimers <cit.>, where each dimer is formed by binding two single-particle excitations that live in distinct legs together. The associated pairing strength depends mostly on the correlated hopping W <cit.>, which drives a BKT type transition at V=0 into the dipole TLL phase. The underlying liquid ground state can be identified by a den Nijs-Rommelse type string order parameter as showed by Chhajlany et al. <cit.>.

Alternatively, we identify here an emergent dipole-like symmetry that constrains the dynamics of single-particle excitations in the dipole TLL state. At W=J the binding strength reaches its maximum <cit.>. At this special point, excitations are strictly confined (see Fig. <ref>) and the dipole symmetry becomes exact as the lattice model Eq. (<ref>) commutes with

    D=∑_i iñ_i,    ñ_i=n_iexp(π∑_j<in_j),

which encodes the local constraint in a nonlocal global operator. We point out that, although clearly grounded in the fixed points provided by the Bariev chain obtained for V=0 <cit.>, the dipole TLL phase is quite robust to a great sort of perturbations, cf. Sec. <ref>, surviving up to finite values of V <cit.>.

Finally, for dominant V coupling we find the phase separated (PS) states PS_c and PS_s. The phases PS_c and PS_s correspond to phase separation of charge and ladder spin <cit.>. We note these are particularly sensitive to boundary conditions (as well as form of interactions, e.g., replacing Ising-like to density-density interactions). For simplicity we only comment on open chains with a double even number of sites, so each sub-lattice is at exact half-filling.

Ferromagnetic interactions favor clustering, giving rise to the PS_c phase at large enough V<0. The two degenerate classical ground states of the Ising-like interaction take the domain-wall form 

    |ϕ_1⟩=| ∙ ∙ ∙ ∙ ∙ ∙ ∘ ∘ ∘ ∘ ∘ ∘ ⟩,    |ϕ_2⟩=| ∘ ∘ ∘ ∘ ∘ ∘ ∙ ∙ ∙ ∙ ∙ ∙ ⟩,

breaking Z_2 mirror reflection across the center link of the chain. The gapped spectrum is composed from the creation of additional domain-walls, obtained by either the full displacement of the cluster or its separation in smaller pieces. The addition of the couplings J and W do not lift the twofold degeneracy of the PS_c ground state, but give kinetic energy to domain-wall excitations renormalizing energy gaps. 

On the other hand, strong antiferromagnetic interactions lead to the so-called PS_s phase, which can be thought of as a Néel state with a domain-wall excitation stuck in the center of the chain <cit.>. As in the ferromagnetic case, the antiferromagnetic Ising-like interaction has two classical ground states:

    |ψ_1⟩=| ∘ ∙ ∘ ∙ ∘ ∙ ∙ ∘ ∙ ∘ ∙ ∘ ⟩,    |ψ_2⟩=| ∙ ∘ ∙ ∘ ∙ ∘ ∘ ∙ ∘ ∙ ∘ ∙ ⟩.

However, while the addition of J is still innocuous, a nonzero W lifts the ground state degeneracy. The favored configuration can be deduced from  the sign of W and the nature of the pair excitation in the center of the state. For positive W the state containing a particle dimer |ψ_1⟩ is favored, while |ψ_2⟩ is selected for negative W. 






 §.§ Duality and exotic dipole constraint


The behavior of the dipole TLL state is reminiscent of the so-called fractonic liquids that have been studied recently in literature <cit.>. To clarify this link, we inspect more closely the point W=J where the dipole-type operator D becomes an exact symmetry of the lattice model.

The operator D has been discussed before in the context of constrained quantum dynamics in one dimension <cit.>. In particular, Ref.  <cit.> introduced D as an ingredient to restrain spinless fermions to move in pairs and produce fractonic dynamics in one-dimensional polaronic systems. The hard-core boson ladder Eq. (<ref>) has a similar behavior when W=J. As illustrated in Fig. <ref>, bosons in one leg only move when assisted by a boson partner in the neighboring chain, binding them into a two-site molecule.



The nonlocal character of D makes it hard to recognize its meaning. We thus move to a dual picture, by performing a Kramers-Wannier-like transformation, defined as

    τ_i^x=(-1)^i∏_j≤ iσ_j^z,    τ_i^z=(-1)^iσ_i^xσ_i+1^x,

where we introduce oscillatory factors for convenience. Then, by replacing n_i=1/2(1+σ_i^z) into the formula for ñ_i in Eq. (<ref>), we learn ñ_i is given by the difference ñ_i=1/2(τ^x_i-1-τ^x_i). This implies that the dipole operator D becomes the dual magnetization along the x axis: 

    D=1/2∑_iτ^x_i,

where we assume an infinite system and drop boundary terms. Translating the hardcore boson model in Eq. (<ref>) as well, we arrive at

    H_KW=∑_i[Wτ_i^yτ_i+1^y+Jτ_i^zτ_i+1^z+τ_i-1^x(Jτ_i^yτ_i+1^y+Wτ_i^zτ_i+1^z)τ_i+2^x+Vτ_i-1^xτ_i+1^x].

The model H_KW has a quite intriguing form. First, we readily recognize it commutes with D when W=J, as H_KW becomes manifestly invariant under rotations along the x axis. The PXP-type constraint is perfectly implemented at the point W=J, where it becomes the folded XXZ model studied by Zadnik and Fagotti <cit.>, who showed the existence of exponentially many jammed states. The W=J model has also been considered by Yang et al. <cit.>, who demonstrated that  the Hamiltonian features Hilbert-space fragmentation and unusal thermalization properties even for nonzero V, away from the integrable point. 

For W≠ J the constraint is no longer perfectly implemented and the Hamiltonian H_KW loses its invariance under U(1) rotations along the x-axis. The model still features a pair of U(1) conservation laws, inherited from the N and M numbers, given by the numbers of ferromagnetic and antiferromagnetic domain walls ∑_iτ_i^xτ_i+1^x and ∑_i(-1)^iτ_i^xτ_i+1^x, which are conserved for all W. We note that Eq. (<ref>) highlights the existence of a strong-weak W coupling duality around W=J, implementing the exchange of τ^y↔τ^z terms in the Hamiltonian.




§ MICROSCOPIC REALIZATIONS OF CONSTRAINED DYNAMICS


In this section we discuss potential realizations of our target Bariev-like model in atomic arrays. We use perturbation theory to examine two scenarios, one with Rydbergs in a linear chain and other where cold atoms move through the sites of an optical lattice. In both cases the central idea is the use of a strong potential bias to split the system into two sublattices, so that the slow, non-equilibrium dynamics preserves the relative number of excitations in each sublattice. Before continuing, let us note that some of the parameter regimes we are interested in can also be achieved following earlier proposals, in particular, Refs. <cit.>. 




 §.§ Rydberg-atom chain


We first illustrate a proposal utilizing Rydberg atoms trapped into optical potentials in the frozen regime, where atomic motion can be neglected <cit.>. We assume atoms, prepared in two different Rydberg states, e.g., |∘⟩=|nS⟩ and |∙⟩=|nP⟩, are placed in the sites of a linear chain. Dipolar coupling between Rydbergs produce a flip-flop (dipolar) exchange interaction t_ij between pairs of atoms that decays approximately as t_ij=t_|i-j|=t/|i-j|^3. We also assume the two atomic states are coupled by an external (microwave) drive. The Rydberg model Hamiltonian then reads

    H_Ryd=Ω/2∑_iσ_i^x+δ/2∑_i(-1)^iσ_i^z+∑_i<jt_ij(σ_i^+ σ_j^-+σ_i^- σ_j^+),

where Ω is the Rabi frequency, and δ is the staggered detuning corresponding to the drive. The staggering can be realized, e.g., by locally changing the Stark shift generated by the optical potential. To engineer the model in Eq. (<ref>), we work in the regime where Ω→0 and δ≫ t. We take the zero Rabi frequency limit Ω→0 so the total number of excitations is preserved, while the large δ limit allow us to freeze the magnetization M and study hopping processes perturbatively. 



  
Perturbative treatment.— We separate hopping terms in two groups. The first group does not change the number M, and comprises the hopping processes within the same sublattice. These tunneling amplitudes are hence not quenched by the detuning bias, popping out directly into the projected effective Hamiltonian. The second group on the other hand includes tunneling processes where an excitation moves from one sublattice to the other. These processes are  associated with a change Δ M=±1, and their leading contribution comes from second-order perturbation theory. Finally, given the rapid decay of the tunneling amplitudes, we truncate long-range hoppings beyond second-neighbors and make use of a Schrieffer-Wolff transformation (see Appendix <ref>) to find

    H_Ryd,eff=J∑_i(σ_i^+ σ_i+2^-+)-W∑_i(-1)^i(σ_i^+σ_i+1^zσ_i+2^-+),

where J=t/8, and W=t^2/2δ. The effective model looks much like the Bariev chain <cit.>, but the correlated exchange term acquires a staggering factor. The oscillatory phase is innocuous to our goal and favors pairing in the antisymmetric channel, giving rise to a dipole TLL made of particle-hole molecules. In fact the staggering phase in W can be easily removed by performing a particle-hole transformation in just one of the legs, say

    C_1 :   σ_2l+1^+↔σ_2l+1^-,    σ_2l+1^z→-σ_2l+1^z.

The resulting model, although with dominant antiferromagnetic correlations, is thus equivalent to Eq. (<ref>) exhibiting a dipole TLL phase.

The Hamiltonian in Eq. (<ref>) can only be taken as the dominant contribution in the more general case once longer-range terms are included. We however anticipate that those perturbations should not be harmful to the dipole TLL liquid. For instance, two perturbations that could arise are

    δ H_Ryd,eff≈ t_4∑_iσ_i^+σ_i+4^-+w_4∑_i(-1)^iσ_i^+(σ_i+1^z+σ_i+3^z)σ_i+4^-++⋯,

where the estimated size of these couplings are t_4=t/64 and w_4∼ t_1t_3/δ=t^2/27δ. These two terms break the integrability of the Bariev model, but are not enough to drive us away from the dipole TLL phase (as can be seen, at weak coupling, by analyzing their bosonized form). We thus argue Rydbergs are a promising platform to observe and study the dipole TLL state in quasi-adiabatic state preparations.




 §.§ Cold atoms with laser driven hopping


We now consider an alternative implementation, based on ultracold atoms  trapped in the sites of an optical lattice <cit.>. We assume atoms are allowed to hop among first and second neighbors sites of a zigzag chain, and repel whenever two or more of them occupy the same site. The microscopic Hamiltonian then takes the form of a Bose-Hubbard (BH) model, with an additional chemical potential bias μ between even and odd sublattices:

    H_BH=-t∑_i(a_i^† a^†_i+1+)-t'∑_i(a_i^† a^†_i+2+)+U/2∑_in_i^a(n_i^a-1)+μ/2∑_i(-1)^in_i^a,

where a_i removes an atom sitting at i, t and t' are the hopping amplitudes, and U is the onsite Coulomb repulsion. The number of atoms is denoted as n_i^a=a_i^† a_i to discern it from the hard-core boson number introduced before. 

We utilize here laser-assisted tunneling <cit.> to induce hopping of atoms with a site-dependent phase. In particular, we consider a nearest-neighbor hopping t→ t^φ_i,i+1 with staggered flux pattern, depicted in Fig. <ref>. We choose the phase so its sole effect is to flip the sign of t every two sites:

    H_t→ H_t=-t∑_l(-1)^l(a_2l-1^† a^†_2l+a_2l^† a^†_2l+1+),

where the l sum is taken over half the system size. This phase dressing helps us to cancel out the oscillatory factor that arises in the perturbation theory, as we argue below.

 





  
Perturbative treatment.— The perturbative treatment is reminiscent of that in the previous subsection. We consider the large-μ limit to effectively enforce spin symmetry, and assume U is strong enough so that every site contains at most one boson. It is worth noting that such strong couplings are not detrimental to a treatment in the single-band Hubbard regime, as the dynamics of single particles along the wires is not affected by those (in fact, similar regimes have been investigated in Ref. <cit.>). 

We then take into account hopping processes perturbatively. As a further simplification, we consider the regime t'≪ t, so the second-neighbor hopping only contributes at first-order in perturbation theory (this assumption is not necessary, but makes computations easier to interpret). Taking into account second-order virtual processes generated by the first neighbor hopping, as seen in Fig. <ref>, we find the effective Hamiltonian governing the slow dynamics of the system takes the form

    H_BH,eff=-∑_lα(J_ασ_2l-2+α^+σ_2l+α^-+W_ασ_2l-2+α^+σ_2l-1+α^zσ_2l+α^-+)+V∑_iσ_i^zσ_i+1^z,

where we use α=1,2 to denote odd and even sub-lattices. The effective Hamiltonian features the desired U(1)_c×U(1)_s symmetry, but lacks leg permutation symmetry. From the Schrieffer-Wolff transformation, we estimate the couplings to be

    J_1,2=t'±t^2/U±μ,    
    W_1,2=t^2/μ±t^2/U±μ,    
    V=-t^2/2(1/U+μ+1/U-μ),

where we take the upper sign for α=1 and the lower sign otherwise. There are two different ways to restore Z_2 leg symmetry.[We note the lack of this Z_2 symmetry is not necessarily incompatible with the dipole conservation. As a simple check one may consider the dipole-preserving, but still leg-anisotropic case where W_1=J_1=J+η and W_2=J_2=J-η, which realizes an alternating hopping pattern for dimers, in a similar fashion to the so called Su-Schrieffer-Heeger model <cit.>.] The first involves adjusting the relative filling in the legs, changing the free Fermi velocity so that they match at some nonzero magnetization M≠0. Another, less fine-tuned possibility, amounts to considering an extra separation of energy scales in the lattice parameters. Either considering U≫μ or U≪μ will do the job. In the first case, for instance, by taking the limit U≪μ, with t'∼ t^2/μ, we can approximate the couplings to J_1,2→ J=t'+t^2/μ and W_1,2→ W=2t^2/μ, while V goes to V→ t^2U/μ^2, and is then assumed to be much smaller than J and W. In this limit, we thus arrive at precisely the model Hamiltonian in Eq. (<ref>).




§ EFFECTIVE FIELD THEORY APPROACH


In this section we present a long-distance, low-energy description for the hardcore boson ladder model. We use bosonization to analyze the effects of interactions starting from the limit of weakly coupled XY chains. As usual for TLLs, we can explore the predictions of the effective theory beyond the perturbative regime by treating the velocities and Luttinger parameters as phenomenological parameters. We shall see that this approach captures the transitions to the  dipole TLL phase as well as to the classically ordered phases, providing a field theory framework for the entire phase diagram sketched in Fig. <ref>. In addition, we examine the representation of the dipole moment operator in the low-energy theory. We argue that the violation of the dipole symmetry is associated with the creation of gapped spin excitations that behave as mobile defects interacting with the gapless charge modes. 




 §.§ Tomonaga-Luttinger liquid theory


Let us write  the hardcore boson ladder model in the limit  of decoupled legs, obtained by setting W=V=0 in Eq. (<ref>). It is convenient to introduce a leg index α=1,2 corresponding to odd and even sites, respectively, and denote the spin operators by σ_α^±(l)≡σ^±_2l-2+α, with l∈ℤ. In this notation, the Hamiltonian for decoupled legs reads

    H_0=-J∑_lα[σ^+_α(l) σ^-_α(l+1)+].

We can then bosonize the low-energy excitations of each XY chain separately <cit.>. The effective Hamiltonian is that of two independent free bosons:

    H_0≈∑_αv_0/2∫ x[(∂_xθ_α)^2+(∂_xϕ_α)^2],

where  v_0=2J is the  velocity of the bosonic modes in each leg  and the bosonic fields obey the commutation relation [θ_α'(x'),∂_xϕ_α(x)]=δ_αα'δ(x-x'). The   fields ϕ_α are  associated with fluctuations of the hard-core boson occupation number by

    δ n_α(l)=1/2σ_α^z(l)≈-1/√(π)∂_xϕ_α(x)+ (-1)^x ×sin[√(4π)ϕ_α(x)].

The staggered part has a nonuniversal prefactor and oscillates with momentum π=π(1+⟨σ_i^z⟩) for a pair of half-filled chains, described by N/L→1/2 and M/L→0. Note that if we allow the numbers N and M to change, the value of the momentum is not fixed and may be even different in each sub-lattice. The continuum expansion of the spin raising and lowering operators reads 

    σ_α^±(x)∝^±√(π)θ_α(x){1+ (-1)^x×cos[√(4π)ϕ_α(x)]}.

We note that the lattice model is invariant under discrete translations i→ i+2,  corresponding to a rigid displacement of one site on each leg. In the low-energy theory, this lattice translation  amounts to x↦ x+1 translations, under which the bosonic fields transform according to

    L:    ϕ_α↦ϕ_α+√(π)/2,    θ_α↦θ_α+√(π).

In addition, the zigzag chain is invariant under a reflection about a site, which acts as site parity for the leg that contains that site but link parity for the other leg. For instance, the reflection about an odd site acts in the low-energy theory as:

    P:     x↦ -x,  ϕ_1↦-ϕ_1,  ϕ_2↦√(π)/2-ϕ_2,   θ_1↦θ_1,  θ_2↦θ_2+√(π).

We can also define  time reversal as the anti-unitary transformation that takes σ_j↦ -σ_j. In the low-energy theory:

    T:    ↦ -,  ϕ_α↦ -ϕ_α,  θ_α↦θ_α.
 

Next, we add interchain interactions perturbatively. We have H=H_0+H_W+H_V, where

    H_W=   -W∑_l[σ_1^+(l)σ_2^z(l)σ^-_1(l+1)+σ_2^+(l)σ_1^z(l+1)σ^-_2(l+1)+],
    
    H_V=    V∑_lσ^z_2(l)[σ^z_1(l)+σ^z_1(l+1)].

The three-spin  interaction H_W preserves ℒ and 𝒫 symmetries, but breaks 𝒯. Using  the continuum expansion of the spin operators, we can combine oscillatory terms from both legs to  produce the  operator δ H_W≈ -2W/π^2∫ x sin[√(4π)(ϕ_1-ϕ_2)] as the most relevant contribution. The  Ising-like interchain interaction H_V   contributes with a marginal operator that couples the uniform magnetization  in the two legs:  
δ H_V≈8V/π∫ x ∂_xϕ_1∂_xϕ_2.   We define the charge and spin fields as the linear combinations

    ϕ_c,s=ϕ_1±ϕ_2/√(2),    θ_c,s=θ_1±θ_2/√(2).

Adding the leading perturbations to  Eq. (<ref>), we obtain a spin-charge-separated  Hamiltonian in the form H=H_c+H_s, where

    H_c   =v_c/2∫ x[K_c(∂_xθ_c)^2+1/K_c(∂_xϕ_c)^2],
    
    H_s   =v_s/2∫ x[K_s(∂_xθ_s)^2+1/K_s(∂_xϕ_s)^2]-λ/2π^2∫ x sin(√(8π)ϕ_s).

At weak coupling, the spin and charge velocities are given  by  v_c,s≈2J(1±4V/π J). The  Luttinger parameters K_c and K_s encode the interactions in the charge and spin sectors, respectively.  To first order in the interleg interaction, we find K_c,s≈1∓4V/π. The sine potential in the spin sector, with   coupling constant λ≈4W, has scaling dimension 2K_s. Note that this operator  is odd under time reversal, as expected for the three-spin operator in H_W.  Importantly, the low-energy Hamiltonian in Eq. (<ref>) remains valid beyond the  regime of small W and V because the sine potential is the leading perturbation compatible with ℒ, 𝒫 and U(1)_c×U(1)_s symmetry. 

The 2TLL phase corresponds to the regime in which both charge and spin sectors in Eq. (<ref>) remain gapless. This can happen with the help of  a repulsive  interaction, V>0, which disfavors the formation of pairs by making K_s>1 and rendering the sine potential irrelevant. We can write the uniform  part of the σ_i^+  operator  in terms of charge and spin fields as 

    σ^+_1,2(l)  ∝^√(π/2)[θ_c(x)±θ_s(x)].

In the 2TLL phase, single-particle correlators display a power-law decay, given by

    σ_i^+σ_i+2r^-∝ r^-(K_c+K_s)/4K_cK_s,

where |0⟩ stands for  the ground state. Note that in Eq. (<ref>) we must take two points that belong to the same chain, otherwise the  correlator vanishes identically. This is a consequence of the  U(1)_c×U(1)_s global symmetry of the ladder and  remains true even if we move away from the weak-coupling limit. 


The transition to the dipole TLL phase is driven by the flow of the λ perturbation  to strong coupling.  For V=0, the coupling is marginally relevant, and, as a result, the spin sector undergoes a BKT-type transition. In the strong-coupling limit, we minimize the potential energy by pinning  the scalar field ϕ_s to  one of its minima:

    √(8π)ϕ_s→π/2+2πZ.

The dipole TLL phase then corresponds to a gapless charge sector  and  a gapped spin sector.  Note that attractive V favors pairing, facilitating the transition to the dipole TLL state. Once the spin sector is gapped out, the single-particle propagator develops an exponential decay as follows:

    σ_i^+σ_i+2r^-∝^-r/ξ/r^1/4K_c.

The correlation length ξ is inversely proportional to the mass gap in the spin sector. In the case of a BKT transition at V=0, the gap is exponentially small at weak coupling, with ξ_BKT^-1∼exp(-/λ) <cit.>. Power-law correlations in the dipole TLL phase are only found by pairing bosons in different sublattices, e.g.,

    σ_i^+σ_i+1^+σ_i+r^-σ_i+r+1^-∝ r^-1/K_c.

