Paper Title: Confined Trions and Mott-Wigner States in a Purely Electrostatic Moiré Potential.








nkiper@phys.ethz.ch
Institute for Quantum Electronics, ETH Zürich, Zürich, Switzerland

Institute for Quantum Electronics, ETH Zürich, Zürich, Switzerland
Institute for Theoretical Physics, ETH Zürich, Zürich, Switzerland

Institute for Quantum Electronics, ETH Zürich, Zürich, Switzerland
Institute for Theoretical Physics, ETH Zürich, Zürich, Switzerland

Institute for Quantum Electronics, ETH Zürich, Zürich, Switzerland

National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan

National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan

Institute for Quantum Electronics, ETH Zürich, Zürich, Switzerland


             

§ ABSTRACT 
Moiré heterostructures consisting of  hetero- and homobilayers have emerged as a promising material platform to study correlated electronic states.  Optical signatures of strong correlations in the form of Mott–Wigner states and fractional Chern insulators have already been observed in  monolayers and their twisted bilayers. In this work, we use a moiré substrate containing a twisted  interface to externally generate a superlattice potential for the  layer: the periodic structure of ferroelectric domains in  effects a purely electrostatic potential for charge carriers.  
We find direct evidence for the induced moiré potential in the emergence of new excitonic resonances at integer fillings, and our observation of  an enhancement of the trion binding energy by ≃ 3 meV. A theoretical model for exciton-electron interactions allows us to directly determine the moiré potential modulation of 30 ± 5 meV from the measured trion binding energy  shift. We obtain direct evidence for charge order linked to electronic Mott–Wigner states at filling factors ν =1/3 and ν=2/3 through the associated exciton Umklapp resonances.

Confined Trions and Mott–Wigner States in a Purely Electrostatic Moiré Potential
    Atac İmamoğlu
    August 5, 2024
================================================================================





§ INTRODUCTION
 

Semiconductor moiré materials have emerged as a fertile platform to explore the physics of strongly correlated electrons <cit.>. Generically,  transition metal dichalcogenide (TMD) moiré materials are realized by interfacing two or more monolayers,  and the moiré pattern emerges due to twist angle or lattice mismatch between the layers. The emerging periodic potential has substantial contribution from lattice reconstruction, in addition to polarization fields and inter-layer tunneling. In these structures, not only charge carriers but also excitons are
typically subject to strong potentials. This feature has been utilized to investigate correlated Mott states of Bose–Fermi mixtures consisting of moiré inter-layer excitons and electrons in heterobilayers with type-II band-alignment <cit.>. 

Due to the complex nature of the moiré potential and its different effect on charge carriers and excitons, the interpretation of optical signatures from TMD moiré structures is often challenging. Thus, designing systems where excitons are not subject to a moiré potential would be advantageous, since any modifications of the optical excitation spectrum would arise exclusively from electronic correlations and their effect on exciton-electron interactions. This scenario would be ensured if the moiré potential is of purely electrostatic origin. Moreover, exposing two identical TMD monolayers  that are separated by a thin tunneling barrier to an identical periodic electrostatic potential would lead to moiré electrons or holes acquiring a layer pseudo-spin degree of freedom with (nearly) SU(2) symmetry. A range of fascinating phenomena ranging from electrically controlled Feshbach resonances <cit.> to chiral layer pseudo-spin liquids <cit.> could be realized using such a structure.

Here, we demonstrate a purely electrostatic moiré potential for itinerant electrons and holes in a  monolayer, which we probe through  spectroscopy. The simple nature of the moiré potential allows us to extract its modulation depth directly from the optical signatures – namely the trion binding energy – using a theoretical model for exciton-electron interactions.  In addition to measuring a clear red-shift of the trion and the associated  resonance, we identify the charge ordered Mott–Wigner states at filling factors 1/3 and 2/3 through the appearance of excitonic Umklapp resonances <cit.>.