Note that the distance r is not restricted to even multiples of the lattice spacing, cf. Eq. (<ref>), since the two-particle operator σ_i^+σ_i+1^+ creates one excitation in each leg, and the correlator always respects the U(1)_c×U(1)_s symmetry.

Instability towards phase separation is deduced from the vanishing of either charge or spin velocities  in the large   V limit. Assuming a monotonic behavior and estimating the interaction dependence from  the weak-coupling expressions for v_c and v_s, we predict V^⋆/J≃±0.78, with positive and negative values corresponding to the transitions towards PS_s and PS_c, respectively. Given that W does not enter into the renormalization of Luttinger parameters at weak-coupling, we expect the critical value V^⋆ to  be roughly  independent of W. We are thus led to the phase diagram shown  in Fig. <ref>.




 §.§ Emergent dipole symmetry


Let us now use our field theory formulation to examine the dipole operator in  Eq. (<ref>). Our goal here is to find its long-distance representation in order to verify it commutes with the low-energy Hamiltonian of the dipole TLL state, and thus represents an emergent symmetry of this phase.

We start from the dual representation in Eq. (<ref>), where the dipole operator is given by the sum of Jordan-Wigner strings. We then expand each τ^x as the product of σ^z strings in each leg. For an odd-site operator τ_1^x(l)=τ_2l-1^x, we get τ_1^x(l)=-S_1(l)S_2(l), where S_α(l) is defined as

    S_1(l)=∏_m<lσ^z_2m,    S_2(l)=∏_m<lσ^z_2m+1.

Likewise, the even-site operator τ_2^x(l)=τ_2l^x is given by τ_2^x(l)=S_1(l)S_2(l+1). We can then rewrite Eq. (<ref>) as 

    D=-1/2∑_l[S_1(l)-S_1(l+1)]S_2(l),

where we sum over half the total number of sites. 

We can now bosonize the dipole operator using the standard expression for the string operators in terms of the bosonic fields in each sublattice. Naive bosonization   yields the complex form S_α(l)≈^π/2x+√(π)ϕ_α(x). To obtain a manifestly Hermitian operator, we symmetrize the string, leading to 

    S_α(l)→ S_α,reg(l)≈cos[π2x+√(π)ϕ_α(x)].

The product of strings at the same site gives S_1(l)S_2(l)≈1/2cos[√(2π)ϕ_s(x)], where we drop oscillatory terms and higher-order corrections. The second term in Eq. (<ref>) is quite similar, but involves a derivative because the fields are at different points:

    S_1(l+1)S_2(l)≈-1/2sin[√(2π)ϕ_s(x)]-√(2π)/4cos[√(2π)ϕ_s(x)]∂_xϕ_c(x)+⋯

Taking both contributions into account, we arrive at the long-distance representation of the dipole operator:

    D∝∫ x[sin(√(2π)ϕ_s+π4)+√(π)/2cos(√(2π)ϕ_s)∂_xϕ_c]+⋯,

where we omit the prefactor and the ellipsis contains higher-order corrections.

From the continuum version of D, we readily learn that the dipole operator can only be a symmetry if ϕ_s condenses. This is clearly not the case in the 2TLL phase, where both ϕ_c and ϕ_s fluctuate and D does not commute with the low-energy Hamiltonian in Eq. (<ref>). In the dipole TLL phase, however, this condition is met as the spin field is pinned to √(8π)ϕ_s→π/2 according to the strong-coupling flow of λ. Plugging this condition into Eq. (<ref>) gives 

    D≈1/√(2π)∫ x ∂_xϕ_c+⋯,

where we drop an unimportant additive constant, and fix the proportionality constant. In this form, the dipole operator is proportional to the total number of particles (half of it if we assume all particles are bound in pairs), and certainly commutes with the low-energy Hamiltonian.

This  analysis tell us that the dipole operator describes an emergent symmetry of the dipole TLL phase. This symmetry is valid at low energies,  below the spin gap, where all particles are confined into pairs but gapless excitations are still possible in the form of  collective  modes in the charge sector. This finding     supports   the dimer picture of the dipole TLL phase. While our bosonization approach started in the weak-coupling limit where the spin gap is small, we expect this picture to become more accurate as we increase W towards the special point W=J, where the dipole symmetry becomes exact and the pairs are strictly confined. 

Single-particle excitations violate the pinning condition on ϕ_s because they carry both charge and spin quantum numbers. In the low-energy theory, the operator  σ^+_i in Eq. (<ref>) creates a kink in the spin field at the position x, shifting ϕ_s as

    √(8π)ϕ_s(y)→√(8π)ϕ_s(y)+2πΘ_H(y-x),

where Θ_H(x) is the Heaviside step function. This means that the kink interpolates between two  ground states of the sine potential in Eq. (<ref>). On the other hand, the kink changes the sign of the dipole operator in Eq. (<ref>) across the position where the single particle is created.  This effect is consistent with the original definition of the dipole operator in Eq. (<ref>), since inserting a single particle changes the sign of the string in ñ_i. Thus, we can view a local single-particle excitation as a defect in the spin field configuration. Far from this defect, we could still pin ϕ_s to a local minimum, but the dipole symmetry is spoiled if the defect is allowed to move through the lattice. In the following we will construct an effective mobile impurity model to describe the dynamics of this defect in the dipole TLL phase. 




 §.§ Mobile impurity model


According to Eq. (<ref>),  at low energies the spin sector of the dipole TLL is described by a sine-Gordon-type model. At weak coupling, i.e., for small spin gap Δ_s≪ J, the elementary excitations are kinks or anti-kinks with relativistic dispersion E_s(k)=√(v_s^2k^2+Δ_s^2) <cit.>. As we increase the Bariev interaction strength W, the spin gap increases and the dispersion relation deviates from the relativistic dispersion. We are now interested in exploring the vicinity of the special point W=J, where the dipole symmetry imposes that single-particle excitations, which are charged under the U(1)_s symmetry, cannot move by themselves. To describe this regime, we restrict the excitation spectrum to allow at most one spin excitation.  This type of problem can  be tackled using  effective mobile impurity models, in which the finite-energy excitation is treated as a distinguishable particle that interacts with the gapless modes of the TLL  <cit.>.

We start by approximating the spin dispersion near its minimum by 
    E_s(k)≈Δ_s+k^2/2m,

where m is the effective mass. For  W=J, we expect m→∞, corresponding to localized spin excitations due to the  exact dipole symmetry. We then treat the single spin excitation as an impurity mode, writing

    σ_i^+∝^√(π/2)θ_c(x)d^†_s(x),

where d^†_s(x) is charge neutral but carries spin quantum number Δ M=+1/2 (-1/2) if i is an odd (even) site. In this representation, the ground state |0⟩=|0⟩_c⊗|0⟩_s is a vacuum of the bosonic charge modes and of the d_s particle. The effective mobile impurity model that includes the spin excitation has the form

    H=H_c+∫ x[ d_s^†(Δ_s-1/2m∂_x^2)d_s-g√(2/π)∂_xϕ_cd^†_sd^†_s]+…,

where H_c is the bosonic Hamiltonian in Eq. (<ref>) and we omit irrelevant interaction terms. Phenomenologically, the coupling g between the impurity   and the charge density can be interpreted as follows. Suppose we perturb the hardcore boson ladder in the  dipole TLL phase by adding a uniform field δ H_Z=-h/2∑_iσ_i^z. In the low-energy theory, this perturbation becomes δ H_Z ≈ h√(2/π)∫ x ∂_xϕ_c. In a grand canonical ensemble formulation, the change in the particle density can be absorbed by shifting the charge boson by ϕ_c(x)→ϕ_c(x)-hK_c/v_c√(2/π) x. Implementing this shift  generates a renormalization of the spin gap in the mobile impurity model in Eq. (<ref>). As a result, we obtain the relation  

    g=∂Δ_s/∂ρ̅_c,

where ρ̅_c=1/2⟨σ^z_i+σ^z_i+i⟩ is the average charge density in the ground state and we use κ=∂ρ̅_c/∂ h=2K_c/π v_c for the charge compressibility of the TLL. Thus,  the coupling constant g depends on how the spin gap changes when we vary the total number of particles. This  coupling is allowed when we have a finite-energy excitation in  either spin or charge sectors <cit.>. 

We can eliminate the interaction between the impurity and the gapless modes using a unitary transformation. We define

    U=exp[-i√(2/π)gK_c/v_c∫ x θ_c d^†_sd^†_s]

and the transformed fields

    d̃_s   =U^† d_sU=d_s^-√(2/π)gK_c/v_cθ_c,
    ∂_xϕ̃_c   =U^†∂_xϕ_cU=∂_xϕ_c-√(2/π)gK_c/v_cd^†_sd^†_s.

In terms of the new fields, the Hamiltonian becomes 

    H= ∫ x[v_cK_c/2 (∂_xθ̃_c)^2+v_c/2K_c(∂_xϕ̃_c)^2+d̃_s^†(Δ_s-1/2m∂_x^2)d̃_s]+…,

where again we drop irrelevant interactions. Importantly, the dressed impurity mode d̃_s   carries a charge proportional to the coupling g because the  charge density operator is given by

    ρ_c=-√(2/π)∂_xϕ_c=-√(2/π)∂_x ϕ̃_c -√(2/π)g K_c/v_cd̃^†_sd̃^†_s .

Since this impurity mode is non-interacting, we obtain the free propagator

    G_d(x,τ)   =d̃^†_s(x,τ )d̃^†_s(0,0)=∫_-∞^∞ k/2π^ kx-Δ_sτ- k^2τ/2m-a^2k^2/2
       =1/√(2π)(a^2+τ/m)^-1/2exp(-x^2/2a^2+2τ/m),

where a^-1 is a momentum cutoff, with a of the order of the lattice spacing. Note that the propagator is a scaling function of τ/m.  
    
   
To probe the time evolution of the defect, let us consider the time-dependent variation in the local density:

    C(j,τ)=⟨Ω|σ_0^-n_j(τ)σ_0^+|Ω⟩-⟨Ω|n_j|Ω⟩,

where |Ω⟩ denotes the ground state of the hardcore boson ladder. In the mobile impurity model, this quantity becomes

    C(x,τ)≈ C_c(x,τ)+C_s(x,τ).
 
The first term involves the charge density. Defining the vertex operator V_c=^-√(π/2)(1-gκ)θ̃_c, we obtain  C_c(x,τ)∝V_c(0)∂_xϕ_c(x,τ)V_c^†(0)_c. This contribution spreads ballistically  with the charge velocity v_c and decays algebraically at long times. The second contribution involves the impurity propagator:

    C_s(x,τ)∝d̃_s(0)d̃_s^†(x,τ)d̃_s(x,τ)d̃_s^†(0)_s =G_d(x,τ)G_d(-x,-τ).

Using Eq. (<ref>) and setting x=0, we find that the density measured at the same position where the defect is created decays with time as C(0,τ)∝ m/τ for any finite mass. In the limit of an immobile defect,  m→∞, C(0,τ) converges to a finite non-universal value for τ→∞, implying that some charge remains at x=0 while the other fraction propagates away with velocity v_c. 

We can also use the mobile impurity model to study the dynamics of the dipole moment within the low-energy theory. To reproduce the properties of the dipole operator in Eqs. (<ref>) and (<ref>), we propose the following expression in the continuum:

    D≈∫_-∞^∞ x xd_s^†(x)d_s(x)+1/√(2π)∫_-∞^∞ x Ŝ(x)∂_xϕ_c(x),

where we define Ŝ(x)=1-2∫_-∞^x x' d_s^†(x')d^†_s(x').  The first term simply accounts for the dipole moment of the single defect. The second term represents the contribution from   bound pairs, where Ŝ(x) implements the  sign change of the string when we cross the position of the defect. In the defect-free sector, the impurity density vanishes identically, and the dipole   operator reduces to the total number of pairs, a conserved quantity within the  low-energy theory. By contrast,  if we consider the initial state   |Ψ(τ=0)⟩=σ_j^+|Ω⟩, we expect the variance of the dipole operator to increase with time as the defect moves  through the system.  We can capture this effect   by calculating the variance due  to the first term in Eq. (<ref>).  We obtain

    ⟨Δ D^2(τ)⟩= ⟨Ψ(τ)|D^2|Ψ(τ)⟩-⟨Ψ(τ)|D|Ψ(τ)⟩^2≈∫ x x^2|G_d(x,τ)|^2,

which can be interpreted as the mean squared displacement of the defect. We find

    ⟨Δ D^2(τ)⟩≈ a^2+(τ/ma)^2.

The  variance is finite  at τ=0 because the initial state is not an eigenstate of the dipole operator. For any finite mass,  the variance is  a function of τ/m and  increases quadratically with time. For m→∞, the variance remains approximately constant, in agreement with the picture of a localized defect enforced by the exact  dipole moment conservation law. 




§ NUMERICAL SIMULATIONS



In this section we present our numerical findings, obtained from a combination of ED and matrix product states (MPS) based methods, such as TDVP and TEBD. We begin with an examination of the phase diagram from the viewpoint of the emergent dipole conservation. We use ED results to provide insight into the conservation of the dipole moment in both ground state and low-energy excited states. We then move to the benchmark of the quasi-adiabatic state preparation. We focus on the more challenging preparation with cold atoms, simulating the dynamical preparation with TDVP. We close our numerical survey by studying the isolated defect dynamics close to the dipole TLL ground state. 




 §.§ Phase diagram and emergent dipole symmetry


We start with a phase diagram characterization of the dipole symmetry. Our goal here is not to precisely determine transition points, but rather to uncover the emergent status of the dipole symmetry in the dipole TLL state.

We first check how the ground state variance of the dipole operator, ⟨Δ D^2⟩=⟨ D^2⟩-⟨ D⟩^2, behaves as we navigate across different portions of the phase diagram. In Fig. <ref>(a) we plot the results obtained for a chain with L=28 sites and open boundary conditions. We observe a great similarity with the phase diagram sketched in Fig. <ref>. Near the origin, where we find the 2TLL state, we see the ground state is far from being dipole symmetric as flagged by the higher variance. Moving either up or down we eventually cross to vanishing dipole variance regions, which we associate to the classically ordered states. Coincidentally, we find the transitions take place around the values V/J≈±0.75, quite close to the ones predicted from the weak-coupling bosonization. Finally, when we leave the 2TLL state moving in the direction of increasing W, we appear to cross a smoother region after which the dipole variance also approaches zero. This would correspond to the BKT-type transition into the dipole TLL state, supporting the picture of an emergent dipole-conserving liquid groundstate. 



The BKT nature of the transition to the dipole TLL phase makes it difficult to use finite-size scaling techniques effectively. This limitation leads to an overestimation of the 2TLL phase in finite-size numerics, as  can be seen in Fig. <ref>(a). This becomes clear when we compute the fidelity susceptibility χ_F, conventionally used to detect critical points via finite-size scaling techniques <cit.> and defined as

    χ_F(η)= lim_δη→ 02/L1-F(η,δη)/(δη)^2,

where F(η,δη)=|⟨ψ(η)||ψ(η+δη)⟩| is the fidelity and η is a parameter of the Hamiltonian.  We show in Fig. <ref>(b) the ground state fidelity susceptibility as a function of W/J for a few  different sizes L. We find  that the transition slowly moves towards weak coupling as we increase the size L, indicating that the 2TLL may actually be much smaller in the thermodynamic limit.



Next, we address the behavior of the dipole operator for excited states. Figure <ref> shows the matrix elements |D_αβ|=|⟨α|D|β⟩| of the dipole operator, computed in the basis of eigenstates for W/J=0, 0.60 and 0.90. Note that we plot absolute values, so we can focus on the strength of nonzero terms. We observe off-diagonal terms are gradually supressed as we move from W/J=0 to W/J=0.90, in support to the emergent status of the symmetry. In particular, for W/J=0.90, we can see a low-energy block, whose nonzero elements are concentrated on the diagonal.




 §.§ State preparation


Let us now consider the state preparation protocol. We start from the simple initial state |Néel×Néel⟩=|01100110…⟩, and evolve it according to the following time-dependent Hamiltonian:

    H_exp(τ)=H_BH(τ)+h(τ)∑_l(-1)^l(n_2l^a+n_2l+1^a),

where H_BH(τ) is a time-dependent variant of the Bose-Hubbard model shown in Eq. (<ref>). At the initial time, τ=0, coupling parameters are chosen so the initial state is the actual ground state of the full Hamiltonian H_exp(0) in the symmetry sector where both sublattices are half-filled. This means, initially, the Bose-Hubbard model only includes the potential terms U=U_0 and μ=μ_0, while the hopping elements are set to zero, t_0=t'_0=0. Note that we also add an extra time-dependent staggering field h(τ), whose initial value h=h_0 favors the |Néel×Néel⟩ configuration.

The preparation then proceeds by slowly tuning the coupling parameters in H_exp(τ). We vary three parameters t, t', and h, while keeping the potentials U and μ static along the evolution. The hopping parameters t and t' are increased up to the terminal values t_f=1 and t'_f=0.1, while h is decreased all the way down to h_f=0. The parameter sweep is shown in Fig. <ref>(a) as a function of τ/T, where T denotes the duration of the sweep. We use ED to verify how the many-body spectrum evolves along our parameter flow. In Fig. <ref>(b) we plot the spectrum evolution of the Hamiltonian (<ref>) for a modest chain with L=16 sites. We observe the low-energy manifold remains separated, below the rest of the spectrum, during the whole evolution. Note that we choose the  parameters to be as close as possible to the strictly confined point of the Bariev-like model, so we perform the ED in the limit of hard-core bosons (U→∞) with potential bias set to μ=10. In this parameter regime, we should ideally end up in the effective model of Eq. (<ref>), with parameters J_α=t', W_α=t^2/μ, and V=0, as estimated from Eq. (<ref>). 



We now consider a quasi-adiabatic protocol with a finite preparation time T <cit.>. To perform the dynamical evolution of the initial state we then resort to TDVP, using a MPS representation of the boson states with maximum occupation number equal to four. We study this preparation as a function of the duration time T and the potential bias μ, fixing the on-site potential to U=50.

Figure <ref>(c) shows the behavior of the overlap between the dynamically prepared and the (DMRG obtained) target state of the Bariev model, F=|⟨Ψ_prep(T)|Ψ_target⟩|. As expected, we see that the fidelity improves with increasing T (or, equivalently, with decreasing sweep rate). We also observe that the maximal overlap is reached at intermediate μ, while small and large values of μ lead to significantly smaller overlaps with the target state. For small μ, higher-order terms in the perturbation theory become sizeable and cannot be neglected. On the other hand, for large μ, the energy gaps, which scale as t^2/μ, become small, such that a higher density of excitations are created in the quasi-adiabatic protocol. We then verify that, while the experimentally prepared state does not conserve the magnetization number M exactly, its expectation value approaches zero with increasing μ as shown in Fig. <ref>(d). 

To conclude, we compare the behavior of the single- and two-particle correlators obtained at the end of the protocol with the ones computed from the ED of the target, hardcore boson model Eq. (<ref>), with W=J and V=0. As shown in panels (e) and (f) of Fig. <ref>, these are in good agreement with the ideal results obtained from the target model. We point out, however, that the single-particle correlator exhibits greater deviations at larger distances, aligning with predictions from the Kibble-Zurek mechanism <cit.>. We have also tried the state preparation protocol in the limit U≪μ. However, we observe that the state does not appear to enter the dipole TLL phase, using experimentally realizable parameter regimes.




 §.§ Dynamics of defects


Finally, we investigate the out-of-equilibrium dynamics of single-particle excitations. The philosophy here is that, after approaching a target ground state in the quasi-adiabatic preparation, one acts locally with an operator that creates an excitation in the center of the chain letting it evolve coherently for some time. With this in mind, we however leave microscopic models behind and concentrate on the ground state and dynamics produced by the effective hardcore boson ladder.

Numerically our quench protocol goes as follows. First, we use DMRG to prepare the ground state |Ω⟩ of the Bariev-like Hamiltonian, Eq. (<ref>). We then act with σ_0^+, where j=0 represents the center site of a chain with an odd number of sites. Finally, we use a three-site gate TEBD to approximate the time evolution |Ψ(τ)⟩=^- Hτ|Ψ_0⟩, with |Ψ_0⟩=σ_0^+|Ω⟩ the prepared initial state.




We apply this recipe to examine how the defect behaves above the dipole TLL ground state. In order to get cleaner results, we choose lattice parameters so we are deep in the dipole TLL phase, close to the strictly confined point W=J. We consider values of W in the range from W/J=1 to W/J=0.92, always with a small V/J=0.1 interaction for generality. We run the DMRG for a chain with L=201 sites and quantum numbers fixed to N=(L-1)/2=100 and M=0. 
For the TEBD part, we use moderately small time steps Jδτ=0.01, and stop the evolution at Jτ=18.0, roughly when the light cone reaches the edges of the chain. The maximum truncation error is set to 10^-8 during the whole numerical experiment. Attained with a maximum bond dimension of χ=1000, the results appear well-converged for the times considered.

In Fig. <ref> we plot the time evolution of the variation in the local density for three different values of W. 
In agreement with the field theory prediction, we observe the defect exhibits two different spreading patterns as it carries both quantum numbers of charge and spin. After emitting the gapless part, which spreads ballistically, leaving a clear light cone signal, the remaining part of the defect features a substantial slowdown in the relaxation towards equilibrium. For W/J=1 the effect is  most dramatic, since the spin part of the defect  lies still at the central site where it was created. However, as we move away from the strictly confined point, we are able to observe a slow spreading of the contribution associated with  defect.

The time evolution of the density at the central site for various values of W is shown in  Fig. <ref>(a). There we can spot two time regimes. At short times we observe a rapid decay due to the formation of the wave front that spreads ballistically. After this initial decay, the density leak slows down. In particular, for W=J the density does not seem to decay significantly for the time scales observed, staying close to n_0≈0.8. This is compatible with the limit m→∞ of the mobile impurity model, where a finite amount of the density excess remains localized at infinitely long times. For W<J we start to observe a slow decay in the density. The inset of Fig. <ref>(a) we show that these curves collapse onto a single curve upon scaling time has been rescaled by (J-W)τ. The data collapse gives  the estimate m∼|W-J|^-1 for the behavior of the effective mass parameter in the mobile impurity model. Note however in the available time windows we could not directly verify the long-time behavior of 1/τ predicted by the impurity model, controlled by the limit τ≫ ma^2, with a a short-distance cutoff.

We show the time evolution of the variance of the dipole operator in Fig. <ref>(b). As expected, we observe that the dipole variance vanishes for W=J  and increases with the difference |W-J|. We use this correlator to cross compare time scales as extracted from the time evolution of the central site density. As showed in the inset of Fig. <ref>(b), the time scales here are also compatible with an effective mass for the mobile impurity model given by m∼1/|W-J|, displaying a quadratic form as predicted in Eq. (<ref>).





To complete the picture, we consider the same quench protocol but with a defect added on top of the 2TLL state. However, due to the large entanglement pattern of the c=2 state, we observe a greater difficulty to prepare well-converged ground states with DMRG. In view of that we reduce the system size and consider a modest chain of L=49 sites. We set the lattice parameters to W=0 and V/J=0.1, setting the truncation error to 10^-12, with maximum bond dimension to χ=600 during the ground state search. The rest of the numerical protocol goes the same as before and we arrive at the results shown in Fig. <ref>. 

Given that the 2TLL phase is adiabatically connected to the fixed point of decoupled chains, the numerical results confirm the natural expectation, and demonstrate the defect thermalizes quickly, spreading ballistically throughout the system. In Fig. <ref>(a) we plot the spacetime dependence of the charge variation ⟨δ n_j(τ)⟩ during the quench with respect to the unperturbed ground state. Note that for the system size considered, and small difference in velocities, it is difficult to tell apart the lightcones associated with the fractionalization of the single-particle excitation into gapless modes of charge and spin. Another contrast is provided in Fig. <ref>(b), where we plot the time evolution for the number occupation at the central site. When compared to the behavior in the dipole TLL phase, we can see the charge leak does not exhibit any sort of slowdown, relaxing at time scales of the order Jτ∼1. In appendix <ref> we include further results obtained for the pair dynamics on top the dipole TLL state.






§ OUTLOOK


We investigated a Bariev-type model for hardcore bosons and its non-equilibrium implementation in biased atomic ladders.  We considered how deviations from a strictly confined point, still leaves significant imprints in the low-energy physics. In particular we uncovered the emergence of a global non-local symmetry in the dipole TLL state, which constrains the dynamics in the ground state, binding excitations into pairs. Through the use of extensive numerical methods, we simulated and verified the effectiveness of the quasi-adiabatic preparation in the context of a Bose-Hubbard model. We further considered the out-of-equilibrium dynamics of single-particle defects created above the dipole TLL state. We showed that they exhibit a substantial  slowdown in the spreading dynamics  and compared their slow motion to that of a heavy particle whose mass diverges as we approach the special point where the dipole symmetry is exact.

We leave some open directions for future works. The continuum-limit description of single-particle defects may be improved by treating the sine-Gordon model in its entirety, which may open the possibility of better understanding the role of the emergent constraint as well as extending the theory to finite densities of such defects. Another promising direction is the exploration of a potential link to the theory of Z_2 lattice gauge theories coupled to spinless fermions <cit.>, which exhibits a Bariev-type dynamics in the strong string-tension limit. Driving inspiration from  works on the folded XXZ model <cit.>, it may also be of interest to explore the role of the emergent dipole symmetry (if there is any) in the realm of finite temperature transport  <cit.>.

Our predictions are of particular relevance to atomic physics experiments, where the protocol we describe could be implemented in a controllable way. We note the use of staggering fields has also been proposed to implement gauge symmetries <cit.> in quantum analog simulators, as an alternative to tilting potentials <cit.> which need to scale with system size. The core idea of splitting a bipartite lattice with a bias potential is quite generic, and one may envision extensions to 2D systems, where the number of excitations is separately conserved in each sublattice. These ideas might also be of relevance to trapped ion chains, where the implementation of three-body terms similar to those discussed here has recently been proposed <cit.>, or via effective dynamics similar to the Rydberg case, but at the price of introducing longer-range interactions <cit.>.
 




§ ACKNOWLEDGEMENTS


We thank Devendra Bhakuni, Michael Knap, Sergej Moroz and Lenart Zadnik for insightful discussions and pointing out key references. H.B.X. expresses his gratitude to Hosho Katsura for stimulating discussions about Bariev-type models and their cold-atom realizations. M.D. thanks Guido Pagano for comments on the manuscript, and enlightening discussions about trapped ion systems. Matrix product state simulations were performed using the ITensor package <cit.>.








  
Funding information

M. D. was partly supported by the QUANTERA DYNAMITE PCI2022-132919, by the EU-Flagship programme Pasquans2, by the PNRR MUR project PE0000023-NQSTI, and by the PRIN programme (project CoQuS). H. B. X. and M. D. were partly supported by the MIUR Programme FARE (MEPH). P. S. T. acknowledges support from the Simons Foundation through Award 284558FY19 to the ICTP.  R. G. P. acknowledges support by
the Conselho Nacional de Desenvolvimento Científico e
Tecnológico  and by a grant from the Simons
Foundation (Grant No. 1023171).


equationsection




§ DETAILS ON THE SCHRIEFFER-WOLFF TRANSFORMATION


Here we offer further details on the Schrieffer-Wolff transformation <cit.> we employ to derive the effective Hamiltonians for the Rydberg and cold atom platforms.




 §.§ Rydbergs


The Rydberg Hamiltonian has the form H_Ryd=2δ M+H_t, where M=1/4∑_i(-1)^iσ_i^z, and H_t=∑_ijt_ijσ_i^+σ_j^- describes long-range tunneling. Given the amplitudes decay quite fast t_ij=t_|i-j|=t/|i-j|^3, we truncate the hopping up to second neighbors. The projection of H_Ryd into a fixed sector of M, say M=0, then gives

    H_Ryd,eff=P_0H_tP_0=t_2∑_i(σ_i^+σ_i+2^-+σ_i^-σ_i+2^+),

where t_2=t/8, and P_0 is the projector onto the M=0 subspace. Second order corrections are obtained from the Schrieffer-Wolff transformation H→ H'=^SH^-S. Treating only the first-neighbor hopping t_1=t, we choose the generator so that SM=-V/2δ, where V is the off-diagonal element that connects the M=0 manifold to the subspaces labeled with M=±1. With the help of projectors onto the relevant manifolds, P_±≡ P_±1, we write V as

    V=P_0H_tP_++P_+H_tP_0+P_0H_tP_-+P_-H_tP_0.

From it we find the suitable choice for S reads

    S=1/2δ(-P_0H_tP_++P_+H_tP_0+P_0H_tP_--P_-H_tP_0).

The second-order term comes from the commutator SV, namely δ H=1/2P_0[S,V]P_0. From the projection operator properties, we can simplify this commutator to 

    δ H_Ryd,eff=-1/2δ(P_0H_tP_+H_tP_0-P_0H_tP_-H_tP_0).

We observe the two signs are different, because excitations increase energy when go from odd to even sublattices, but lower energy when move in the opposite direction. Writing out the explicit formula for the hopping, we fix initial and final sites with the use of projection operators, and the expression becomes

    δ H_Ryd,eff=-t^2/2δ∑_ij(σ_2i-1^++σ_2i+1^+)σ_2i^-σ_2j^+(σ_2j-1^-+σ_2j+1^-).

Applying the commutator identity ABCD=ABCD+ACBD+ACDB+CADB, we then arrive at

    δ H_Ryd,eff=-t^2/2δ∑_i(-1)^i(σ_i^+σ_i+1^zσ_i+2^-+).

We draw attention to the fact that Ising terms cancel out due to the staggering behavior of second-order processes. Together with Eq. (<ref>) this leads us to the formula presented in the main text.




 §.§ Cold atoms


We write the Bose-Hubbard model, Eq. (<ref>), as H_BH=UC+μ M_a+H_t+H_t'. The first two terms are dominant. They represent the on-site Coulomb repulsion, C=1/2∑_in_i^a(n_i^a-1), and the staggered potential, M_a=1/2∑_i(-1)^in_i^a. We factor the couplings out to ease the power counting. The other two terms describe first- and second-neighbor site hopping with tunneling amplitudes t and t'. Projection onto the subspace manifold where all sites are either empty or singly-occupied, and there is an equal number of atoms in the two sublattices gives

    H_BH,eff
    =P_s0 H_BH P_s0
    =-t'∑_i(σ_i^+σ_i+2^-+σ_i^-σ_i+2^+),

where σ_i^-=P_s0a_iP_s0, with P_s0 the projector onto the C=M_a=0 subspace. We now assume t'≪ t, so at second-order we only consider the effect of H_t. The off-diagonal elements generated by the H_t in Eq. (<ref>) can be organized as

    V   =P_s0H_tP_s++P_s+H_tP_s0
    +P_s0H_tP_s-+P_s-H_tP_s0
       +P_s0H_tP_d++P_d+H_tP_s0
    +P_s0H_tP_d-+P_d-H_tP_s0.

The first line describes hopping processes where the atom hops from one sublattice to the other landing on an empty site. This matrix element involves a change Δ M_a=±1 but not on Δ C=0, as indicated by the projectors P_s±. In contrast the second line represents tunneling processes where the atom arrives at a site that is already occupied. Hence both quantum numbers change: Δ C=1 and Δ M_a=±1, so we use P_d± to indicate we are within the manifold with one doubly-occupied site. To perform the Schrieffer-Wolff transformation, we pick the generator

    S   =1/μ(-P_s0H_tP_s++P_s+H_tP_s0
    +P_s0H_tP_s--P_s-H_tP_s0)
       +1/U+μ(-P_s0H_tP_d++P_d+H_tP_s0)
    +1/U-μ(-P_s0H_tP_d-+P_d-H_tP_s0).

It satisfies SUC+μ M_a=-V, so we turn to second-order corrections computed from the power-series expansion of the Schrieffer-Wolff transformation: δ H=1/2P_s0[S,V]P_s0. In particular, we find

    δ H_BH,eff   =-1/μ(-P_s0H_tP_s+H_tP_s0
    +P_s0H_tP_s-H_tP_s0)
       -1/U+μP_s0H_tP_d+H_tP_s0-1/U-μP_s0H_tP_d-H_tP_s0.

Let us now go term a term. We note the first term describes a hard-core process, as the one considered in the Rydberg case. Apart from a normalizing factor of two, the key distinction comes from the phase dressing in Eq. (<ref>). When we expand the formula for the hopping, we obtain

    δ H_BH,eff^(μ)=t^2/μ∑_ij^'(-1)^i+j(σ_2i-1^++σ_2i+1^+)σ_2i^-σ_2j^+(σ_2j-1^-+σ_2j+1^-),

where we use the P_s0 projection to replace boson for spin-1/2 operators. From the commutator, we then find

    δ H_BH,eff^(μ)=-t^2/μ∑_i(σ_i^+σ_i+1^zσ_i+2^-+),

where we drop additive constants. The other two terms in the second line of Eq. (<ref>), come from virtual processes containing a doubly-occupied site. When the doubly-occupied site happen to be in the even sublattice, we get 

    δ H_BH,eff^(U+μ)   =-t^2/U+μ∑_i^'[σ_2i-1^+(1+σ_2i^z)σ_2i+1^-+σ_2i-1^-(1+σ_2i^z)σ_2i+1^+]
       -t^2/2(U+μ)∑_i^'(σ_2i-1^z+σ_2i+1^z)σ_2i^z.

Likewise, when the virtual doubly-occupied site belongs to the odd sublattice, 

    δ H_BH,eff^(U-μ)   =+t^2/U-μ∑_i^'[σ_2i^+(1+σ_2i+1^z)σ_2i+2^-+σ_2i^-(1+σ_2i+1^z)σ_2i+2^+]
       -t^2/2(U-μ)∑_i^'(σ_2i^z+σ_2i+2^z)σ_2i+1^z.

Collecting results from Eqs. (<ref>), (<ref>) and (<ref>), we arrive at the effective Hamiltonian (<ref>). Note that the doubly-occupied processes have different couplings for elements in the odd and even sublattices.




§ DYNAMICS OF AN EXCITATION PAIR


In this appendix we report results obtained for the quench dynamics of a pair of defects. The two excitations are created upon the action of σ_i^+σ_j^+ onto the half-filled ground state. The numerical details follow the prescription of Sec. <ref>.

We remain close to the strictly confined point, i.e., |W-J| small, and consider two different scenarios. In the first case we create a pair in adjacent sites σ_i^+σ_i+1^+, while in the second case we take them far apart σ_i-r^+σ_i+1+r^+, with r=20. Figure <ref> shows the results obtained for a chain with L=200 sites. In Fig. <ref>(a) we observe a clear light-cone signal formed upon the time evolution of a creation of the local pair. Note that the ballistic spreading of the local pair is in agreement with the continuum limit field theory, featuring a motion compatible with the emergent dipole symmetry. This is further showcased in Fig. <ref>(c), where we observe that the variance of the dipole does not evolve significantly in time; cf. the variance for a single defect in Fig. <ref>(d). Finally, in Fig. <ref>(b) a separated pair exhibits the approximate behavior of independent single-particle defects. This is further reinforced in Fig. <ref>(c) where we see the dipole variance changes with time whenever |W-J|≠0.



 







 Paper Title: Interpretable correlator Transformer for image-like quantum matter data.









Department of Physics, Indian Institute of Science Education and Research, Pune
Institut für Theoretische Physik, Universität Regensburg, D-93035 Regensburg, Germany


Fakultät für Physik, Munich Quantum Center, Ludwig-Maximilians-Universität München
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany


ORIGINS Data Science Lab
Technical University of Munich, München, Germany


Institut für Theoretische Physik, Universität Regensburg, D-93035 Regensburg, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany



§ ABSTRACT 
Due to their inherent capabilities of capturing non-local dependencies, Transformer neural networks have quickly been established as the paradigmatic architecture for large language models and image processing. Next to these traditional applications, machine learning methods have also been demonstrated to be versatile tools in the analysis of image-like data of quantum phases of matter, e.g. given snapshots of many-body wave functions obtained in ultracold atom experiments. While local correlation structures in image-like data of physical systems can reliably be detected, identifying phases of matter characterized by global, non-local structures with interpretable machine learning methods remains a challenge. Here, we introduce the correlator Transformer (CoTra), which classifies different phases of matter while at the same time yielding full interpretability in terms of physical correlation functions. The network's underlying structure is a tailored attention mechanism, which learns efficient ways to weigh local and non-local correlations for a successful classification. We demonstrate the versatility of the CoTra by detecting local order in the Heisenberg antiferromagnet, and show that local gauge constraints in one- and two-dimensional lattice gauge theories can be identified. Furthermore, we establish that the CoTra reliably detects non-local structures in images of correlated fermions in momentum space (Cooper pairs) and that it can distinguish percolating from non-percolating images.




Interpretable correlator Transformer for image-like quantum matter data
    Annabelle Bohrdt
    August 5, 2024
=======================================================================






§ INTRODUCTION


Machine learning (ML) techniques provide powerful tools for image analysis <cit.>. Since the dawn of deep neural networks, considerable progress has been made in image classification (computer vision) <cit.>. In recent years, this has led to an increasing interest in applying machine learning based analysis tools to a wide range of physically relevant input data <cit.>. For instance, it has been demonstrated that neural networks can efficiently classify and differentiate between various phases of matter <cit.>, including topologically ordered states <cit.> and systems with strong correlations <cit.>. Furthermore, progress has been made towards discovering novel quantum phases <cit.>; further unsupervised and supervised learning of quantum phases and their transitions have been studied in Refs. <cit.>. 





With the development of highly controllable analog quantum simulation platforms, e.g. ultracold atoms in optical lattices <cit.>, the application of neural networks to projective measurements of the many-body wave function constitutes one particular field of interest <cit.>. These snapshots of the wave function contain a plethora of information that goes beyond local observables <cit.>, making them a promising platform for machine learning tools <cit.>. 


In order to gain physical insights, interpretable machine learning techniques that are able to detect underlying order in a given set of measurements are desirable. 
Examples for physically interpretable algorithms include finding order parameters using support vector machines (SVM)  <cit.>, interpretability in terms of principle component analysis (PCA) <cit.> and analyzing the Hessian matrix of the loss function <cit.>. 
Using network theory, wave function networks have been developed to study such quantum simulation snapshots <cit.>. 
In relatively shallow neural networks, insights can potentially be gained by looking into learned weights for a given classification task <cit.>. 

Typically, however, despite their effectiveness, neural networks tend to operate as black boxes. 
Interpretable neural network architectures ideally combine a high performance with the possibility to uncover patterns and correlations in physical systems with complex dependencies. 
For this purpose, the correlation convolutional neural networks (CCNN) <cit.> has been developed, which allows to extract the most defining correlations in the Fock basis of a given snapshot. However, the convolutional structure of the CCNN architecture prevents an efficient detection of global structures and hidden or non-local order.

In order to access long-range dependencies in a snapshot, a different choice of architecture is more natural:
The introduction of the Transformer architecture <cit.> marked a groundbreaking moment in the development of large language models. Although conceived as a sequence-to-sequence model, this was later adapted to the vision Transformer <cit.> to be compatible with image data. The attention mechanism, which aims to mimic human-like attention, provides a map of pertinent portions and correlations of the image, which may help to gain a physical understanding of (global) underlying structures. It has been shown that the Transformer architecture can be used to learn for example entanglement growth and state complexities from experimental data <cit.>. In <cit.>, the Transformer architecture was applied to quantum snapshots of the Heisenberg model, however, without giving a physical interpretation of the resulting attention mechanism. Here, we demonstrate that the vision Transformer can reliably classify images of correlated systems, with full interpretability in terms of local and non-local correlation functions. In particular, we introduce a tailored attention mechanism that enables us to isolate and capture specific orders of correlations from different Transformer layers.

This paper is organized as follows. In Section <ref>, we introduce the correlator Transformer architecture. We discuss how higher-order correlators are constructed in each subsequent layer and demonstrate that by adding tailored L_1 regularization, one can obtain the most relevant order of correlation. After obtaining the latter, an analysis of the learned weights of that particular Transformer layer (attention maps) enables further insights into the specific correlator. In Section <ref>, we show our results for local correlations upon applying the correlator Transformer to different physical models: we demonstrate how the Transformer learns local correlations in the two-dimensional (2D) antiferromagnetic (AFM) Heisenberg model, the one-dimensional (1D) ℤ_2 lattice gauge theory (LGT) as well as the 2D ℤ_2 Ising LGT, using local correlations up to fourth order. In Section <ref>, we apply the Transformer to problems involving non-local correlations, in particular in the analysis of momentum space images of Cooper pairs and in the classification of percolating and non-percolating snapshots. Finally, we summarize our results in Section <ref>.




§ CORRELATOR TRANSFORMER

The correlator Transformer (CoTra) architecture is inspired by the vision Transformer, a state-of-the-art image classification model. Like the vision Transformer, the CoTra is a versatile model that can handle different types of images. In this work, the input to our model is image-like quantum matter data, including, for instance, projective measurements in real or momentum space of quantum gas microscopes <cit.> or tunneling microscope images <cit.>.

For a given depth n of the CoTra (corresponding to the number of layers in the Transformer architecture), the model has access to n+1 order weighted correlations (from now on denoted by X^n+1) with respect to these input measurements. These weights then give insights into the specific correlations that the model uses for its classification.



 §.§ Architecture

Consider a snapshot in ℝ^(h,h,c) where h,c are the height (width) and number of channels. As done in standard vision Transformer workflows, we first divide this snapshot into equal patches of dimension p× p, where p is the patch size; hence, the transformed snapshot is of dimension (h/p)^2 × cp^2, where M = (h/p)^2 denotes the number of patches. Afterwards, a linear projection is performed that transforms the patches to vectors of hidden dimension d. We label this projected snapshot as X^1 ∈  ℝ^((h/p)^2,d); it can be directly used as input of the Transformer encoder or it can be supplemented with additional learnable positional encoding depending on the input data, as further commented on below. As the above only consists of reordering and linear transforming the original data, the resulting matrix corresponds to first order terms X^1, see Fig. <ref> (a). By default, we do not use any Layer  Norm, and it is used only if it is crucial for the model's performance. This choice is made to keep the model as simple as possible for better interpretability.

The general idea of the CoTra is as follows (see Fig. <ref>): In the first layer, both the query and key projections are computed from X^1, which, together with a positional embedding, returns the second-order terms X^2. In a second Transformer layer, the query projection remains the same, while the key projection is now computed from the output of the first layer, i.e. X^2. This second layer now computes the third-order terms X^3 and so on. Therefore, using n Transformer layers enables us to compute correlations up to order (n+1). 

In particular, the query and key for the first Transformer layer can be written as 

    Q_1    = X^1 W^1Q
    
                K_1    = X^1 W^1K,

Where W^1Q, W^1K are the query and key projection matrices of the first layer. Meanwhile, the value is a linear projection of the sine-cosine encoding, defined by <cit.>

    PE_(i, 2j)   = sin(i/100000^2j/d) 
    PE_(i, 2j+1)   = cos(i/100000^2j/d).