§ ELECTROSTATIC MOIRÉ POTENTIAL GENERATED BY TWISTED HEXAGONAL BORON NITRIDE
 

We generate a purely electrostatic periodic potential using ferroelectric domains in twisted hexagonal boron nitride (), which is a layered insulator with a large band gap. In bulk,  naturally occurs with AA’ stacking, where neighboring layers are rotated by 180^∘ with respect to each other [see Fig. <ref>(a)]: inversion symmetry of the structure ensures a weak paraelectric response to applied out-of-plane fields. In AB-stacked or 0^∘ aligned , however, the two neighboring layers have the same orientation. The lattice can now take one of two possible stacking orders: AB and BA stacking. Both break inversion and out-of-plane mirror symmetry [see Fig. <ref>(b)]. Therefore, an out-of-plane electric polarization develops spontaneously and the bilayer becomes ferroelectric <cit.>. A small twist angle between the  layers leads to a spatial modulation of the local stacking order, resulting in alternating AB and BA domains with opposite out-of-plane electrical polarization [see Fig. <ref>(c)]. These alternating electric dipoles create electric fields that extend beyond the twisted bilayer and effect a superlattice potential for charges in any target material placed close to the twisted interface [see Fig. <ref>(d)] <cit.>.

We place a  monolayer directly below a  bilayer in a dual-gated, charge tunable van der Waals heterostructure [Fig. <ref>(e)-(f)]. The purely electrostatic nature of the moiré potential for electrons and holes in   is ensured by two key features of the   interface: namely, (i) the lattice mismatch exceeding 20 %, and (ii)  the large difference between the conduction (valence) band minimum (maximum) of the two materials. The sizeable lattice mismatch ensures that no large scale moiré pattern is formed between the  and  layers directly, while the band alignment ensures that charges are injected solely into the  layer. Based on these features, we assume the electronic moiré potential to have a simple shape determined purely by the local stacking order of the  layers, with a single minimum per moiré unit cell and no secondary local minima. Up to unity filling of the moiré lattice, we expect charged particles to populate the moiré flat band originating from hopping between these minima. We also note that we expect electrons and holes to experience potentials of equal shape and depth, but of opposite sign. These features render -generated moiré potentials an ideal platform to investigate strong electronic correlations in  moiré materials.



§ BASIC CHARACTERIZATION
 

We use   measurements of the  monolayer to determine the strength of the periodic potential experienced by charge carriers through the modification of the trion binding energy. We designed our sample to ensure that the   layer extends beyond the  region, allowing us to directly compare the experimental signatures with (region I) and without (region II) moiré potential in a single device (see Appendix <ref> for details on sample fabrication). In the charge neutral regime, the DR spectrum is indistinguishable between regions I and II, verifying the absence of a moiré potential for excitons. This is in stark contrast to conventional hetero-bilayer TMD moiré systems <cit.>, where the presence of a moiré superlattice leads to dramatic modifications of the exciton spectrum. 



Figures <ref>(a)-(b) compare the  spectrum  of the  monolayer with and without an electrostatic moiré potential as a function of the chemical potential V_μ=0.42V_TG+0.58V_BG:  depicted in (a) and (b) is obtained at a spot in region II and I, respectively. The two measurement spots are indicated by the orange circle and blue triangle in Fig. <ref>(f). We calculate the charge density corresponding to the applied gate voltage using the capacitor model, and using the integrated DR around the exciton resonance energy to determine the doping onset (see Appendix <ref> for details). We keep the applied out-of-plane displacement field V_E=0.5(V_TG-V_BG) constant at V_E=0. 
In the charge neutral regime (0≤ V_μ≤ 0.26 V), there is a single resonance in the spectrum at 1.632 eV (a) and 1.634 eV (b), corresponding to the A exciton of ; the small (2 meV) difference in the resonance energy between (a) and (b) is likely to stem from local strain variation, which is known to result in a shift of the neutral exciton energy by as much as 5 meV in -encapsulated  monolayers free of external potentials. 

The absence of additional excitonic resonances and lack of reduction in the DR contrast of the A exciton in Fig. <ref>(b) verify that excitons in our sample are not subject to a significant moiré potential. We emphasize that while electrons and holes experience a moiré potential whose strength is linearly proportional to the out-of-plane electric field (E_z,moiré) generated by the ferroelectric domains, we would expect charge-neutral excitons to experience a much weaker moiré potential that originates from the dc-Stark shift with a strength proportional to the square of the in-plane electric field <cit.>. Since we do not observe any additional exciton resonances in charge neutrality, we conclude that the energy splitting arising from dc Stark effect is smaller than the exciton linewidth γ=2.3 meV. Therefore, we estimate a maximal in-plane field |E_||,max|=25 mV/nm, assuming an exciton polarizability α=6.5 eV nm^2 V^-2 <cit.>. 

Upon doping the monolayer in region II with electrons for V_μ≥0.26 V, the  resonance appears, while the exciton evolves into a  which shifts to higher energies and broadens with increasing electron density [Fig. <ref>(a)]. In the limit of vanishing doping, the  is separated from the neutral exciton by the trion binding energy E_T=24 meV.  The  gains oscillator strength with increased electron doping and exhibits first a slight redshift up to V_μ≈ 1.5 V followed by an overall blueshift. 



§ EXTRACTION OF MOIRÉ DEPTH FROM INCREASED TRION BINDING ENERGY
 

The  in region I [Fig. <ref>(b)] shows striking differences as compared to the electron-doped monolayer in region II [Fig. <ref>(a)]: by focusing on the low electron doping regime, we find that the trion binding energy is increased by 2.75 meV to E_T,moiré=26.75 meV.  
Figure <ref>(c) shows trion binding energies versus exciton energies on six different -adjacent spots in region I of our device, as well as one  spot in region II. Additionally, we show trion binding energies on five different simple -encapsulated  monolayer devices for comparison. Although the A exciton resonance energies vary by more than 10 meV between different devices due to different local strain and dielectric environments, the moiré-free trion binding energy is very uniform (within ∼1 meV). In stark contrast, E_T,moiré is clearly enhanced on every -adjacent spot measured on this sample. This increase in  E_T,moiré in comparison to E_T is remarkable, given the uniformity of the trion binding energy despite the exciton energy variation.

The increased trion binding energy E_T,moiré is a direct consequence of the localization of the electron, and consequently of the trion, wave function in the superlattice potential. Intuitively, since the electron is already confined by the moiré potential, the kinetic energy cost of correlating its motion with the exciton is reduced as compared to the free electron case. To verify this explanation, we carried out a calculation quantifying the modification of the trion binding energy in the presence of a moiré potential for the electron (see Appendix <ref>). Assuming that the exciton Bohr radius is much smaller than all other relevant length scales, we use an effective short range exciton-electron interaction potential that correctly describes the binding energy of the trion in the absence of the moiré potential. We find that the 2.75 meV increase in trion binding energy is consistent with a moiré potential modulation (Δ_m) in the range 30 meV ≲Δ_m≲ 40 meV, assuming a moiré lattice constant of a_m = 10.2 nm [see Fig. <ref>(d) and Fig. <ref>]. The main contribution to the uncertainty in the potential depth is the spread of measured trion binding energies. The size of the electron Wannier orbital for such a moiré potential is a_W≃ 3.6 nm [We estimate the size of the Wannier orbital as the variance of the numerically calculated electron Wannier function.], which is comparable to but larger than the trion Bohr radius a_T≃ 2.1 nm. We remark that the calculated E_z,moiré based on previous reports <cit.> predicts a factor of 4 larger Δ_m value as compared to what we determine from our measurements. We tentatively attribute the difference to the significant dielectric screening of E_z,moiré by the  monolayer. This value of Δ_m combined with a domain length scale of l∼ 10 nm is also consistent with the absence of splitting of the exciton due to the dc Stark effect, as |E_||| ∼Δ_m/l≪ |E_||,max|=25 mV/nm. 