Here, i ∈{0,1,...N-1} is the position of N patches and j ∈{0,1,....,d-1} where d is the hidden dimension. 
 Since these are not learnable parameters, they do not increase the complexity of the model. The value, then, is
 
    V_1 = (PE) W^1V.


The Transformer output of the first layer (X^2) can now be written as [see Fig. <ref> (b)]

    X^2    =1/√(d) Q_1 K^T_1V_1 
       = 1/√(d)X^1 W^1Q W^1K^TX^1^T (PE) W^1V.


In particular, X^2 and the corresponding attention map a^2 = 1/√(d)Q^1K^1^T both contain only 2^nd order terms. This can be directly seen by writing out the matrix elements [see Eq. (<ref>) in Apendix <ref>] of the 2^nd order attention map (with patch labels i,j),  




    a^2_ij  = ∑_rs=1^cp^2∑_klsm=1^d x_irx^T_sjP_ikP^T_lj   W^P_rk
       W^P^T_lsW^1Q_kmW^1K^T_ml.

Here, x denotes the pixel values (projective spin measurements, phase space measurements etc.) of a given snapshot,  W^P is the linear projection matrix to the hidden dimension, W^1Q, W^1K are the query and key projection matrices of the first layer and P is an additional learnable positional encoding. Depending on the dataset, this learnable positional encoding is included at X^1 along with the sine-cosine encoding to improve accuracy. 
As the indices in Eq. (<ref>) i,j go through all the patches and r,s  through all the pixels inside the patches, we can obtain all the 2^nd order local and non-local correlators and their corresponding attention weights. 

Similarly, the n^th layer output [see Fig. <ref> (b)] is defined as

    X^n+1 = 1/√(d)Q^1_n K^T_n V_n,

where Q^1_n denotes the query projection of X^1 by W^nQ, K_n is the key projection of X^n and V_n is the value projection of the positional embedding at each layer n. This attention can be thought of as weighing the different correlations in QK^T with their positional information. In this manner, we can build up further orders of weighted correlations using the Transformer, and by inspecting the weights of the corresponding attention maps, the most relevant correlators can be obtained. 

For each order X^n, we compute the mean X̅^n over all dimensions, which is used as input for the final classification, 𝐱̃_𝐧 = [X̅^1,X̅^2, …, X̅^n]. A linear transformation on 𝐱̃_𝐧 is then performed to form the logits for the cross-entropy loss function coupled with a regularization loss of strength α. In particular, the loss function reads

    ℒ≡ -∑_i=1^N y_i log(ŷ_i) + α𝐱̃_𝐧,

where y is the true distribution, ŷ is the predicted distribution (logits), and ||.|| is the norm.

This modified architecture contains a similar number of parameters as a traditional vision Transformer and is able to provide direct interpretability in terms of local as well as non-local correlations.
After training the CoTra with a particular hidden dimension (d), learning rate (lr), regularization strength (α) and other hyperparameters, we collect the learned correlators and perform a regularization path analysis to understand which order terms influence the classification the most, as described next.
 


 §.§ Regularization Path Analysis

Regularization path analysis is a common technique used for feature selection <cit.>. Here, as done in <cit.>, we use the L_1 regularization. The idea is to collect the learned mean correlators X̅^n and perform a lasso regression to see which of the n-point correlators are the most important for phase classification. We only re-train the architecture's logistic classifier part (final linear projection) rather than the whole model.




The loss function for the regularization path analysis is given by

    ℒ_path≡ -∑_i y_i log(ŷ_i) + λβ,

where β≡ [β^1,β^2,…, β^n] are the logistic weights of the corresponding (mean) n-point correlator. 
We use the Scikit-learn <cit.> logistic regression library to optimize the classifier over a range of λ starting from a very high value, where the classification fails, to gradually smaller strengths where the classification is successful (see also <cit.>). By looking at the strengths of β^n over the range of λ one can see which particular β^n (and hence which order of correlation) is responsible for a successful classification. Afterward, a closer look into the attention weights inside that particular Transformer layer tells us which specific n-point correlators are used by the network for classification. For local correlations, we can fix a patch and look at the attention weights inside that patch to see how each correlator is weighed. On the other hand, the attention map itself gives information about the dependencies between patches for long-range correlations.



§ LEARNING LOCAL CORRELATIONS AND GAUGE CONSTRAINTS

In this section, we discuss the results of our model in learning local correlations. In the following, we consider three exemplary datasets: the 2D Heisenberg model, a 1D ℤ_2 lattice gauge theory (LGT) and the 2D ℤ_2 Ising LGT.



 §.§ 2D Heisenberg model

We start by analyzing snapshots of a prototypical spin model, namely the Heisenberg model. The Hamiltonian of the system is given by

    Ĥ = J ∑_⟨ i,j ⟩( Ŝ_i^x Ŝ_j^x + Ŝ_i^y Ŝ_j^y + Ŝ_i^z Ŝ_j^z ),

where J is the nearest neighbour coupling constant and S^x,y,z are spin-1/2 operators. We train the Transformer on snapshots from an equilibrium finite temperature state at T/J = 0.6 with AFM couplings (J>0) against random snapshots with vanishing magnetization. This ensures that the model does not compare magnetization values directly, but instead looks for higher-order correlations when classifying the snapshots.
We use a lattice size of 16 ×16 and sample Heisenberg snapshots  using stochastic series expansion quantum Monte Carlo <cit.>. Fig. <ref> (a) depicts one of the Heisenberg snapshots as well as a random configuration. The chosen hyperparameters for this data are d=16, patch size =2,λ=10^-3, and lr=5×10^-3. The inset in the right panel of Fig. <ref> (a) shows how the spins inside a single 2× 2 patch are numbered, which is important when evaluating the correlations at a later stage. We obtain an accuracy of 96.4%.



Fig. <ref> (b) shows the regularization path, which indicates that the model is capable of classifying the two datasets by evaluating 2-point correlations. We take the mean over the 2^nd order attention map a^2 = 1/√(d)Q^1K^1^T for each snapshot in the dataset and plot a histogram. Indeed, Fig. <ref> (c) shows that the distributions of the mean 2^nd order attention for the AFM and random snapshots separate to a large degree, corroborating that the network uses 2^nd order correlations for its classification. Now, to see which particular 2^nd order correlators are used by the network, we take a more detailed look into the attention weights. The learned weights of the 2^nd order correlator (labels inside a patch are given by r,s) can be written as

    C^^(2)_rs =|1/M∑_i=1^M∑_klm=1^d P_ik   P_li^TW^P_rk
       W^P^T_lsW^1Q_kmW^1K^T_ml|,

where we have taken the average over all patches i. P is the learnable positional encoding, W^P is the linear projection matrix to the hidden dimension and W^nQ, W^nK denotes the query, key projections in the n^th Transformer layer, see also Appendix <ref> for an explicit derivation of Eq. (<ref>).

Fig. <ref> (d) depicts this 2-point correlation matrix where r,s runs through all four spins inside the patch. We can see that the model puts its emphasis on nearest neighbour correlations, cf. the labeling of spins within the patch in the inset of Fig. <ref> (a). With the combined information of the regularization path and the learned weights, we conclude that the correlator Transformer is calculating second-order (intra-patch) correlations to make its classification decision. 



 §.§ 1D ℤ_2 LGT

Originally proposed in the context of non-perturbative theories in particle physics, lattice gauge theories have become a paradigmatic class of systems also in the field of strongly correlated phases of matter <cit.>. Their defining feature is the presence of a local gauge symmetry.  Here, we consider a (1+1)D ℤ_2 LGT where hard-core bosonic matter is coupled to ℤ_2 gauge fields. The Hamiltonian of the system is described as <cit.> 

    Ĥ_0 = J ∑_j=1^L-1( â^†_j τ̂^z_j,j+1â_j+1 + H.c.) - h ∑_j=1^Lτ̂^x_j,j+1,

where â^†_j,â_j are the bosonic ladder operators acting on matter site j, τ^z_j,j+1 denotes the gauge field between the matter site j and j+1 and  τ^x_j,j+1 is the electric field between the matter site j and j+1. 
The ℤ_2 electric field is subject to Gauss' law (GL), defined by the local generator 

    Ĝ_j = (-1)^n̂_jτ̂^x_j-1,jτ̂^x_j,j+1,

which has eigenvalues g_j=± 1. In the following, we test whether the CoTra can identify the local gauge structure. In order to do so, we generate two classes of snapshots: One set consists of states with g_j=+1 ∀ j in the lattice. The network is then given the task to distinguish this set from snapshots with random gauge sectors g_j = ± 1 ∀ j.

We consider a chain of length 16 with open boundaries. Fig. <ref> (a) depicts the left end of the 1D chain with matter sites n_j and link variables τ^x_j,j-1. In the following, a single patch consists of two matter and link variables, with labelling as depicted in the inset of Fig. <ref> (a). For better interpretability, we describe both matter and gauge fields on the same footing, i.e., we convert the occupancy of the matter site (which is measured in the Fock basis as 0 or 1) to ± 1. We note that this transformation changes the local generator of gauge invariance to 

    Ĝ_j = n̂_j τ̂^x_j-1,jτ̂^x_j,j+1.

For the matter site corresponding to n_1 within the unit cell depicted in Fig. <ref> (a), only the elements 0,2,3 contribute to the calculation of Eq. (<ref>).



The hyperparameters when training the CoTra for this data are d=12,  patch size =2 (contains 2 matter and 2 links), λ=2×10^-4, lr=10^-2 and we obtain an accuracy of 97%. After training, we again perform a regularization path analysis, see Fig. <ref> (b). Indeed, third-order correlations are used to distinguish the two different classes as expected from Eq. (<ref>). When evaluating the mean third-order attention maps for each snapshot, we see that states that fulfil the constraint g_j = 1 form a sharp peak, while random snapshots have a broad distribution [Fig. <ref> (c)]. Indeed, the two histograms only have marginal overlap, underlining that the network is making its decision based on three-point correlations.

In order to analyze which particular three-point correlator within a patch is used for classification, we construct the third-order attention weights a^3 = 1/√(d)Q^1K^2^K. The weights for the different possible three-point correlations are given by (see Appendix <ref>)

    C^^(3)_ers =    |1/M∑_i=1^M∑_kln=1^d∑_abcm=1^dP_iaP^T_kiP_ilW^P_eaW^P^T_kr
       W^P_slW^1Q^T_mkW^2Q_acW^1K_lmW^2K^T_cbW^1V^T_bn(PE)_ni|,

where we again average over all patch labels i and the indices e,r,s go through the 4 elements inside each patch. We note that in contrast to Eq. (<ref>), the third-order correlator weight matrix explicitly includes the absolute positional embedding, as it involves terms (X^2) that have passed through the first Transformer layer. 

Fig. <ref> (d) shows that significant weight is only on the correlator with indices (e,r,s)=(0,2,3), which exactly corresponds to evaluating Eq. (<ref>). We note that choosing a larger patch size leads to identical conclusions. For instance, for a patch size of four, there are three possible combinations that contribute to the Gauss law; correspondingly, this leads to the appearance of three peaks in the correlation weights, i.e., the network calculates identical correlations to perform its classification. We were also able to classify states with 1-2 Gauss's law violations along the chain against the completely fulfilling snapshots with the addition of Layer Norm for the query and key projections. 



 §.§ Classical 2D ℤ_2 Ising LGT


Lastly, we focus on the classical ℤ_2  Ising lattice gauge theory in 2D, where Gauss law is given by a 4^th order term. The Hamiltonian is defined as <cit.>

    Ĥ = -J ∑_p ∏_l ∈ pσ̂^z_l,

where J denotes the magnetic coupling constant, σ̂^z is the z Pauli matrix, and the summation covers all plaquettes p in the lattice. In the ground state, each plaquette p of the lattice satisfies the Gauss' law:

    G_p = ∏_l ∈ pσ̂^z_l = 1

The system has a highly degenerate ground state, comprising all possible configurations where G_p = 1 is obeyed.  At finite temperature, the gauge constraint is violated locally. Excited states are thus characterized by 

    G_p = ∏_l ∈ pσ̂^z_l = ±1.

As in the 1D LGT, we generate two classes of snapshots: One where Gauss's law (G_p = +1) is obeyed globally and one where it is violated randomly.

In particular, we consider a 16×16 lattice, with each lattice point having one horizontal and vertical link. The left-hand side of the inset in Fig. <ref> (c) depicts an exemplary configuration of the ground state, where on each plaquette, the product of Pauli z operators is positive. On the other hand, the products are both positive and negative in a finite temperature snapshot; see the right-hand side of the inset in Fig. <ref> (c). It is important to note that a single patch of patch size 2 contains 8 links. We arrange the input snapshots into 2 channels, one for the horizontal and one for the vertical links. By construction, out of the 8 links in a patch, only the links 2,5,6,7 contribute to the central plaquette of the patch; see the explicit labeling in the inset of Fig. <ref> (c).

The hyperparameters for this data are d=6, patch size =2, λ=2×10^-4, lr=5×10^-3. For these parameters, the network is able to fully classify the snapshots with an accuracy of 100%.
A regularization path analysis for the given choice of hyperparameters shows that the model is focusing on fourth-order correlators, Fig. <ref> (a). Note that we do not employ any additional learnable positional encoding here because the model is able to achieve perfect accuracy with absolute positional encoding. This is corroborated by the distributions of mean attention scores, as shown in Fig. <ref> (b). Indeed, both distributions fully separate, making a full distinction between the two data sets possible. 



In order to check which particular four-point correlator the model learns, we construct the fourth-order attention weights a^4 = 1/√(d)Q^1K^3^T, given by (see Appendix <ref>)

    C^^(4)_hers = |1/M∑_i=1^M   ∑_kln=1^d∑_abcm=1^d∑_pqto=1^dW^P_hpW^P^T_aeW^P_rk
         W^P^T_ls W^1Q_kmW^2Q^T_ca
        W^3Q_ptW^1K^T_mlW^2K_bc
       W^3K^T_qt
        W^1V_nbW^2V^T_qo(PE)_in(PE)^T_oi|,

where the indices h,e,r,s run through the 8 elements inside a single patch, and we take an average over all the patches to compute C^^(4)_hers. Note that the only dependency on patches comes from absolute positional embedding in this case. We show the intra-patch weights as given in Eq. (<ref>) in Fig. <ref> (c), and find that the model is giving the most significant weight to the correlator corresponding to the term (h,e,r,s) = (2,5,6,7), which matches Gauss' law [cf. the labeling in each unit cell as shown in the inset of Fig. <ref> (c)].

We note that for the case of the ℤ_2 Ising LGT, we additionaly use a Layer Norm after the query and key projections, which we identify as crucial for a successful classification (without it, the network fails to identify the gauge structure, leading to an accuracy of 50%). We further find that the network's output depends on the choice of hyperparameters: runs with various choices of parameters filter out 2^nd, 3^rd as well as 4^th as the most important correlations. Nevertheless, in Appendix <ref>, we show that for all cases, corresponding results are interpretable and that, indeed, classification can be achieved in various ways by looking at different orders of correlations for the classical ℤ_2 LGT.    



§ LEARNING NON-LOCAL STRUCTURES

So far, we have seen that the CoTra can detect local correlations as well as local gauge structures through the evaluation of local n-point correlations. In this section, we consider datasets containing snapshots that can be distinguished only by detecting non-local structures. This demonstrates the ability of the Transformer to capture long-range dependencies, which constitutes a challenge for many architectures and thus establishes the CoTra as a versatile analysis tool for quantum matter data.



 §.§ Cooper Pairs

Our first testing case is the detection of non-local structures of Fermi surfaces in BCS-type states that feature bound Cooper pairs <cit.>.  
This is in particular motivated by recent experimental observations of Fermi surface correlations in a mesoscopic 2D Fermi gas <cit.>. There, using a 2D optical dipole trap-optical tweezer setup, momentum-resolved snapshots have been obtained through time of flight (TOF) expansion measurements. In Ref. <cit.>, single particle momentum correlations are analyzed in the experimental data. At weak attractions, the particles show no correlations; however, at sufficiently strong interaction strengths, they form opposite momentum opposite spin pairs across the Fermi surface. Here, we generate synthetic TOF data akin to the experimental results in Ref. <cit.> and train the CoTra to analyze whether it can successfully capture these long-range correlations in continuous image-like data.

To this end, we again generate two classes of data: In both cases, the momentum space snapshots are generated by distributing +1 for spin-up and -1 for spin-down fermions on a 30×30 grid with a Fermi momentum of 10 units. The paired state features two Cooper pairs on the Fermi surface and four particles randomly
distributed inside the Fermi surface. Images in the second class also have two spin-up and two spin-down fermions on the Fermi surface, but none of these form a pair. Exemplary snapshots for paired and unpaired states are shown in the left-hand side of Fig. <ref> (a) and (b), respectively.

The hyperparameters for this data are d=16, patch size =2, λ=10^-4,  and  lr=10^-2; we obtain an accuracy of 98.1%. The regularization path analysis shown in Fig. <ref> (c) indicates that the Transformer is using second-order correlations to make its decision. The distributions of mean second order attention maps (mean over the whole attention map matrix) are shown in Fig. <ref> (d). Similar to earlier examples, the paired state has a sharp distribution of attention values, which is largely separated from a broad distribution of the unpaired state.

We obtain the attention map with respect to a given patch δ through 

    a^2_iδ = ∑_rs=1^cp^2∑_klsm=1^d   x_irx^T_sδP_ikP^T_lδ
       W^p_rkW^p^T_lsW^1q_kmW^1k^T_ml.

The index i runs through all the patches, thereby giving us correlations between patch δ and all others. Thus, by focusing on quantities like Eq. (<ref>), long-range correlations can be analyzed. The right-hand side of Fig. <ref> (a) and (b) show the resulting second-order attention maps.



For the paired snapshot, the attention is clearly given to the Cooper pair; see Fig. <ref> (a). In contrast, no such structure exists in the unpaired state, see Fig. <ref> (b). A particular feature of the attention maps is their sign structure: In the paired state, attention weights of the Cooper pair are of opposite sign, while in the random state, all weights are negative. This, in turn, leads to a separation of the mean attention values, and hence the separation of the corresponding histograms, cf. Fig. <ref> (d). We note that the above holds independent of the reference patch δ. In Appendix <ref>, we show attention maps when using a reference patch containing a particle inside the Fermi surface. In this case, the attention is on both Cooper pairs, while no particular structures exist for the unpaired state. 



 §.§ Percolation

Percolation theory, first introduced by
Flory and Stockmayer in the 1940s <cit.>, considers the formation of clusters that eventually span an extended system in a percolating cluster <cit.>. Naturally, the distinction of whether a particular snapshot is percolating or not constitutes a problem that requires the evaluation of non-local correlations. In this section, we train the CoTra with synthetic data of percolating and non-percolating binary images on a 2D lattice.

In particular, we focus on a simplified percolating system with only one percolating path (directed percolation). We generate a 16×16 lattice of random spins directed from one edge to the opposite edge (i.e. we generate random percolating structures). Half of the generated percolating images are then randomly shuffled such that they become non-percolating. The left-hand sides of Fig. <ref> (a) and (b) show exemplary percolating (non-percolating) snapshots. We note that due to our construction of the snapshots, the total number of grey pixels is the same; this prevents the network from simply counting the number of active pixels.

We train the CoTra with hyperparameters d=16, patch size =2, λ=10^-3,  and lr=10^-2. We obtain an accuracy of 99.3%. After training, a regularization path analysis shown in Fig. <ref> (a) indicates second-order correlations to be crucial for the classification. Mean second-order attention values are seen to separate, whereby percolating (non-percolating) images have a sharp (broad) distribution, see Fig. <ref> (b). We emphasize the generality of this observation and suspect that a global underlying structure typically leads to distributions of the CoTra mean attention map values that have small widths.  

We plot second-order attention maps given by Eq. (<ref>) on the corresponding right-hand sides of Fig. <ref> (a). The reference patch δ is given by the black square in the snapshot images. In the percolating case, the attention mechanism indeed puts emphasis on the percolating structure with uniform sign. In case of the non-percolating image, the attention seems to be localized relatively close to the reference patch, with a seemingly random sign structure. We note that, by utilizing the capability of analyzing the dependencies between all patches, second-order inter-patch (i.e., long-range) correlations allow the network to make a classification. We note that when choosing a reference patch that does not contain any pixel of the percolating cluster, the attention maps is zero. 

For both Cooper pairs and percolation, the sign of the attention map is seen to be an important factor contributing to the classification. In both cases, we see that the classification boundary lies around zero [see the black dashed lines in Fig. <ref> (d) and Fig. <ref> (d)]. By assigning a certain sign structure to the inter-patch attention, the CoTra is able to reliably detect non-local structures through inter-patch correlations.  



§ SUMMARY AND OUTLOOK

We have introduced a Transformer-inspired machine learning framework to analyze image-like quantum matter data with a tailored attention mechanism that yields full interpretability in terms of correlation functions. Combined with a regularization path analysis, we have shown that the network is able to identify both local and non-local structures in a variety of testing cases. We have demonstrated that the network can learn local correlations, such as present in the AFM Heisenberg model, and that it can identify gauge constraints, e.g. in 1D and 2D lattice gauge theories. We further demonstrated the network's capabilities of detecting non-local dependencies and correlations, by training on images of correlated Fermi surfaces featuring BCS pairs as well as percolating and non-percolating images. 