§ MOTT–WIGNER STATES AT FRACTIONAL FILLINGS OF THE MOIRÉ LATTICE
 

At n≈ 3.7·10^11 cm^-2 and n≈7.4·10^11 cm^-2, two new, much weaker resonances appear at slightly higher energy than the  [see Fig. <ref>(a)]. We associate them with Umklapp resonances arising from the scattering of the  from the periodic potential that arises from electronic charge order <cit.>. The sudden appearance and disappearance of the Umklapp peaks indicates that the electrons form a Mott–Wigner state at these specific densities. Based on previous reports of insulating states at fractional fillings of TMD moiré lattices <cit.>, we assign them to be the ν=1/3 and ν=2/3 states, as these are usually the most robust fractionally filled states. Although the electron density at ν=2/3 is twice that at ν=1/3,
the two Umklapp resonances appear approximately at the same blue detuning. This confirms the expectation that at ν=2/3, electrons form a hexagonal lattice with the same unit cell size as the triangular lattice they form at ν=1/3, and is in agreement with prior theoretical predictions <cit.> (see also Appendix <ref>). The Umklapp resonances allow us to accurately determine the electron density corresponding to integer fillings. We find n|_ν=1=(1.1±0.2)·10^12 cm^-2, yielding a moiré lattice length a_m=10.2±0.7 nm. The corresponding  twist angle of θ_m,exp = 1.4±0.1^∘ is in relatively good agreement with the angle θ_m,tar=1^∘ we targeted during the fabrication of the heterostructure. The onset of doping and n|_ν=1 are indicated by the dashed gray lines in Fig. <ref>(a).



§ CHARACTERIZATION OF THE MOIRÉ-INDUCED POLARON RESONANCES
 

The  spectrum in Fig. <ref>(b) shows that there are two new resonances that appear upon increasing the electron density, which we term the second attractive polaron () and the middle polaron (MP). The MP attains its maximum strength at n|_ν=1≃ 1 ·10^12 cm^-2, where we also see the onset of , strongly indicating that they are linked to the electronic Mott state at ν=1. We emphasize that we observe the same new spectral features – the  and MP resonances – when the monolayer is hole doped (see Appendix <ref>).

We extract the oscillator strengths of the  and  resonances by fitting the spectra with dispersive Lorentzian functions [see Fig. <ref>(b) and Appendix <ref>]. We find that the oscillator strength of the  resonance increases linearly up to ν=1, where it saturates.  Meanwhile, the  resonance  increases linearly with n between ν=1 and ν=5/3. Interestingly, both  resonances' oscillator strengths exhibit nonlinear behaviors at ν=5/3, hinting at the presence of correlated states at higher fractional fillings that are not detectable by Umklapp spectroscopy.




To better understand the new features induced by the -moiré potential, we apply a magnetic field of B_z=7 T in the out-of-plane direction. The optical selection rules and spin-valley locking due to spin-orbit coupling in  make it possible to optically probe the density of spin-up and spin-down electrons using circular polarization-resolved DR spectroscopy. Since a bound trion state composed of two electrons and one hole is only formed when the two bound electrons have a symmetric orbital wavefunction centered around the hole, only electrons occupying K (-K) valley can dress a -K (K) valley exciton to form an  <cit.>. Figures <ref>(a) and (b) show  spectra for K (σ_+) and -K (σ_-) valley exciton-polaron resonances at B_z = 7 T.

For σ_+-polarization [Fig. <ref>(b)], we see an  originating from a trion with binding energy of E_T=26.6 meV; the measured E_T is identical to the one measured at B_z=0 T, confirming that the confinement from the  electrostatic moiré potential in the limit of vanishing electron density is unaffected by the external magnetic field. 

The σ_--polarized exciton cannot form a bound state with the electrons, but experiences a blueshift due to phase space filling and broadening due to intravalley electron scattering. We observe a cusp-like feature in the energy and a local maximum in integrated DR spectrum of the σ_- exciton at V_μ=1.45 V [see Fig. <ref>(a) and (c)]: such features have been previously reported for excitonic excitations out of incompressible electronic states <cit.>. Assuming that this cusp is associated with the ν=1 Mott state, we find a twist angle θ_m^exp=1.36±0.09^∘, which in turn is (within the measurement uncertainty) identical to the one we determined independently at B_z=0 T. We also observe the Umklapp features at B_z = 7 T [see Fig. <ref>(d) and Fig. <ref>]. 

The absence of an  resonance in σ_--polarization for n≤1.9·10^12 cm^-2 [We attribute the weak AP resonance in σ_--polarization we observe for n ≤ 0.4×10^12 cm^-2 to probe-light-induced spin-valley depolarization <cit.>.]
indicates that, similarly to the moiré-free case <cit.>, the electrons at these densities are completely spin-polarized due to the valley-Zeeman effect <cit.>. This observation in turn indicates that for 1.0 < ν < 1.8  the electrons in doubly occupied sites are in the m_z = 1 triplet (T_+) state. The fact that we can polarize the spins completely even above ν=1 implies that the valley-Zeeman energy is larger than the singlet-triplet energy splitting in doubly occupied sites. This conclusion is further supported by exact Diagonalization (ED) calculations for two electrons confined to a single moiré unit cell, where we account for neighboring electrons using a Hartree potential (see Appendix <ref>). In particular, owing to the strong Coulomb interactions, we find that the triplet exceeds the singlet in energy by only 0.38 meV at ν=2. 

The AP' resonance disappears when the electrons are valley polarized at B_z = 7 T and when ν<1 at B_z = 0 T. This suggests that AP' is the optical excitation out of doubly occupied moiré sites where the electrons form a singlet or m_z=0 triplet (T_0) state, measured with respect to the quantization axis set by the spin of the exciton (see Appendix <ref> for details). 
On these sites, only one electron can participate in trion formation, and the other electron leads to a blueshift of the polaron resonance due to phase space filling. The energy difference between the AP and AP' resonances is approximately reproduced by a phenomenological model at ν=2, based on ED calculations of an exciton and two electrons in a single unit cell. We provide a detailed discussion of the theoretical model in Appendix <ref>.

The relative AP and AP' spectral weights for a given exciton spin/polarization can be understood as follows. For B_z= 0 T and 1 < ν < 2, 50% of the doubly occupied sites are either in singlet or T_0 states, and yield the blue shifted AP' resonance. The singly occupied sites and doubly occupied T_+ sites give rise to AP. T_+ sites contribute doubly to the oscillator strength, since the exciton can bind to either electron. On T_- sites, no attractive polarons or trions can be formed.
At B_z= 7 T, electrons in all doubly occupied sites are in T_+ states (for σ_+-polarization), and we only observe the AP resonance.

The MP resonance also appears only in σ_+ polarization, indicating that it originates from excitons interacting with opposite valley electrons. Given the experimental signatures at B_z=0 T and B_z=7 T, we expect that the MP resonance arises from the attractive Hartree potential that excitons are subject to when the electrons form a Mott state at  ν≃ 1. Theoretical modeling of this Hartree potential indeed gives rise to a new negative-energy excitonic band, and the large oscillator strength observed experimentally is consistent with delocalized excitons. What remains to be better understood is the character of MP at B_ z =0 T, where we expect that the effective Hartree potential fluctuates sizeably from site to site due to the strong dependence of the exciton-electron interaction on the electron spin.






§ CONCLUSION
 

We have presented unequivocal evidence for the realization of a purely electrostatic moiré potential for charge carriers, and the formation of generalized Wigner states. Using the enhanced trion binding energy, we have determined the moiré potential generated by twisted  is at least a factor of two weaker than those reported in TMD heterobilayers <cit.>. This system hosts a rich interplay between electronic Coulomb interactions and the moiré potential, revealing qualitatively new features such as the second attractive polaron and the middle polaron. Owing to the electrostatic nature of the moiré potential, we can use the charge-neutral excitons to gain valuable insights into these features, such as finding strong evidence for the origin of AP' in the spin-structure of doubly occupied sites in the Hartree-moiré potential. 

This new platform for moiré provides a number of interesting avenues for future work, such as placing two TMD layers above a  interface, giving the electrons an additional layer pseudo-spin degree of freedom with weakly broken SU(2) symmetry. This structure could be used to explore a number of exotic phenomena, ranging from spin-polaron formation through kinetic pairing <cit.>, to chiral layer-pseudospin liquids <cit.>.