Our proposed architecture can readily be used for experimental image-like data, for instance, real- and momentum-space snapshots of ultracold atoms <cit.> or scanning tunneling microscopes. This may help to reveal hidden order, topological phases, and other global, non-local structures such as symmetries of e.g. charge density wave phases <cit.>. The CoTra may further be used for learning and interpreting disentangling gates, akin to the reinforcement learning protocol demonstrated in <cit.>. Recently, it has been proposed that percolation can be used as an effective order parameter for confinement in ℤ_2 LGTs <cit.>. It would be interesting to see how the CoTra performs when given finite temperature snapshots of confined and deconfined phases, and if any connection to percolation can be made from the resulting attention maps. It would further be interesting to combine the CoTra with a confusion learning scheme akin to <cit.>, which allows for a fully unsupervised phase detection and classification. This, in particular, can be useful to analyze doping and temperature scans of the strongly correlated Fermi-Hubbard model, where non-local, hidden structures are believed to play an essential role in the low-doping regime <cit.>. 
 


Acknowledgements.–

We thank Simon M. Linsel for valuable comments. This research was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC2111 – 390814868, as well as the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 948141 — ERC Starting Grant SimUcQuam). AS acknowledges support by the Graduiertenkolleg of the University of Regensburg (IRTG SFB 1277), as well as the KVPY program of DST (Department of Science and Technology) Govt of India.  BH acknowledges support by Excellence Cluster ORIGINS, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094-390783311.






§ CORRELATOR WEIGHTS

We here derive the expressions for the attention maps as well as correlator weights presented in the main text, see i.p. Eqs. (<ref>), (<ref>), (<ref>).


 §.§ 2^nd Order Correlators



Let x be the input image in ℝ^(h,w,c), where h,w,c represents the height, width and number of channels. First, we patch this image with a patch size of p; the flattened image will then be of size (hw/p^2,cp^2) for a 2D snapshot and (h/p,cp) for a 1D snapshot. We denote the number of patches (hw/p^2) with M. 

Now we linearly project these flattened patches to a hidden dimension d and multiply with a learnable position encoding P in an element-wise fashion,

    X^p    = P*W^pX 
    
        X^p_ij   = ∑_r^cp^2 x_irW^p_rjP_ij.

Here, W^p is the projection matrix in ℝ^(cp^2,d) and X denotes the flattened patches. The learnable position encoding helps in more accurate classification since it encodes each patch with a certain weight.

After this, we introduce three independent linear projections by W^α Q, W^α K, W^α V to form the query, key and value, as in standard Transformer architectures <cit.>. α denotes the particular Transformer layer that the projection is applied to.


    Q^1_ij   = ∑_k^d∑_r^cp^2 x_irW^p_rkP_ikW^1Q_kj
    
        K^1_ij   =  ∑_l^d∑_s^cp^2 x_isW^p_slP_ilW^1Q_lj
    
        V^1_ij   =  ∑_n^d PE_inW^1V_nj

Where the value (V^1) is derived from the sine-cosine positional encoding, see Eq. (<ref>). Now, we calculate the 2^nd order attention map a^2 (setting the constants (d) to 1 for simplicity) by

    a^2 =    Q^1K^1^T
    
        a^2_ij  =    ∑_rs=1^cp^2∑_klsm=1^dx_irx^T_sjP_ikP^T_ljW^p_rk
       W^p^T_lsW^1q_kmW^1k^T_ml

The 2^nd order weight matrix for a given patch can be constructed from the coefficients of a^2 by fixing i=j and then taking an average over all M patches i. In the analysis in the main text, we focus on the absolute value of all the n^th order correlator weights C^^n.


    C^^(2)_rs =|1/M∑_i=1^M∑_klm=1^d P_ikP_li^TW^P_rkW^P^T_lsW^1Q_kmW^1K^T_ml|,


The output of the first layer X^2 can be written as

    X^2 =    a^2V^1 
    
        X^2_ij =    ∑_g=1^M∑_rs=1^cp^2∑_klsmn=1^d x_irx^T_sgP_ikP^T_lgW^p_rk
       W^p^T_lsW^1q_kmW^1k^T_ml(PE)_gnW^1v_nj




 §.§ 3^rd Order Correlators


The key projection in the 2^nd layer is derived from X^2 while the query projection remains the same, ie from X^1 itself but with a different projection matrix W^2Q,

    K^2=    X^2W^2k
     
        K^2_ij =    ∑_g=1^M∑_rs=1^cp^2∑_klsm=1^d∑_nb=1^d x_irx^T_sgP_ikP^T_lgW^p_rkW^p^T_sl
       W^1q_kmW^1k^T_ml(PE)_gnW^1v_nbW^2k_bj.


The 3^rd order attention map can be constructed by


    a^3  =    Q^2K^2^T
    
        a^3_ij =   ∑_g=1^M∑_ers=1^cp^2∑_klsn=1^d∑_abcm=1^dx_iex^T_rjx_ngP_iaP^T_kjP_glW^p_ea
       W^p^T_krW^p_sl 
        W^1q^T_mkW^2q_acW^1k_lmW^2k^T_cbW^1v^T_bn(PE)^T_ng,

where Q^2 is the 2^nd layer query projection of X^1.
The 3^rd order correlator weights for a given patch can be constructed from learnable coefficients of the 3rd order attention map by fixing i=j=g and then taking a mean over all the M patches.

    C^^(3)_ers =    |1/M∑_i=1^M∑_kln=1^d∑_abcm=1^dP_iaP^T_kiP_ilW^P_eaW^P^T_kr
       W^P_slW^1Q^T_mkW^2Q_acW^1K_lmW^2K^T_cbW^1V^T_bn(PE)_ni|,

The output of the 2^nd layer X^3 can be written as 


    X^3_ij=   ∑_gf=1^M∑_ers=1^cp^2∑_klsn=1^d∑_abcmo=1^dx_iex^T_rfx_gsP_iaP^T_kf
       P_glW^p_eaW^p^T_kr W^p_slW^1q^T_mk
         W^2q_acW^1k_lmW^2k^T_cbW^1v^T_bn
       (PE)^T_ngW^2v_oj(PE)_fo.



 §.§ 4^th Order Correlators


Similar to constructing the previous order correlator weights, we can obtain the key projection from X^3 while keeping the query projection the same. Note that we do not use any learnable positional encoding for the Ising LGT.


    K^3 =    X^3W^3k
    
        K^3_ij =   ∑_gf=1^M∑_ers=1^cp^2∑_klsn=1^d∑_abcm=1^d∑_oq=1^dx_iex^T_rfx_ns
       W^p_ea W^p^T_krW^p_slW^1q^T_mkW^2q_acW^1k_lm
         W^2k^T_cbW^1v^T_bn
       (PE)^T_nfW^2v_oq(PE)_foW^3k_qj

The 4^th order attention map is written as

    a^4 =    Q^3K^3^T
    
        a^4_ij =   ∑_gf=1^M∑_ers=1^cp^2∑_klsn=1^d∑_abcm=1^d∑_pqto=1^dx_ihx^T_ejx_qrx^T_sn
       W^p_hgW^p^T_ae W^p_rkW^p^T_lsW^1q_kmW^2q^T_ca W^3q_gpW^1K
       k^T_ml
       W^2k_bcW^3k^T_pqW^1v_nbW^2v^T_qo(PE)_gn(PE)^T_of.

Where Q^3 is the 3^rd layer query projection of X^1.
The 4^th order correlator weights can then be written as


    C^^(4)_hers = |1/M∑_i=1^M   ∑_kln=1^d∑_abcm=1^d∑_pqto=1^dW^p_hpW^p^T_aeW^p_rk
         W^p^T_ls W^1q_kmW^2q^T_ca
        W^3q_ptW^1k^T_mlW^2k_bc
       W^3k^T_qt
        W^1v_nbW^2v^T_qo(PE)_in(PE)^T_oi|.



§ DISCUSSION ON HYPERPARAMETER DEPENDENCE


As noted in the main text, we find that especially for the Ising ℤ_2 LGT, the results depend on the choice of hyperparameters. One possible explanation for this is our use of the Layer Norm, which is crucial for a successful classification in the case of the Ising LGT. The use of Layer Norm complicates the interpretability of the network because we do not account for the weights and bias of the normalization in our analysis. By varying the hyperparameters, we indeed observe that in different runs the Transformer focuses on 2^nd, 3^rd and 4^th order correlators to classify the Ising LGT. In the following, we look in more detail into the output of the Transformer for all of the above cases and show that the network's decisions remain interpretable.





Fig. <ref> shows the results for the hyperparameters:  d=4, patch size =2,  λ=1e^-3, lr=0.005. For this set of hyperparameters, the network uses 2^nd order correlations to discriminate the phases. Though this may seem surprising, it is indeed possible to discriminate the two phases by adding certain 2-point correlations: If we look at the sum of the (horizontal) product of two links - call it g^2 - on the plaquette, we get the following criteria (see Fig. <ref>):  

    g^2    = ±2   for states that globally fulfil GL
     
        g^2    = 0,±2   for excited states.



Indeed, this is similar to Gauss' law Eq. (<ref>). In the weight matrix C^2_rs, shown in Fig. <ref> (b), we see that 2-point correlations involving the plaquette sites 2,5,6,7 are used. By adding these correlations, the Transformer is able to reliably detect the local gauge constraints.

Fig. <ref> shows the results for the hyperparameters  d=28, patch size =2, λ=1e^-4, lr=0.01; here, the network utilizes 3-point correlations for classification.

The reasoning for the 2^nd correlations can also be extended to the 3^rd order. The sum of 3-point terms g^3 on a plaquette can be categorized as

    g^3    = 0,±4   for ground states
     
        g^3    = ±2   for excited states.

As we can see from the correlation weights for the 3^rd order terms, Fig. <ref> (b), the larger weights are associated with the terms 256, 257, 267 and 567. Thus, by summing over these 3-point correlations, perfect accuracy can be achieved.  


§ VARYING REFERENCE PATCH FOR COOPER PAIR DATA





Fig. <ref>
shows the attention map w.r.t a particle inside the Fermi surface. For a snapshot that features Cooper pairs, the attention is - regardless of the reference patch - seen to focus on the paired electrons on the Fermi surface, see Fig. <ref> (a). On the other hand, for an unpaired snapshot, Fig. <ref> (b), no such structures are observable.



§ ADDITIONAL RESULTS



 §.§ Percolation

In this section, we train the model with slightly modified percolation data in which the percolating path is broken at very few points. We get similar results to the main text data with 2^nd order correlations to be the most important.

The hyperparameters used are d=16, patch size =2, λ=2×10^-3,  and  lr=5×10^-3; we obtain an accuracy of 95.78%. This indicates that the CoTra is able to precisely classify percolation in this dataset.
Fig. <ref> (a,b) shows 2^nd attention maps for the percolating and non-percolating states. While Fig. <ref> (c,d) shows the regularization path and histogram of mean attention.



 §.§ Staggered Magnetization

In this section, we train the model with snapshots of 16×16 lattice having staggered magnetization equal to zero against snapshots with staggered magnetization equal to 0.015625. 

The hyperparameters used are d=4, patch size =2, λ=10^-4,  and  lr=10^-1. The model is able to successfully classify the dataset with 99.38% accuracy. 

Fig. <ref> (a) shows the 2^nd order regularization path, and Fig. <ref> (b) shows the corresponding order weights. Though we get more than 99% accuracy, the regularization paths and the weights depend on the choice of hyperparameters (we show the results for the lowest hidden dimension).


§ REFERENCES





 Paper Title: Provably Efficient Adiabatic Learning for Quantum-Classical Dynamics.






Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Simons Institute and Department of Mathematics, University of California, Berkeley, CA 94704, USA

Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

correspondence: diluo@mit.edu
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
The NSF AI Institute for Artificial Intelligence and Fundamental Interactions
Department of Physics, Harvard University, Cambridge, MA 02138, USA



§ ABSTRACT 
Quantum-classical hybrid dynamics is crucial for accurately simulating complex systems where both quantum and classical behaviors need to be considered. However, coupling between classical and quantum degrees of freedom and the exponential growth of the Hilbert space present significant challenges. Current machine learning approaches for predicting such dynamics, while promising, remain unknown in their error bounds, sample complexity, and generalizability. In this work, we establish a generic theoretical framework for analyzing quantum-classical adiabatic dynamics with learning algorithms. Based on quantum information theory, we develop a provably efficient adiabatic learning  (PEAL) algorithm with logarithmic system size sampling complexity and favorable time scaling properties. We benchmark PEAL on the Holstein model, and demonstrate its accuracy in predicting single-path dynamics and ensemble dynamics observables as well as transfer learning over a family of Hamiltonians. Our framework and algorithm open up new avenues for reliable and efficient learning of quantum-classical dynamics.



Provably Efficient Adiabatic Learning for Quantum-Classical Dynamics
    Di Luo
    August 5, 2024
====================================================================










Introduction.— Efficient simulation of quantum-classical hybrid dynamics is crucial to multi-scale modelings of a wide range of physical systems, opening new avenues for advancements in material science, chemistry, and drug discovery by providing a more comprehensive understanding of molecular interactions <cit.>. A common hybrid dynamics approach relies on the adiabatic approximation, where two well-separated timescales of a system allow one to treat the slow dynamics classically while quantum calculations are used to solve the fast, often electronic, degrees of freedom that adiabatically follow the classical dynamics. A well-known example is the Born-Oppenheimer approximation which is widely used in ab initio molecular dynamics <cit.>.
The significance of quantum-classical hybrid dynamics lies in its potential to revolutionize how we model and predict the behavior of complex systems, ultimately pushing the frontiers of both fundamental research and practical applications.


The simulations of quantum-classical dynamics, however, is computationally challenging due to not only an exponentially large Hilbert space of quantum sub-systems and repeated time-consuming quantum calculations at every time step, but also the nonlinear differential equation coupled both the quantum and the classical variables. In the past two decades, machine learning (ML) has emerged as a powerful tool in developing force fields and inter-atomic potentials for ab initio molecular dynamics <cit.>. ML force-field approaches have recently been generalized to enable large-scale dynamical simulations of condensed-matter lattice systems <cit.>. 
This approach leverages massive datasets of quantum mechanical results to train models that can predict the potential energy surfaces with high precision.  ML-enhanced force fields facilitate the simulation of large intricate systems by accurately capturing the essential quantum mechanical effects while maintaining computational efficiency. Despite intensive studies and wide applications of ML force field models over past decades, critical questions on the error bound, sample complexity and generalizability of the ML methods have remained unresolved. 

In this work, we establish a generic theoretical framework for analyzing quantum-classical adiabatic dynamics with learning algorithms. We start with the analysis of the approximately constant linear model, derive the error bounded condition for the non-linear model, and introduce the relaxation method to check the error bounded property for a generic model, which lays down a solid foundation for the reliability of learning algorithms in quantum-classical adiabatic dynamics. Inspired by the recent development of quantum information theoretic learning theory for quantum many-body systems <cit.>, we develop a provably efficient adiabatic learning (PEAL) algorithm for quantum-classical dynamics, which offers a sample complexity scaling logarithmically with the system size and favorable scaling of evolution time. We benchmark our algorithm on the Holstein model and demonstrate accurate prediction of the single path dynamics and ensemble dynamics observables, as well as transfer learning across different couplings between quantum and classical degrees of freedom. 



 


Adiabatic Quantum-Classical Dynamics Learning.— We consider a general quantum-classical Hamiltonian:


    Ĥ = Ĥ_q + ∑_α,i g_αÔ_α,i G_α,i(P,Q) + H_cl(P,Q),

where Ĥ_q is the Hamiltonian operator for the quantum degrees of freedom, g_α is the quantum-classical coupling coefficient, α is the index of different types of couplings, i∈{1,…,L} is the label of local regions (e.g. lattice sites), Ô_α,i is the quantum operator that enters the coupling, G_α,i(P,Q) is a general function of the classical degree of freedom Q=(Q_1,…,Q_L) and its canonical momentum P=(P_1,…,P_L), and H_cl(P,Q) is the classical Hamiltonian.


We study the adiabatic evolution dynamics driven by the quantum-classical Hamiltonian in Eq. (<ref>). In this adiabatic limit, similar to the Born-Oppenheimer approximation in quantum molecular dynamics, the quantum subsystem is assumed to quickly relax to the ground state of the total Hamiltonian Ĥ, while the classical degrees of freedom follow the Hamilton's equations of motion (EOM):

    d/dt Q_j    = ∑_α,i g_α⟨Ô_α,i⟩∂/∂ P_j G_α,i + ∂/∂ P_j H_cl, 
        
    d/dt P_j    = -∑_α,i g_α⟨Ô_α,i⟩∂/∂ Q_j G_α,i - ∂/∂ Q_j H_cl - γ P_j,

where j∈{1,…,L}, ⟨·⟩ is the ground state expectation, and γ > 0 is the damping coefficient due to dissipation.

The conventional way to solve the above equations is through an iterative scheme, where one first updates {Q_j(t), P_j(t)} based on the classical EOM, and then solve the ground state from Ĥ. The updated ground state is used to compute expectation values ⟨Ô_α,i⟩ which determine the driving terms of the EOM for the next step. However, repeated ground-state calculations of Ĥ at every time step could be computationally costly using quantum state solvers (QSS), such as exact diagonalization (ED) <cit.>, density functional theory <cit.>, DMRG <cit.>, neural network <cit.>, and quantum computers <cit.>  . 


We consider a general quantum-classical ML model for learning such dynamics, which we call Adiabatic Dynamics Model Learning (ADML). ADML consists of two components, which uses machine learning to predict the quantum observables and evolves the classical observables using classical numerical schemes. The ML force-field models widely used in quantum molecular dynamics can be viewed as special classes of ADML <cit.>. Our goal here is to predict the adiabatic dynamics of Eq. (<ref>) with a learning-based approach. Given access to a dataset ⋃_s=1^N_s{(g_α, P(t), Q(t), ⟨Ô⟩(t))_s : t∈𝒯_s}, where 𝒯_s is the set of sampled time steps, N_s is the number of initial conditions {(g_α, P(t_init), Q(t_init))_s}_s=1^N_s sampled from a distribution 𝒟_init, and Ô stands for Ô_α,i in Eq. (<ref>) or other operators of insterest but not in the Hamiltonian, the task is to design ADML for predicting the dynamics starting with other { (g_α, P(t_init), Q(t_init))} from 𝒟_init. 
In the following, we analyze and derive the error bounded conditions for ADML.



(i) Approximately constant linear model. To serve as a starting point, we consider a simple example of Eq. (<ref>),
Ĥ = Ĥ_q + ∑_i g Ô_i Q_i + ∑_i( 1/2M P_i^2 + 1/2 k Q_i^2 ),
where we only consider one type of quantum-classical coupling with G_α,i(P,Q)=Q_i, and the classical Hamiltonian H_cl(P,Q) is for simple harmonic oscillators with mass M and spring constant k. Further assuming that during the dynamical process we are interested in, the response ∂⟨Ô_i⟩/∂ Q_i is approximately a constant, and the off-diagonal response ∂⟨Ô_i⟩/∂ Q_j (i≠ j) is approximately zero, the EOM of the system is then reduced to that of independent simple harmonic oscillators. 


Focusing on the classical degree of freedom Q_i, we can view the quantum-classical coupling g Ô_i Q_i as a driving force on the oscillator. Since the EOM in Eq. (<ref>) and (<ref>) are approximately linear in this case, the accumulated momentum and position errors between ML and the exact simulation using QSS, p(t) = P_i,ML(t)-P_i,Exact(t), q(t) = Q_i,ML(t)-Q_i,Exact(t) (i index suppressed), also satisfy a similar EOM:

    d/dt q(t)    = 1/M p(t),
        
    d/dt p(t)    = F(t) - K q(t) - γ p(t) + o(q(t))
        ,

where we define the error force F(t) = - g δ⟨Ô_i⟩(t) and the error stiffness K = g ∂⟨Ô_i⟩/∂ Q_i + k. δ⟨Ô_i⟩(t) = ⟨Ô_i⟩_ML(Q_ML(t)) - ⟨Ô_i⟩_Exact(Q_ML(t)) is the ML prediction error at the t-th time step. o(q(t)) means higher order terms in q(t) (See Supplemental Material for the derivation). Given a bounded F(t), even if it is tuned to drive the oscillator optimally, as long as K,γ > 0, the oscillator cannot be driven to infinite amplitude, i.e. the accumulated momentum and position errors are bounded. For F(t) to be bounded, it suffices to have δ⟨Ô_i⟩(t) bounded, which motivates us to define an Error Bounded Property such that when δ⟨Ô_i⟩(t) is bounded, p(t) and q(t) are also bounded. More precisely, we define the following


    A model satisfies the Error Bounded Property with respect to δ⟨Ô_i⟩(t) for t∈[t_init,t_end], if and only if the following claim is true: ∃  C_q, C_p>0 such that ∀ϵ>0, if ∀ t∈[t_init,t_end], |δ⟨Ô_i⟩(t)|^2≤ϵ, then there are |q(t)|≤ C_q√(ϵ) and |p(t)|≤ C_p√(ϵ), ∀ t∈[t_init,t_end].



    The approximately constant linear model satisfies the Error Bounded Property if K>0.

The proof of Prop. <ref> is in Supplemental Material.



(ii) Non-linear model. We can generalize the approximately constant linear model to allow non-linearity. We drop the assumption that the response ∂⟨Ô_i⟩/∂ Q_i is approximately a constant, allow G_α,i(P,Q) to be a non-linear function of Q_i, and allow the potential 1/2k Q_i^2 to include non-quadratic component in Q_i, as long as we can Taylor expand the potential at its minimum. These generalizations can be absorbed by a redefined F(t) and a time-dependent error stiffness K(t). Unlike the approximately constant linear model, the oscillator could have infinite amplitude even if K(t)>0 for all the time. We present a condition in Supplemental Material which guarantees a bounded bounded in the worst case scenario. Summarized as an informal theorem, we have:



 



    If K(t)>M(γ/2)^2+C with a positive constant C for all t, and the error stiffness K(t)'s fluctuation, as well as G_α,i(P,Q) and its first derivatives are bounded, then the non-linear model satisfies the Error Bounded Property in Def. <ref>.




The proof for Thm. <ref> is provided in Supplemental Material. We can further apply it to the Hamiltonian in normal mode with the quadratic momentum under Fourier transform, which could appear in a more general setup. 




(iii) General Relaxation Method. We now come back to the most general ADML. We allow arbitrary G_α,i(P,Q) and H_cl(P,Q) in Eq. (<ref>), and we make no assumption on the response ∂⟨Ô_i⟩/∂ Q_j. Because in the most general case the classical degrees of freedom are no longer decoupled, we restore the i index of the accumulated momentum and position errors, p_i(t) = P_i,ML(t)-P_i,Exact(t), q_i(t) = Q_i,ML(t)-Q_i,Exact(t). 


For a general ADML, the EOM for the errors q_i(t), p_i(t) can in general be derived from Eq. (<ref>) and (<ref>):

    d/dt q_i    = ∑_j ( 𝒦_q_j,q_i q_j + 𝒦_p_j,q_i p_j ) + ℱ_q_i + o(q,p),
        
    d/dt p_i    = ∑_j ( 𝒦_q_j,p_i q_j + 𝒦_p_j,p_i p_j ) + ℱ_p_i - γ p_i + o(q,p),

where the error stiffness matrix 𝒦_q(p)_j,q(p)_i and the error force vector ℱ_q(p)_i depend on (P,Q), and ℱ_q(p)_i is linear in δ⟨Ô_α, i⟩(t) (see Supplemental Material for details). 

While it is difficult to write down an error bounded condition for a general ADML, we propose a relaxation method to provide a sufficient (but not necessary) condition to check the Error Bounded Property in Def. <ref>.
The idea is to consider the worst case scenario. If in the worst case scenario the error still converges, then it is safe to use ADML to accelerate our dynamical simulation. The relaxation method is as follows. First, we require 𝒦 elements, as well as G_α,i(P,Q) and its first derivatives, are bounded. Second, assuming |δ⟨Ô_α,i⟩(t)|^2≤ϵ, we identify possible upper and lower bounds for the elements in 𝒦 and ℱ. 
The bounds just only need to be effective during the time range of the simulation. Third, to achieve the worst case scenario, we want to maximize d q_i/dt when q_i is positive, and minimize when negative (the same for p_i). Therefore, we insert the upper bound of 𝒦_q(p)_j,q(p)_i when q(p)_i and q(p)_j have the same sign, and the lower bound if the opposite sign. ℱ_q(p)_i are adjusted to their upper or lower bounds accordingly. Fourth, we perform a classical simulation of the EOM in Eq. (<ref>) and (<ref>), with the worst case scenario stated above. Finally, if the simulation shows there exist constants C_q_i and C_p_i such that for any ϵ>0, there are |q_i(t)|≤ C_q_i√(ϵ) and |p_i(t)|≤ C_p_i√(ϵ) during the time range of interest, then the Error Bounded Property is verified with the relaxation method. 





Provably Efficient Adiabatic Learning.— Next, we present a provably efficient learning algorithm for the above ADML models based on quantum information theory, which we call Provable Efficient Adiabatic Learning (PEAL). The PEAL algorithm, equipped with a learning model ℳ and a classical ordinary differential equation (ODE) solver, consists of the following steps. 

(i) Data collection for training. We sample a set of N_s initial conditions {(g_α, P(t_init), Q(t_init))_s}_s=1^N_s from a distribution 𝒟_init. 
We evolve the system with QSS and ODE solver to get (P(t), Q(t), ⟨Ô⟩(t))_s. 
For each s, we sample a set of time steps 𝒯_s uniformly from [t_init, t_end]. 
(ii) Model training. We use the dataset ⋃_s=1^N_s{(g_α, P(t), Q(t), ⟨Ô⟩(t))_s : t∈𝒯_s} to train a model ℳ:(g_α, P, Q)↦⟨Ô⟩ with the learning algorithm developed in <cit.>. 
(iii) Prediction. For any unseen new initial condition (g_α, P(t_init), Q(t_init))_new∼𝒟_init, PEAL outputs the dynamical trajectory (P(t), Q(t), ⟨Ô⟩(t))_new for t∈[t_init, t_end], by alternatively updating ⟨Ô⟩(t) with ℳ and P(t), Q(t) with ODE solver, integrated with our symmetry-preserving techniques shown later. The sample complexity and error bounds are summarized in the following informal theorem.




    
   When the Error Bounded Property in Def. <ref> is satisfied, for T time steps quantum-classical adiabatic dynamics of an n-qubit gapped system, with sample complexity O((n)), 
   PEAL gives rise to controllable accumulated errors of classical variables and all k-local, bounded quantum observables scaling as (i) O(√(T)) for generic model ℳ (ii) O(√(log T)) for sub-Gaussian ℳ's prediction error (iii) independent on T for bounded ℳ's prediction error. 




We leave the proofs for the above theorem in Supplemental Material. For Theorem <ref>, it can also be applied to an n-qubit system with ground states of exponential-decay correlation functions using recent results <cit.>. We highlight that PEAL works for unseen couplings g_α,new∉{g_α,s}_s=1^N_s, demonstrating transfer learning over a family of Hamiltonian. Furthermore, PEAL can predict any k-local observable Ô even if it does not appear in the Hamiltonian. 





Numerical Experiments.— We consider the Holstein model <cit.> which describes the electron phonon interaction as follows:

    H = - ∑_i,j c_i^† c_j - g ∑_i (c_i^†c_i - 1/2) Q_i + ∑_i(P_i/2M + k Q_i^2/2),

where c_i is related to the fermionic degree of freedom, while Q_i and P_i are the position and momentum of the phonon as classical degrees of freedom. 

In the numerical experiment, we study such model on a 1D periodic chain with L=50 sites. We set M=k=1 and the damping coefficient γ=0.1. The electronic degree of freedom at time t will be the ground state of H_e({Q_i(t)}) due to the fast relaxation, showing a 1D strong Anderson localization <cit.>. Therefore, PEAL's requirement of a ground state with exponential-decay correlation functions is satisfied. Analytically, the 1D adiabatic Holstein model always cools down to a charge density wave (CDW) in zero temperature. In Supplemental Material, we derive the relation between phonon amplitude and CDW response function, showing the error stiffness K is always positive when it's close to a checkerboard configuration. We also numerically measure the error stiffness K(t) during the time range of interest, confirming the Error Bounded Property (see Supplemental Material). Therefore, we can apply PEAL to the 1D adiabatic Holstein model with a bounded error guarantee during the dynamics. 

 



Since Eq. (<ref>) has the U(1) and translation symmetry, we develop a symmetry-preserving PEAL. The U(1) global symmetry is respected by conserving the total electron density, and the translation symmetry by applying the same model on all sites. The U(1) symmetry preserving is a new feature in our PEAL that does not exist in previous literature and we have shown that the symmetry-preserving PEAL maintains provably efficient error bound in Supplemental Material.

 



We begin with the single-path prediction task by training a model using some initial conditions and certain g values. The goal is to predict observable dynamics from different initial conditions under both the training g values (standard learning) and unseen g values (transfer learning). Here, we choose g∈ G_SL = {1.3, 1.32, 1.34, 1.36, 1.38, 1.4} for training. In Figure <ref>, we demonstrate the single-path prediction by PEAL. The solid curves are the exact simulation using exact diagonalization (ED) for QSS and RK4 for classical ODE solver <cit.>, and the dashed curves are with PEAL. Blue curves are for g=1.4 (standard learning), and orange curves for g=1.39 (transfer learning). We present the time evolution of six different observables during the dynamics, which are the total charge density wave (CDW) ∑_i(-1)^i n_i, the electron density at the first site n, the phonon amplitude at the first site Q, the phonon momentum at the first site P, the hopping term ⟨ c_0^† c_1⟩, and the next-nearest-neighbor correlation ⟨ c_0^† c_2⟩. We note that CDW is a sum of local observables, ⟨ c_0^† c_1⟩ does not enter classical EOM, and ⟨ c_0^† c_2⟩ does not even appear in the Hamiltonian. Nevertheless, all observables in the PEAL curves agree well with their corresponding exact simulations, demonstrating that PEAL guarantees a controllable error for k-local observable and well performs in transfer learning. 

In Figure <ref>, we present the sample complexity of PEAL. We use the same training data in the single-path prediction, build the test set with g∈ G_SL but with different initial conditions, and demonstrate transfer learning with g∈ G_TL = {1.31, 1.33, 1.35, 1.37, 1.39}. For illustration, we consider the root mean square of |n_PEAL-n_Exact| as the test error. The test errors for both standard learning and transfer learning show a logarithmic scaling law with the number of samples. The inner panels also demonstrate that n_PEAL in standard learning and transfer learning match well with n_Exact.



Generically, the ensemble dynamics (i.e. the dynamics average over an ensemble of paths from different initial conditions) can provide more information of the underlying physics. In Figure <ref>, we compare the ensemble averages of the correlation function Q_i(t)Q_j(t) at time t=100 for g=1.4 and g=1.7. Here, g=1.4 prediction is based on PEAL training in g ∈ G_SL, and g=1.7 prediction is based on another model from PEAL training in g∈{1.6,1.62,1.64,1.66,1.68,1.7} where long-lasting domain walls exist. For g=1.4, the system has cooled down to a pure checkerboard configuration, and shows an oscillating correlation function. For g=1.7, since the system has domain walls, it shows a decaying correlation function and a large variance. PEAL provides a good agreement with the exact simulations for both the mean and the variance over different g values. 






Conclusion.— In this work, we establish a comprehensive theoretical framework for analyzing quantum-classical adiabatic dynamics using learning algorithms. We provide a systematic analysis for the error bounded properties of the approximately constant linear model, the non-linear model, and generic models, ensuring the reliability of ADML. We develop a provably efficient adiabatic learning algorithm PEAL, demonstrating logarithmic scaling of sample complexity with system size and favorable evolution time scaling. Benchmarking PEAL on the Holstein model, we achieve accurate predictions of single-path dynamics and ensemble dynamics observables, with effective transfer learning across various quantum-classical coupling strengths. Our framework and algorithm opens up new directions for efficient learning in quantum-classical dynamics, including applying PEAL to quantum experiments, integrating advanced machine learning techniques, and extending to generic non-equilbrium processes.





Acknowledgements—  The authors acknowledge helpful discussion with Max Metlitski and Zhi Ren. DL acknowledges support from the NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI). JPL acknowledges support by the National Science Foundation (PHY-1818914, CCF-1729369), the NSF Quantum Leap Challenge Institute (QLCI) program (OMA-2016245, OMA-2120757), and a Simons Foundation award (No. 825053). GWC acknowledges the support of the US Department of Energy Basic Energy Sciences under Contract No. DE-SC0020330.
The authors acknowledge the MIT SuperCloud for providing HPC resources that have contributed to the research results reported within this paper.



  
  

 

  






    § SUPPLEMENTAL MATERIAL FOR PROVABLY EFFICIENT MACHINE LEARNING FOR ADIABATIC QUANTUM-CLASSICAL DYNAMICS
    
    
    § I. DIMENSIONLESS MODEL
    
    
    Consider the standard spinless Holstein model:
    
        H = -t_nn∑_⟨ ij ⟩ c_i^† c_j - g ∑_i ( c_i^† c_i - 1/2) Q_i + ∑_i ( P_i^2/2M + kQ_i^2/2).
    
    
    And the equation of motion for the phonons is
    
        dQ_i/dt = P_i/M,   dP_i/dt = g n_i - k Q_i,
    
    where n_i = ⟨ c_i^† c_i ⟩ is the on-site fermion number. The mass m and elastic constant k are related by the familiar formula,
    
        ω = √(k/M),
    
    The inverse ω^-1 gives a characteristic time scale for the dynamical problem. Next, one can introduce a “length scale”Q_0 for the displacement of the simple harmonic oscillator. The energy related to Q at a given site is
    
        E(Q) = -g n_i Q_i + k Q^2/2.
    
    
    Assuming electron number n ∼ 1, minimization with respect to Q gives ∂ E/∂ Q |_Q_0 = 0:
    
        Q_0 ∼g/k.
    
    
    From this one can then introduce a scale for the momentum via the relation dQ/dt = P/M
        ω Q_0 ∼P_0/M P_0 = M ω Q_0 = M ω g/k.
    
    
    We can now define the dimensionless time, displacement and momentum as
    
        t̃ = ω t,   Q̃_i = Q_i/Q_0,   P̃_i = P_i/P_0.
    
    
    In terms of dimensionless quantities, the equation of motion is then simplified to
    
        dQ̃_i/dt̃ = P̃_i,   dP̃_i/dt̃ = n_i - Q̃_i.
    
    
    Next, we consider the tight-binding Hamiltonian for the fermions for a given {Q_i} configuration. We factor out the nearest-neighbor hopping constant t_nn and use it as the unit for energy. Also, we use the dimensionless Q̃_i:
    
        H_TB = t_nn[ -∑_⟨ ij ⟩ c_i^† c_j + g Q_0/t_nn∑_i Q̃_i c_i^† c_i ].
    
    
    The coefficient of the second term above gives an important dimensionless parameter for Holstein model. Instead of t_nn, we can introduce the bandwidth of the tight-binding model: W = 4t_nn for the 1D model. We then define a dimensionless electron-phonon coupling
    
        λ = g Q_0/W = g^2/kW.
    
    
    The dimensionless tight-binding Hamiltonian then becomes
    
        H̃_TB = -∑_⟨ ij ⟩ c_i^† c_j + 4λ∑_i Q̃_i c_i^† c_i.
    
    
    One can see that, using these dimensionless quantities, the only adjustable parameter of the adiabatic dynamics of the Holstein model is this dimensionless λ. In general, for real materials λ≲ 1. For example, we can set it to λ = 0.5 or 0.25 in the simulations.
    
    
    
    
    § II. HOLSTEIN MODEL CHARGE DENSITY WAVE RESPONSE ANALYSIS
    
    
    Consider a tight binding model on a 1D lattice:
    
        H    = -t ∑_i (c_i^† c_i+1 + h.c.) - g ∑_i (-1)^i Q c_i^†c_i,
    
    where i∈{0,1,…,L-1}, L is the lattice size and is even. The lattice has periodic boundary condition. The effective potential on the lattice is staggered: [-gQ, +gQ, -gQ, +gQ, …]. The unit cell consists of 2 lattice sites. 
    
    We can solve the single-particle wavefunctions using the ansatz
    
        |ψ_k⟩ = (a, b e^ik, a e^2ik, b e^3ik, …)^T,    k∈{0, 2π/L, …, 2π/L(L/2-1)}.
    
    Note that the range of k is halved because the unit cell is doubled.
    
    The eigenvalue equation H|ψ_k⟩ = E|ψ_k⟩ becomes
    
        - g Q a - 2(cos k) b    = E a, 
        
            - (2cos k) a + g Q b    = E b,
    
    which requires E=±√((gQ)^2+(2cos k)^2) to have nontrivial solutions. 
    
    The solutions form two bands. At half filling, all the states in the lower band, which has negative E, are filled with a particle. The charge density wave amplitude for |ψ_k⟩ is n_k=(a^2-b^2)/2. From the eigenvalue equation and the normalization (L/2)(a^2+b^2)=1, we can solve the CDW amplitude for |ψ_k⟩:
    
        n_k = sgn(gQ)/L√((2cos k/g Q)^2 + 1),
    
    where sgn(gQ) is the sign of gQ.
    
    The total charge density wave at half filling is 
    
        n = ∑_k n_k = ∑_k sgn(gQ)/L√((2cos k/g Q)^2 + 1).
    
    The range of k has been described above.
    
    In the infinite lattice limit, L→∞, the sum over k turns into an integral 
    
        n →∫_0^πdk/2πsgn(gQ)/√((2cos k/g Q)^2 + 1) = sgn(gQ) EllipticK[-(2/gQ)^2]/π,
    
    where EllipticK[m]≡π/2_2 F_1(1/2, 1/2;1;m) is the complete elliptic integral of the first kind, _2 F_1(a, b;c;x) is the hypergeometric function <cit.>.
    
    When gQ→ 0, the result says n→ 0. However, the derivative ∂ n/∂ (gQ) diverges at zero as log (1/gQ). More precisely, n∼ -gQ log (gQ) for small gQ.
    
    The property of n as a function of gQ is important, especially for gQ close to zero. This is because when the system stabilizes, the equation of motion tells us k_springQ = g n, i.e. the forces are balanced. This means a straight line n=(k_spring/g^2)gQ that crosses the origin. The number of crossing points between this straight line and the n vs. gQ function determines whether the system have a stable CDW in the long time. Figure <ref> shows the function curves for different lattice size.
    
    
     
    
    For example, in the infinite lattice limit, the derivative diverges at zero. Therefore, any straight line crossing the origin with a finite slope will cross the function at some other points. This means that there always exists a stable CDW for the infinite lattice. However, we can estimate how large the CDW amplitude should be for a small g. Combine the straight line n=(k_spring/g^2)gQ and the asymptotic behavior n∼ -gQ log (gQ), we can solve a non-trivial crossing point at the phonon amplitude Q∼ e^-k_spring/g^2/g. We see that the amplitude decays exponentially when g→0.
    
    However, for a finite system size L, the situation is different. For L=4N, the function n has a finite limit ±1/L when gQ→0, and has a discontinuity at 0. Therefore, a straight line crossing the origin always has non-trivial intersections with the function curve. i.e. There exists CDW with amplitude at least 1/L for all values of g. For L=4N+2, the function n goes to zero when gQ→0, and has a finite slope at 0. When g is small, (k_spring/g^2) exceeds this slope, the straight line only has intersection with the function curve at zero. When g is large, (k_spring/g^2) smaller than this slope, there are non-trivial intersections. Therefore, there is a phase transition due to the finite system size: When g<g_crit, no CDW; when g>g_crit, there is CDW. The system size we study, L=50, is in this situation.
    
    We can analytically solve g_crit. We can take the derivative of Eq. <ref> at the limit to zero:
    
        .∂ n/∂(gQ)|_gQ→ 0^+ = ∑_k 1/2L|cos k|.
    
    By identifying (k_spring/g_crit^2) with this slope, we get
    
        g_crit = √(2k_springL/∑_k 1/|cos k|).
    
    Some numerical values for k_spring=1 are listed in Table. <ref>
    
    For our choice L=50, g_crit=1.1524. The domain wall formation g value for L=50 is around 1.6. Therefore, we pick g∈[1.3,1.4] in our numerical experiment, to avoid the finite size effect and the domain walls.
    
    In the main text, we see the bounded error requires k_spring - g∂ n/∂ Q > 0. We can prove this is true when the system is close to the stable CDW configuration. We apply the self-consistency equation k_spring=gn/Q. From the concavity of the n vs. gQ function when gQ>0, and the convexity when gQ<0, we see n/gQ>∂ n/∂ (gQ) as long as gQ≠ 0. Therefore, k_spring - g∂ n/∂ Q = g^2 (n/gQ - ∂ n/∂ (gQ)) > 0 almost surely, because the gQ=0 case has measure 0. This completes our proof.
    
    
    
    § III. FROM THE EOM OF Q, P TO THE EOM OF Q, P
    
    
     §.§ Approximately constant linear model
    
    
    In the main text, we introduced the “approximately constant linear model”, which has the Hamiltonian: 
    
        Ĥ = Ĥ_q + ∑_i g Ô_i Q_i + ∑_i( 1/2M P_i^2 + 1/2 k Q_i^2 ).
    
    It has the EOM for Q_i and P_i:
    
        d/dt Q_i(t)    = 1/M P_i(t), 
        d/dt P_i(t)    = - g ⟨Ô_i⟩ -k Q_i(t) - γ P_i(t),
    
    where the quantum observable ⟨Ô_i⟩ is computed based on the configuration Q(t).
    
    The EOM above is equally valid for both the exact dynamical path and the ML dynamical path:
    
        d/dt Q_i,Exact(t)    = 1/M P_i,Exact(t), 
        d/dt P_i,Exact(t)    = - g ⟨Ô_i⟩_Exact -k Q_i,Exact(t) - γ P_i,Exact(t),
    
    and 
    
        d/dt Q_i,ML(t)    = 1/M P_i,ML(t), 
        d/dt P_i,ML(t)    = - g ⟨Ô_i⟩_ML -k Q_i,ML(t) - γ P_i,ML(t).
    
    
    Take the difference between the ML EOM and the exact EOM, define the accumulated position and momentum errors q(t) = Q_i,ML(t)-Q_i,Exact(t), p(t) = P_i,ML(t)-P_i,Exact(t), we get 
    
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = - g (⟨Ô_i⟩_ML - ⟨Ô_i⟩_Exact) -k q(t) - γ p(t).
    
    
    However, here the observables depend on different configurations, ⟨Ô_i⟩_ML = ⟨Ô_i⟩_ML(Q_ML(t)), and ⟨Ô_i⟩_Exact = ⟨Ô_i⟩_Exact(Q_Exact(t)). When we take their difference, we not only need to take care about the difference between the prediction methods, we also need to take care about the difference between the configurations Q_ML and Q_Exact. 
    
    We do a Taylor expansion and apply the assumption that the off-diagonal response ∂⟨Ô_i⟩/∂ Q_j (i≠ j) is approximately zero. We can make the difference between the observables into two terms:
    
        ⟨Ô_i⟩_ML(Q_ML(t)) - ⟨Ô_i⟩_Exact(Q_Exact(t))    = ⟨Ô_i⟩_ML(Q_ML(t)) - ⟨Ô_i⟩_Exact(Q_ML(t)) + ∂⟨Ô_i⟩/∂ Q_i q(t) + o(q(t)) 
            
           = δ⟨Ô_i⟩(t) + ∂⟨Ô_i⟩/∂ Q_i q(t) + o(q(t)),
    
    where δ⟨Ô_i⟩(t) = ⟨Ô_i⟩_ML(Q_ML(t)) - ⟨Ô_i⟩_Exact(Q_ML(t)) is the single-step prediction error.
    
    Insert the two terms back to the EOM, we get the same result as in the main text:
    
        d/dt q(t)    = 1/M p(t),
            
        d/dt p(t)    = - g δ⟨Ô_i⟩(t) - g ∂⟨Ô_i⟩/∂ Q_i q(t) - k q(t) - γ p(t) + o(q(t)).
    
    
     §.§ Non-linear model
    
    
    
    In the main text, we then generalized a bit to the non-linear model, which has the Hamiltonian: 
    
        Ĥ = Ĥ_q + ∑_i g Ô_i G_i(Q_i) + ∑_i( 1/2M P_i^2 + V_i(Q_i) ).
    
    It has the EOM for Q_i and P_i:
    
        d/dt Q_i(t)    = 1/M P_i(t), 
        d/dt P_i(t)    = - g ⟨Ô_i⟩ G_i^'(Q_i) - V_i^'(Q_i) - γ P_i(t),
    
    where the quantum observable ⟨Ô_i⟩ is still computed based on the configuration Q(t).
    
    The EOM above is equally valid for both the exact dynamical path and the ML dynamical path:
    
        d/dt Q_i,Exact(t)    = 1/M P_i,Exact(t), 
        d/dt P_i,Exact(t)    = - g ⟨Ô_i⟩_Exact G_i^'(Q_i,Exact) -V_i^'(Q_i,Exact) - γ P_i,Exact(t),
    
    and 
    
        d/dt Q_i,ML(t)    = 1/M P_i,ML(t), 
        d/dt P_i,ML(t)    = - g ⟨Ô_i⟩_ML G_i^'(Q_i,ML) -V_i^'(Q_i,ML) - γ P_i,ML(t).
    
    
    Take the difference between the ML EOM and the exact EOM, define the accumulated position and momentum errors q(t) = Q_i,ML(t)-Q_i,Exact(t), p(t) = P_i,ML(t)-P_i,Exact(t), expand to first order of q(t), we get:
    
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = - g (⟨Ô_i⟩_ML - ⟨Ô_i⟩_Exact) G_i^'(Q_i,ML) - g ⟨Ô_i⟩_Exact G_i^”(Q_i,Exact) q(t) - V_i^”(Q_i,Exact) q(t) - γ p(t) + o(q(t)).
    
    
    We apply the same technique as in the approximately constant linear model. Insert
    
        ⟨Ô_i⟩_ML - ⟨Ô_i⟩_Exact = δ⟨Ô_i⟩(t) + ∂⟨Ô_i⟩/∂ Q_i q(t) + o(q(t)),
    
    we get
    
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = - g δ⟨Ô_i⟩(t) G_i^'(Q_i,ML) - g ∂⟨Ô_i⟩/∂ Q_i G_i^'(Q_i,ML) q(t) - g ⟨Ô_i⟩_Exact G_i^”(Q_i,Exact) q(t) - V_i^”(Q_i,Exact) q(t) - γ p(t) + o(q(t)).
            
           = F(t) - K(t) q(t) - γ p(t) + o(q(t)),
    
    where we define the effective driving force and the effective spring constant as
    
        F(t)    = -g δ⟨Ô_i⟩(t) G_i^'(Q_i,ML(t)), 
        
            K(t)    = g ∂⟨Ô_i⟩/∂ Q_i G_i^'(Q_i,ML(t)) + g ⟨Ô_i⟩_Exact G_i^”(Q_i,Exact(t)) + V_i^”(Q_i,Exact(t)).
    
    
    This EOM is equivalent to a damped spring with time dependent driving force and spring constant. We can prove the amplitude of the spring will not diverge if F(t) is bounded and K(t) does not fluctuate largely.
    
    
    
     §.§ Generic model
    
    
    The generic model has the Hamiltonian 
    
        Ĥ = Ĥ_q + ∑_α,i g_αÔ_α,i G_α,i(P,Q) + H_cl(P,Q),
    
    and the EOM
    
        d/dt Q_j    = ∑_α,i g_α⟨Ô_α,i⟩∂/∂ P_j G_α,i + ∂/∂ P_j H_cl, 
            
        d/dt P_j    = -∑_α,i g_α⟨Ô_α,i⟩∂/∂ Q_j G_α,i - ∂/∂ Q_j H_cl - γ P_j.
    
    
    We can repeat the techniques above: writing down the EOM for ML and the exact simulation, making a difference of the EOM, defining q_i and p_i (here the index i cannot be suppressed), doing Taylor expansion, and combining terms up to the first order. After similar derivations, we can get the result
    
        d/dt q_i    = ∑_j [ q_j 𝒦_q_j,q_i(P,Q) + p_j 𝒦_p_j,q_i(P,Q) ] + ℱ_q_i + o(q,p),
            
        d/dt p_i    = ∑_j [ q_j 𝒦_q_j,p_i(P,Q) + p_j 𝒦_p_j,p_i(P,Q) ] + ℱ_p_i - γ p_i + o(q,p),
    
    where
    
        𝒦_q_j,q_i   = ∑_α,k g_α( ∂⟨Ô_α,k⟩/∂ Q_j∂ G_α,k/∂ P_i + ⟨Ô_α,k⟩∂^2 G_α,k/∂ Q_j ∂ P_i) + ∂^2 H_cl/∂ Q_j ∂ P_i,
            
        𝒦_p_j,q_i   = ∑_α,k g_α( ∂⟨Ô_α,k⟩/∂ P_j∂ G_α,k/∂ P_i + ⟨Ô_α,k⟩∂^2 G_α,k/∂ P_j ∂ P_i) + ∂^2 H_cl/∂ P_j ∂ P_i,
            
        𝒦_q_j,p_i   = - ∑_α,k g_α( ∂⟨Ô_α,k⟩/∂ Q_j∂ G_α,k/∂ Q_i + ⟨Ô_α,k⟩∂^2 G_α,k/∂ Q_j ∂ Q_i) - ∂^2 H_cl/∂ Q_j ∂ Q_i,
            
        𝒦_p_j,p_i   = - ∑_α,k g_α( ∂⟨Ô_α,k⟩/∂ P_j∂ G_α,k/∂ Q_i + ⟨Ô_α,k⟩∂^2 G_α,k/∂ P_j ∂ Q_i) - ∂^2 H_cl/∂ P_j ∂ Q_i,
            
        ℱ_q_i   = ∑_α,j g_αδ⟨Ô_α,j⟩(t)∂/∂ P_iG_α,j(P,Q),
            
        ℱ_p_i   = -∑_α,j g_αδ⟨Ô_α,j⟩(t)∂/∂ Q_iG_α,j(P,Q).
    
    
    § IV. PROOF OF PROPOSITION <REF>
    
    
    Here we prove that the approximately constant linear model satisfies the Error Bounded Property in Def. <ref> if K>0. We use the analogy to a damped harmonic oscillating spring, phrasing the error force F(t) in the main text as the “driving force” and the error stiffness K in the main text as the “spring constant”. For a damped spring, given a bounded driving force, even if the force is tuned to drive the spring optimally, as long as K, γ>0, the spring cannot be driven to infinite amplitude. This can be seen by the following worst-case analysis. In a worst case scenario, the driving force is set to be the maximal value in the direction of the spring movement. We know a constant driving force means a shift of the reference point of the spring. Therefore, by shifting the reference point back and forth, the driving force at most linearly increases the amplitude with the motion cycles of the spring. However, a finite damping decreases the amplitude of the spring by a constant factor in each cycle. Therefore, for a large enough initial amplitude, the decrement of the amplitude due to the damping must exceed the increment of the amplitude due to the driving force in the cycle, and thus the amplitude in the next cycle must be smaller than the initial one, which tells us that the damped spring cannot be driven to infinite amplitude.
    
    Now we analyze the relation between the maximal amplitude q, the maximal momentum p, and the maximal driving force F, using the dimensional analysis. At t_init, there are no accumulated errors, i.e. q(t_init)=p(t_init)=0. Therefore, the only physical quantity that carries the dimension of length is the maximal driving force [F]=[M][L][T]^-2, where [M], [L], and [T] are the dimension of mass, length, and time, respectively. Note that the maximal amplitude [q]=[L] and the maximal momentum p=[L][T]^-1 both carries power one of the length dimension [L]. Therefore, by dimensional analysis, they are both proportional to the maximal driving force F. i.e. ∃ C_q, C_p such that q = C_q F, p = C_p F.
    
    For any ϵ>0, if |δ⟨Ô_i⟩(t)|^2≤ϵ, by the definition F(t)=-gδ⟨Ô_i⟩(t), we have |F(t)|≤ |g|√(ϵ). By the analogy above, it means that the maximal force F=|g|√(ϵ). Therefore, by the meaning of the maximal amplitude and the maximal momentum, there are |q(t)|≤q=C_q F=C_q|g|√(ϵ) and |p(t)|≤p=C_p F=C_p|g|√(ϵ). Define two new constants C_q = |g|C_q and C_p = |g|C_p, we get |q(t)|≤ C_q√(ϵ) and |p(t)|≤ C_p√(ϵ), which is what we want to show. This completes the proof of Prop. <ref>. 
    
    § IV. PROOF OF THEOREM <REF> (ERROR BOUNDED CONDITION FOR NON-LINEAR MODEL)
    
    
    
    Here we present the formal version of Thm. <ref> in the main text. We use the analogy to a damped harmonic oscillating spring, phrasing the error force F(t) in the main text as the “driving force” and the error stiffness K(t) in the main text as the “spring constant”.
    
    
    
        Consider a damped harmonic oscillator with the following EOM:
        
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = F(t) - K(t) q(t) - γ p(t) + o(q(t)),
    
        where q is the classical coordinate, p is the classical momentum, t is time, M is the mass of the oscillator, F(t) is a time-dependent driving force, K(t) is a time-dependent spring constant, γ is the damping coefficient.The initial condition is q(t_init)=p(t_init)=0.
    
        Given K_max>K_min>M(γ/2)^2 satisfying the following inequality:
        
        K_max/K_min < exp[2(arctan ω_min/ω_min + π - arctan ω_max/ω_max)],
    
        where ω_max=√(K_max/M(γ/2)^2-1), ω_min=√(K_min/M(γ/2)^2-1).
    
        If K(t)∈[K_min,K_max] for all t, and G_α,i(P,Q) and its first derivatives are bounded, then the Error Bounded Property in Def. <ref> is satisfied, i.e. ∃ C_q, C_p>0 such that ∀ϵ>0, if |δ⟨Ô_i⟩(t)|^2≤ϵ for all t, then there are 
        
        |q(t)| ≤ C_q√(ϵ),    |p(t)| ≤ C_p√(ϵ),   for all  t.
    Proof. Consider a damped harmonic oscillator with the following EOM:
    
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = F(t) - K(t) q(t) - γ p(t) + o(q(t)),
    
    where q is the classical coordinate, p is the classical momentum, t is time, M is the mass of the oscillator, F(t) is a time-dependent driving force, K(t) is a time-dependent spring constant, γ > 0 is the damping coefficient. We assume F(t)∈[-F,F], K(t)∈[K_min, K_max], where K_max > K_min > M(γ/2)^2. Note that with Eq. <ref>, F(t) = -g δ⟨Ô_i⟩(t) G_i^'(Q_i,ML(t)), the assumption |δ⟨Ô_i⟩(t)|^2≤ϵ implies F≤ O(√(ϵ)) given bounded G_i^'.
    
    We consider the worst case scenario. The mass on the spring starts from one side with a large amplitude. As the mass moving towards the other side, the driving force always keeps the maximal value towards the other side. Before the mass crosses the origin, the spring constant is set as K_max to maximize the drag towards the other side. After the mass crosses the origin, the spring constant is set as K_min to minimize the burden for its moving as far as possible. In this worst case scenario, the mass will stop at its largest amplitude on the other side. If this amplitude on the other side is smaller than the one it started with, than the spring cannot have diverging amplitude, and therefore the error converges.
    
    We now analyze this worst case scenario. Assume the mass starts at q(-t_0)=-q_0, q(-t_0)=0. At time t=0 it crosses the origin, q(0)=0, with a velocity q (0) = v. At time t=t_1 it stops on the other side at q(t_1)=q_1, q(t_1)=0.
    
    Before the mass crosses the origin, the spring constant is K_max. The EOM is
    
        q(t) = - γq(t) -K_max/M q(t) + F/M,    t∈[-t_0, 0].
    
    The EOM has a general solution
    
        q(t) = A e^-γ/2 tsinΩ_max t + B e^-γ/2 tcosΩ_max t + F/K_max,
    
    where Ω_max = √(K_max/M - (γ/2)^2).
    
    Inserting the boundary condition at t=0, we get equations
    
        q(0)    = 0 = B + F/K_max, 
        q(0)    = v = Ω_max A - γ/2 B,
    
    from which we solve
    
        A    = 1/Ω_max(v-γ/2F/K_max), 
        
            B    = -F/K_max.
    
    
    Inserting the boundary condition at t=-t_0, we get equations
    
        q(-t_0)    = -q_0 = - A e^γ/2 t_0sinΩ_max t_0 + B e^γ/2 t_0cosΩ_max t_0 + F/K_max, 
        q(-t_0)    = 0 = A (γ/2 e^γ/2 t_0sinΩ_max t_0 + Ω_max e^γ/2 t_0cosΩ_max t_0 ) + B( - γ/2 e^γ/2 t_0cosΩ_max t_0 + Ω_max e^γ/2 t_0sinΩ_max t_0 ).
    
    We can solve t_0 from the second equation:
    
        t_0    = 1/Ω_maxarccot(γ/2)A+Ω_maxB/-Ω_maxA+(γ/2)B
           = 1/Ω_maxarccot(-(γ/2)v - F/M/Ω_maxv),
    
    where we have inserted the solution of A and B. We can then insert everything into the first equation to solve q_0, but we do not do it now.
    
    Let's also take a look at the time after the mass crosses the origin. The spring constant changes to K_min. The EOM is 
    
        q(t) = - γq(t) -K_min/M q(t) + F/M,    t∈[0, t_1].
    
    The EOM has a general solution
    
        q(t) = C e^-γ/2 tsinΩ_min t + D e^-γ/2 tcosΩ_min t + F/K_min,
    
    where Ω_min = √(K_min/M - (γ/2)^2).
    
    Inserting the boundary condition at t=0, we get equations
    
        q(0)    = 0 = D + F/K_min, 
        q(0)    = v = Ω_min C - γ/2 D,
    
    from which we solve
    
        C    = 1/Ω_min(v-γ/2F/K_min), 
        
            D    = -F/K_min.
    
    
    Inserting the boundary condition at t=t_1, we get equations
    
        q(t_1)    = q_1 = C e^-γ/2 t_1sinΩ_min t_1 + D e^-γ/2 t_1cosΩ_min t_1 + F/K_min, 
        q(t_1)    = 0 = C ( - γ/2 e^-γ/2 t_1sinΩ_min t_1 + Ω_min e^-γ/2 t_1cosΩ_min t_1 ) + D( - γ/2 e^-γ/2 t_1cosΩ_min t_1 - Ω_min e^-γ/2 t_1sinΩ_min t_1 ).
    
    We can solve t_1 from the second equation:
    
        t_1    = 1/Ω_minarccot(γ/2)C+Ω_minD/Ω_minC-(γ/2)D
           = 1/Ω_minarccot(γ/2)v - F/M/Ω_minv,
    
    where we have inserted the solution of C and D. Note that comparing to the case before, there is no minus sign in the arccot function. 
    
    To have a bounded error, we want that for large enough q_0, there is q_1/q_0<1, where q_0 is the starting amplitude and q_1 is the stopping amplitude. i.e. The amplitude never diverges. Iteratively we consider q_1 as the next starting amplitude, and we can see the sequence of amplitudes converges in the long time. Note that the amplitude has the dimension of length, and among all given parameters, only the maximal force F carries the dimension of length. Using dimension analysis, we see that the maximal amplitude the system could reach will be proportional to the maximal force F. The maximal momentum the system could reach will also be proportional to F from dimension analysis. 
    i.e. q(t) ≤ O(F), p(t) ≤ O(F). Combining with the assumption F≤ O(√(ϵ)), it leads to q(t) ≤ O(√(ϵ)), p(t) ≤ O(√(ϵ)), which is what we want to show.
    
    
    Now the final step is to simplify the condition q_1/q_0<1. Note that for arbitrarily large q_0, the velocity v can also be arbitrarily large. Therefore, we can consider the limit F/v→ 0. Under this limit,
    
        A/v    →1/Ω_max, 
        
            B/v    → 0, 
        
            C/v    →1/Ω_min, 
        
            D/v    → 0, 
        
            t_0    →1/Ω_maxarccot(-(γ/2)/Ω_max) = π - arctan(2Ω_max/γ)/Ω_max, 
        
            t_1    →1/Ω_minarccot(γ/2)/Ω_min = arctan(2Ω_min/γ)/Ω_min.
    
    
    Finally,
    
        q_1/q_0   = C e^-γ/2 t_1sinΩ_min t_1 + D e^-γ/2 t_1cosΩ_min t_1 + F/K_min/A e^γ/2 t_0sinΩ_max t_0 - B e^γ/2 t_0cosΩ_max t_0 - F/K_max
           →(1/Ω_min) e^-γ/2 t_1^* (Ω_min/√(K_min/M)) /(1/Ω_max) e^γ/2 t_0^* (Ω_max/√(K_max/M)) 
           = √(K_max/K_min) e^-γ/2 (t_0^* + t_1^*),
    
    where we have used the equality: sin[arccot(-(γ/2)/Ω_max)] = Ω_max/√(K_max/M), sin[arccot(γ/2)/Ω_min] = Ω_min/√(K_min/M). Here t_0^* and t_1^* are the limiting values in Eq. <ref> and <ref>, respectively.
    
    We define ω_max = Ω_max/(γ/2) = √(K_max/M(γ/2)^2 - 1) and ω_min = Ω_min/(γ/2) = √(K_min/M(γ/2)^2 - 1). After rearranging the terms, we simplifies the condition q_1/q_0<1 into
    
        K_max/K_min < exp[2(arctan ω_min/ω_min + π - arctan ω_max/ω_max)],
    
    which is the inequality we see in Eq. <ref> in Thm. <ref>. This completes the proof of Thm. <ref>. 
    
    
    Note that the inequality in Eq. <ref> in Thm. <ref> is satisfied in the limit K_max-K_min/K_min→ 0. We can see this by take a log on both sides of the inequality and get
    
        log(1+K_max-K_min/K_min) < 2(arctan ω_min/ω_min + π - arctan ω_max/ω_max),
    
    whose left hand side is O(K_max-K_min/K_min) and right hand side is O(1) because ω_min≈ω_max. Therefore, in the limit K_max-K_min/K_min→ 0, the inequality always holds. This agrees with the Error Bounded Property of the approximately constant linear model stated in Prop. <ref>. (Actually the approximately constant linear model satisfies the Error Bounded Property under a more general condition: instead of K>M(γ/2)^2 we only need K>0 for the approximately constant linear model.)
    
    
    
    
    If we transform the coordinates of the system to their normal modes, because the normal modes are also described by damped harmonic oscillators, all the analysis works the same for normal modes. This addresses the comment under Thm. <ref> in the main text.
    
    In Figure <ref>, we show an example of the fluctuation of the error stiffness K(t)=k-g∂ n/∂ Q during the Holstein model dynamics. We see that the error stiffness K(t)>M(γ/2)^2=0.0025, where M=1 and γ=0.1. The highest and lowest points in the plot are 0.7104 and 0.5011. We insert these numbers into Eq. <ref>. The left hand side of the condition results in 1.418, and the right hand side 1.501. Therefore, we confirm that the inequality in Eq. <ref> in Thm. <ref> is satisfied during the time range we are interested in, and thus the Error Bounded Property is satisfied in the Holstein model.
    
    
    
    § V. PROVABLY EFFICIENT ADIABATIC LEARNING
    
    
    Here we provide more details about how the PEAL algorithm is performed.
    
    Step I: Collecting data. To be able to train a learning model, we first need to collect data samples from the distribution we are interested in. For example, one can sample a few different initial conditions, and use either classical solver or quantum computer to obtain a few dynamical evolution paths. The “configuration”-“observable” pairs sampled from those paths can serve as the data set we need. If one wants to predict other k-local observable that is not involved in the dynamics, one can compute that observable along the paths and collect the data for the next training step as well. 
    
    Step II: ML training. A provable efficient ML algorithm for predicting ground state properties was presented recently <cit.>. Here we apply this algorithm to adiabatic dynamics. On the data set collected in Step I, a nonlinear feature map with geometrically local region information is performed. A model is trained with an l_1-regularized regression (LASSO) on the features. Hyperparameters in the model are properly chosen. One can train a single model for one type of observable if the system has translation symmetry. The same training process can be applied to the data set of observable that is not involved in the dynamics. Moreover, the learning model can be transferred to unseen physical parameter g_α in the Hamiltonian. An example is shown in our numerical experiment, also illustrated in Fig. <ref> in the main text.
    
    Step III: ML prediction. For dynamical evolution paths with unseen initial conditions, we can use the PEAL prediction to speed up the simulation, instead of repeatedly using costly classical solver or quantum computer. The dynamical simulation is realized by iteratively updating the classical degree of freedom and the quantum degree of freedom. The classical updating step is done by classical ODE solver. The quantum updating step is done by ML prediction. In each iteration, we can add an extra correction step to achieve symmetry-preserving PEAL (see details in the following section). The provably efficiency and controllable error of the PEAL algorithm will be presented later in Thm. <ref>.
    
    Before we formally present Thm. <ref>, we would like to mention two other theorems introduced in other works. In Ref. <cit.>, the authors prove two theorems:
    
    
    Consider any family of n-qubit geometrically-local Hamiltonians  
    {H(x): x ∈ [-1,1]^m } in a finite spatial dimension, such that each local term in H(x) depends smoothly on x, and the smallest eigenvalue and the next smallest eigenvalues have a constant gap ≥Ω(1) between them. Then the ground state properties can be efficiently predicted. 
    
        
    Given n, δ > 0, 1/e > ϵ > 0 and a training data set {x_t, y_t}_t=1^N of size
    
        N = log(n/δ)2^polylog(1/ϵ),
    
    where x_t is sampled from an unknown distribution D and |y_t - Tr(O ρ(x_t))| ≤ϵ for any observable O with eigenvalues between -1 and 1 that can be written as a sum of geometrically local observables. With a proper choice of the efficiently computable hyperparameters δ_1, δ_2, and B, the learned function h^*(x) = 𝐰^* ·ϕ(z) satisfies
    
        𝔼_x ∼ D| h^*(z) - Tr(O ρ(z)) |^2 ≤ϵ
    
    with probability at least 1 - δ. The training and prediction time of the classical ML model are bounded by 𝒪(nN) = n log(n/δ) 2^polylog(1/ϵ).
    
    
    
    The request in the first one, that the Hamiltonian has a constant gap, can be soften into that the correlation length has a finite upper bound. 
    
    
    
    
    Now we present the formal version of Thm. <ref> in the main text.
    
    
        Given n,δ,η>0, 1/e>ϵ>0 and a training data set {(g_α, P,Q)_l,⟨Ô_α,i⟩_l}_l=1^N of size
        
        N=log(n/δ)2^polylog(1/ϵ),
    
        where l∈{1,2,…,N} is the index of data points, (g_α)_l is the coupling constant used when collecting the data, (P,Q)_l is classical variables sampled from exact simulation data, Ô_α,i is an observable with eigenvalues in [-1,1] that can be written as a sum of geometrically local observables, ⟨Ô_α,i⟩_l is the ground state expectation value with an n-qubit gapped geometrically local Hamiltonian Ĥ((P,Q)_l). Apply an ML predicted model ℳ learned with a proper choice of the efficiently computable hyperparameters. When the Error Bounded Property in Def. <ref> is satisfied, a T-step PEAL prediction Q_i,PEAL(T) comparing to the exact dynamical process Q_i,Exact(T), i.e. the accumulated error, has an error bound
        
        |Q_i,PEAL(T) - Q_i,Exact(T)| ≤ O(√(Tϵ/η))
    
        with probability ℙ≥ 1-η-δ.
    
        If further assume the learning error of model ℳ is sub-Gaussian distributed, the error bound can be improved into 
        
        |Q_i,PEAL(T) - Q_i,Exact(T)| ≤ O(√(2ϵlog(2T/η)))
    
        with probability ℙ≥ 1-η-δ.
    
        If further more assume the learning error of model ℳ is bounded by √(ϵ) almost surely, the error bound can be further improved into
        
        |Q_i,PEAL(T) - Q_i,Exact(T)| ≤ O(√(ϵ))
    
        with probability ℙ≥ 1-δ, i.e. the error is bounded by a constant.
    
        The accumulated error of classical variables P_i and all k-local, bounded quantum observables also have the same scaling as the accumulated error of Q_i.
    Proof. To be able to apply the Error Bounded Property in Def. <ref>, we request that the prediction errors |δ⟨Ô⟩| = |⟨Ô⟩_PEAL(P_PEAL,Q_PEAL) - ⟨Ô⟩_Exact(P_PEAL,Q_PEAL)| are upper bounded for all T number of time steps. From the Thm. <ref> above, we learn
    
        𝔼 |δ⟨Ô⟩|^2 ≤ϵ,
    
    from which we can estimate the probability of a single-shot prediction error going beyond a threshold:
    
        ℙ[|δ⟨Ô⟩| ≥ A √(ϵ)]    = ℙ[|δ⟨Ô⟩|^2 ≥ A^2 ϵ] 
            
           ≤𝔼 |δ⟨Ô⟩|^2/A^2 ϵ
           ≤1/A^2,
    
    where A ≥ 0 is a constant factor we choose to describe the threshold. In the second line, we apply the Markov's inequality because |δ⟨Ô⟩|^2 is non-negative.
    
    For a dynamical simulation process with T number of prediction stpes, the probability of any prediction error going beyond the threshold is controlled by the union bound:
    
        ℙ[⋃_i=1^T{|δ⟨Ô⟩(t_i)| ≥ A √(ϵ)}] ≤∑_i=1^Tℙ[|δ⟨Ô⟩(t_i)| ≥ A √(ϵ)] ≤T/A^2.
    
    
    Taking into account that the trained model could have at most δ probability of failure, we get the probability of not being able to apply our PEAL method is at most (T/A^2 + δ) by another union bound. 
    
    When we are able to control all the prediction errors under the threshold A√(ϵ), we have 
        |q(t)|=|Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(A√(ϵ))
    
    by the Error Bounded Property in Def. <ref>.  From the analysis above, we see this is applicable with a probability at least (1 - T/A^2 - δ).
    
    For given η>0, we choose the threshold factor A=√(T/η), such that η = T/A^2. Therefore, we get our most general bound
    
        |Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(√(Tϵ/η))
    
    with probability ℙ≥ 1 - η - δ.
    
    This (1 - η - δ) behavior in the probability is expected, because we have an adiabatic hybrid algorithm with both quantum and classical updates. The δ failure comes from approximating the quantum algorithm, and the η failure comes from the adiabatic quantum-classical dynamics.
    
    This most general bound is showing a √(T) behavior, which happens to be the same as a diffusion model. 
    
    We can improve this bound if we can acquire more knowledge on the distribution of the prediction error δ⟨Ô⟩. For example, if δ⟨Ô⟩ has a sub-Gaussian distribution with a moment-generating function (MGF) 
    
        M_δ⟨Ô⟩(s) = 𝔼[ e^sδ⟨Ô⟩] ≤expϵ s^2/2,
    
    then we can apply the Chernoff bound and get
    
        ℙ[|δ⟨Ô⟩| ≥ B √(ϵ)] ≤ 2exp(-(B√(ϵ))^2/2ϵ) = 2 exp(-B^2/2),
    
    where B is also a threshold factor.
    
    Repeat the union bound argument, we can get |Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(B√(ϵ)) with a probability at least (1 - 2 T exp(-B^2/2) - δ). For given η>0, we choose B = √(2log(2T/η)), such that η = 2 T exp(-B^2/2). Therefore, we get our bound with assuming sub-Gaussian prediction error
    
        |Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(√(2ϵlog(2T/η)))
    
    with probability ℙ≥ 1 - η - δ.
    
    This improves the general √(T) behavior to a better √(log(T)) behavior.
    
    We can improve this bound even further if we add even stronger assumption on the distribution of the prediction error δ⟨Ô⟩. If we assume there exists a constant factor C such that |δ⟨Ô⟩| ≤ C√(ϵ) almost surely, then we can directly apply our Thm. <ref> and get
    
        |Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(√(ϵ))
    
    with probability ℙ≥ 1 - δ.
    
    This is a constant bound which is independent on T. i.e. The accumulated position error of the dynamical simulation is bounded by a constant.
    
    Because we also have a controlled |p(t)| in Thm. <ref>, all the arguments above are also true if replacing Q by P. 
    
    With the proof of the following lemma, we complete the proof of Thm. <ref>.  
        All k-local, bounded observables have the same provably efficient bound under PEAL, no matter whether they are involved in the dynamics or not. For such an observable Ω̂, given a training data set {(P,Q)_l,⟨Ω̂⟩_l}_l=1^N of size N same as in Thm. <ref>, under various assumptions in Thm. <ref>, the accumulated error |⟨Ω̂⟩_PEAL(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))| holds similar bounds as in Eq. <ref>, <ref>, and <ref>.
    Proof. The accumulated error for an observable is defined as 
    
        |⟨Ω̂⟩_PEAL(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))|.
    
    
    There are two source of error in this operator learning: one from the PEAL path (P_PEAL(T),Q_PEAL(T)) deviate from the exact simulation (P_Exact(T),Q_Exact(T)), the other from the single-step ML prediction error. In terms of formula, we have
    
        |⟨Ω̂⟩_PEAL(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))| 
        ≤    |⟨Ω̂⟩_PEAL(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_PEAL(T),Q_PEAL(T))|
            
            + |⟨Ω̂⟩_Exact(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))|.
    
    
    We first analyze the second term by the perturbation theory. We have
    
        ⟨Ω̂⟩_Exact(P_PEAL(T),Q_PEAL(T))    = ⟨GS_PEAL|Ω̂|GS_PEAL⟩, 
            
        ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))    = ⟨GS_Exact|Ω̂|GS_Exact⟩,
    
    where |GS_PEAL/Exact⟩ = |GS(P_PEAL/Exact(T),Q_PEAL/Exact(T))⟩ is the ground state wavefunction for the Hamiltonian with classical parameters (P_PEAL/Exact(T),Q_PEAL/Exact(T)), respectively.
    
    Note that (P,Q) are served as parameters of the Hamiltonian. For slightly different (P_PEAL(T),Q_PEAL(T)) and (P_Exact(T),Q_Exact(T)), the corresponding Hamiltonians are slightly different, and thus the difference between the ground states can be analyzed by first order perturbation
    
        |GS_PEAL⟩ - |GS_Exact⟩ = ∑_n≥1⟨ n_Exact|V̂|GS_Exact⟩/E_GS,Exact - E_n,Exact |n_Exact⟩ + o(p, q),
    
    where |n_Exact⟩ is the n-th excited state for the Hamiltonian with classical parameters (P_Exact(T),Q_Exact(T)). E_GS,Exact and E_n,Exact are energies of |GS_PEAL⟩ and |n_Exact⟩, respectively. We define the accumulated errors of classical parameters p=P_PEAL(T)-P_Exact(T), q=Q_PEAL(T)-Q_Exact(T). V̂ is the first order Taylor expansion of the Hamiltonian
    
        V̂ = ∑_i (∂Ĥ/∂ Q_i q_i + ∂Ĥ/∂ P_i p_i).
    
    
    Note the the dependence on (p, q) in Eq. <ref> only appears in V̂, which is linear in (p, q). Moreover, Eq. <ref> is bounded by looking at its left hand side. Therefore, we see that overall |GS_PEAL⟩ - |GS_Exact⟩ is of order O(p, q), and is controlled by the various bounds in Thm. <ref>.
    
    Therefore, the second term can be controlled by
    
        |⟨Ω̂⟩_Exact(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))| 
            
        
            =     |⟨GS_PEAL|Ω̂|GS_PEAL⟩ - ⟨GS_Exact|Ω̂|GS_Exact⟩|
            
        ≤    |⟨GS_PEAL|Ω̂|GS_PEAL⟩ - ⟨GS_PEAL|Ω̂|GS_Exact⟩| + |⟨GS_PEAL|Ω̂|GS_Exact⟩ - ⟨GS_Exact|Ω̂|GS_Exact⟩|
            
        
            =     |⟨GS_PEAL|Ω̂ (|GS_PEAL⟩ - |GS_Exact⟩ )| + | (⟨GS_PEAL| - ⟨GS_Exact| ) Ω̂|GS_Exact⟩|,
    
    where (|GS_PEAL⟩ - |GS_Exact⟩ ) and (⟨GS_PEAL| - ⟨GS_Exact| ) is of order O(p, q) and everything else is O(1).
    Therefore the last line in Eq. <ref> is of order O(p, q) because both terms are of order O(p, q). Hence, the second error term satisfies the same bounds as Q does in Thm. <ref>, which in the proof we analyze the probability of all T steps' predictions having errors under a certain threshold.
    
    The first term is the single-step prediction error δ⟨Ω̂⟩ at T-th step, which is analyzed in the proof of Thm. <ref>. Combining the first term and the second term is equivalent to requiring (T+1) steps' prediction having errors under a certain threshold, which can be analyzed with the same technique we used in the proof of Thm. <ref>. More precisely, we have (T+1) terms in the union bound in Eq. <ref>. This is equivalent to having results in Thm. <ref> by changing T into (T+1). However, such change does not affect the scaling behaviors of the bounds with T.
    
    Therefore, the first and second terms combined also satisfy the same bounds as Q does in Thm. <ref>. This completes the proof of Lemma <ref>, that the accumulated error of any k-local observable satisfy the same bounds in Thm. <ref>. With this lemma proved, we complete the entire proof of Thm. <ref>. 
    
    § VI. PEAL IMPLEMENTATION
    
    
    Here we provide more details about how we implement the PEAL algorithm in our numerical experiment on the Holstein model.
    
    PEAL Step I: Collecting data. For each of the six training values g∈{1.3, 1.32, 1.34, 1.36, 1.38, 1.4}, we randomly sampled 18 independent initial conditions Q(t=0)iid∼𝒩(0,Q_v)^⊗ L with the standard deviation Q_v=0.2. L=50 is the system size. We set P(t=0)=0. For each initial condition, We iteratively perform quantum steps and classical steps, for 10,000 epochs, to generate a dynamical evolution path. We set the simulation time step dt=0.01 and the total time is 100. In the quantum steps, we use exact diagonalization (ED) to obtain the quantum observables. In the classical steps, we use fourth order Runge-Kutta method. For each path, we randomly sampled 500 pairs of (Q_shift-i(t), n_i(t)) data, where t is a random time step and i is a random site. Q_shift-i is the Q vector with elements rolled by a shift of i, so that the i-th element is put on the first place. Overall, we have 6×18×500=54,000 data pairs, which effectively serve as 1,080 samples on the size L=50 system.
    
    The purpose to use this kind of site-shifting data pairs is to respect the translation symmetry of the system. With periodic boundary condition, the system has translation symmetry. Therefore, we can use the same model to predict n_i on any location in the system, as long as we also translate the Q vector accordingly.
    
    PEAL Step II: ML training. We use the provably efficient ML algorithm introduced in <cit.>. For each length-50 vector Q, we generate 50 local regions, each local region having radius 1 and consisting of 3 elements. We perform a random Fourier feature map on the local regions, generate R number of cosine features and R number of sine features for each local region. R is a hyperparameter chosen from {5, 10, 20, 40, 80, 160}. The frequency factor γ_ω for the random Fourier feature map is another hyperparameter chosen from {0.3, 0.6, 1, 2, 3, 6, 10, 20}. We trained an l_1-regularized regression (LASSO) on the non-linear features. The strength of regularization α is determined by LassoLars method with 4-fold cross validation <cit.>. For 1,024 samples, the grid-search best hyperparameters are R=20, γ_ω=6, and α=4.98×10^-6. The best model has 316 non-zero linear coefficients after the LASSO feature selection. 
    
    PEAL Step III: ML prediction. To speed up the dynamical simulation, we replace the ED steps by the prediction from the learning model. We take care of the U(1) global symmetry of the system. After each ML boosted quantum step, we subtract the electron density n_i by the mean of their excess over 1/2. Therefore, we can make the total electron number conserved at half filling, and respect the U(1) global symmetry of the system. See more details in the following section.
    
    
    
    § VII. DETAILS OF SYMMETRY-PRESERVING PEAL
    
    
    In the application of PEAL to specific systems with symmetries, we would like to preserve the symmetries during our numerical simulation. Here we present symmetry-preserving PEAL for two types of symmetries: The U(1) global symmetry and the translation symmetry.
    
    To preserve the U(1) global symmetry, we would like to have the total charge conservation at each time step during the dynamics. When we sum up the model's prediction of the electron charge density at each site of the system, often case the result is not the same as the total charge in the previous time step. We can make a correction on the electron charge density at each site during each time step of the dynamics, in order to conserve the total charge.
    
    There are two possible ways to do the correction. The first one is to uniformly add or subtract a constant on the electron charge density at each site. The second is to uniformly multiply a factor to the electron charge density at each site. However, the second way does not treat an electron (charge density equals to 1) and a hole (charge density equals to 0) equivalently. Therefore, we use the first way to do the correction.
    
    Suppose the learning model ℳ predicts the electron charge density n_i=⟨ c_i^† c_i⟩ on each site i. In order to preserve the U(1) global symmetry, we would like to have the corrected electron charge density n_i such that ∑_i n_i = L/2, assuming at half filling for a system with L sites. It is straightforward to see that we can achieve this charge conservation by doing the correction n_i→n_i = n_i + Δ n with Δ n = 1/L∑_i(1/2-n_i).
    
    Now we prove the error bounds for our PEAL algorithm are still valid with this correction. Suppose each n_i has a prediction error δ n_i=n_i-n_i^*, where n_i^* is the electron charge density computed with QSS, which satisfies ∑_i n_i^* = L/2. We can see Δ n=-1/L∑_i δ n_i, and the error of the corrected electron charge density is δn_i = n_i - n_i^* = δ n_i + Δ n = δ n_i - 1/L∑_i δ n_i.
    
    In the proof of PEAL error bounds, we analyzed all situations by turning the problem into the calculation of the probability of |δ⟨Ô⟩| ≤ D √(ϵ), where the constant D stands for A, B, or C in the proof of Thm. <ref>. Note that when all |δ n_i|≤ D√(ϵ), we have 
    
        |δn_i| = |δ n_i - 1/L∑_i δ n_i| ≤ |δ n_i| + 1/L∑_i |δ n_i| ≤ 2  D √(ϵ),
    
    which means that we are safe to apply every error bounds after we raise the corresponding constant D by a factor of 2. This completes the proof.
    
    When the system has a translation symmetry, such as the one for the periodic chain we studied in the main text, every site i is on equal footing. Therefore, we will get the identical error distribution when we apply a model ℳ_i trained for n_i to another site j to predict n_j. This means that we can save our cost by applying one model repeatedly on every site i instead of training an independent model for every n_i, and we have the error bounds unaffected. Moreover, by doing so, we preserve the translation symmetry in the PEAL prediction: If instead of the original initial condition Q(0)=(Q(0)_0, Q(0)_1, …, Q(0)_L-2, Q(0)_L-1) we input a shifted initial condition (Q(0)_1, Q(0)_2, …, Q(0)_L-1, Q(0)_0), then all outputs from the symmetry-preserving PEAL algorithm will be also shifted correspondingly comparing to the original outputs.
    
    
    
    § VIII. DETAILS OF NUMERICAL EXPERIMENTS
    
    
    We are interested in the quantum dynamics of the Holstein model in Eq. <ref> with a random initial distribution of Q_i(0), and we set initial momentum P_i(0)=0. Under the adiabatic approximation, the quantum dynamics can be further described by three equations:  
    
        Q^'_i(t)    = 1/M P_i 
        
            P^'_i(t)    = -k Q_i + g (n_i - 1/2) - γ P_i 
        
            H_e({Q_i(t)})    = -t ∑_i,j c_i^† c_j - g ∑_i (c_i^†c_i - 1/2) Q_i
    
    where n_i = ⟨ c_i^†c_i ⟩ and γ is the damping coefficient.
    
    In the experiment, we observed three regimes of g values. For a small g value, the final stable configuration of the system has no CDW. This is due to the finite size effect. For the system size L=50 we used, the critical value to generate stable CDW is g_crit=1.152. This critical value can be analytically solved and we present the calculation already in earlier section. For a large g value, domain walls can be generated in the system. The characteristic length of the domain walls decreases when g increases. Therefore, when the system size is much larger than the domain wall characteristic length, domain walls can be relatively far from each other, which makes their interaction exponentially small and the equilibrium time exponentially large. For the system size L=50 we used, the value to generate long-lasting domain walls is g_DW∼ 1.6. For the value g_crit < g < g_DW, we can see a clear CDW developing during the dynamics. Therefore, we collect a data set for g∈{1.3, 1.32, 1.34, 1.36, 1.38, 1.4}, mimicking a uniform distribution in [1.3,1.4].
    
    For each g value, we generate 18 ED dynamical evolution paths with random initial conditions, serving as the training data. We also generate 10 more ED dynamical evolution paths with random unseen initial conditions for each g∈{1.3, 1.32, 1.34, 1.36, 1.38, 1.4} and for each g∈{1.31, 1.33, 1.35, 1.37, 1.39}, serving as the standard learning test set and the transfer learning test set, respectively. In Figure <ref> in the main text, we show how the test error scales with the number of samples used in model training. We trained models with number of samples from 1, 2, 4, ..., to 1,024. (Each sample contains 50 data pairs based on the system size we used.) We see an error scaling law agrees with the predicted logarithmic scaling. The transfer learning test error is only slightly higher than the standard learning test error. The scatter plots of the target values show the model has learned nicely for both the normal case and the transfer learning case.