The data that supports the findings of this paper is available in the ETH Research Collection <cit.>.

Acknowledgements This work was supported by the Swiss National Science Foundation (SNSF) under Grant Number 200021-204076.
H.S.A acknowledges support from the Swiss Government Excellence Scholarship. A.C. was supported by an ETH Fellowship.
K.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Numbers 20H00354, 21H05233 and 23H02052) and World Premier
International Research Center Initiative (WPI), MEXT, Japan. We thank Tomasz Smolenski, Ivan Morera and Livio Ciorciaro for useful discussions.





§ SAMPLE FABRICATION
 

All flakes were mechanically exfoliated from bulk crystals ( from HQ Graphene,  from our collaborators at NIMS, natural graphite) on Si substrates with a 285 nm SiO_2 capping layer. The thicknesses of all flakes were determined by their optical contrast in bright field microscopy. The monolayer nature of the  that was used to make the twisted interface was further confirmed by measuring its second harmonic generation. The heterostructure was assembled with a standard dry transfer technique using a poly(bisphenol A carbonate) (PC) covered hemispherical polydimethylsiloxane (PDMS) stamp and deposited on a Si/SiO_2 (285 nm) substrate. The stacking was done at high temperature (140 ^∘C) inside a glove box in inert Ar atmosphere. The PC was delaminated from the PDMS by heating it to 180 ^∘C and subsequently dissolved in chloroform. The twisted  was fabricated from a single  monolayer using the tear-and-stack technique. Metallic contacts (5 nm Ti sticking layer, 85 nm Au) to the gate and contact graphite flakes were prepared using optical lithography and electron beam evaporation. See Fig. <ref> for the detailed layout of the device.





§ EXPERIMENTAL SETUP
 

All DR measurements were carried out using a confocal microscope setup with the sample at T = 4.2 K. The entire setup is schematically depicted in Fig. <ref>. The sample was mounted on top of x-y-z piezoelectric positioners at the bottom of a stainless steel tube containing 20 mbar He exchange gas, which was submerged in a liquid He bath cryostat equipped with a superconducting magnet coil. The light source for DR measurements was a single-mode fiber-coupled broadband (3 dB bandwidth of 20 nm) light emitting diode (LED) centered at 760 nm (1.631 eV), from Exalos. After exiting the fiber, the excitation light was collimated before entering the cryostat insert through a window at the top. The light was then focused on the sample surface by a long working distance apochromatic cryogenic objective (LT-APO/LWD/VISIR/0.65, NA=0.65). The reflected light from the sample was collected by the same objective and separated from the excitation light by a 90 (R) : 10 (T) beam splitter. It was then coupled into a single mode fiber which brought it to a 0.75 m spectrometer equipped with a liquid nitrogen cooled CCD camera and a 1200 groove/mm diffraction grating.
Two pellicle beam splitters on flip mounts allowed us to selectively inject white light for imaging into the detection path, and image the sample using a CCD camera in the excitation path. 
For the circular-polarization resolved DR measurements at B_z=7 T, we set the polarization to be parallel to the axis of the beam splitter using a linear polarizer in the excitation path, before converting it to a circular polarization using a quarter-wave (λ/4) plate in the common excitation on detection path. The λ/4 plate was mounted on a motorized rotation stage, allowing us to switch between σ_+ and σ_- polarization. The circular polarization was set by suppressing the reflection of the excitation light from the sample on the imaging CCD.
For all DR measurements, the excitation power was kept below 10 nW. 





§ ANALYSIS OF DIFFERENTIAL REFLECTION SPECTRA
 

To obtain the differential reflection Δ R/R_0, we measured the reflection spectrum R_0 from an excitation spot on our device away from the MoSe_2 layer. Together with our bare reflection spectrum R, we then calculated Δ R/R_0=R-R_0/R_0. This  method of background subtraction is known to introduce systematic errors, due to the sensitivity of the reflection spectrum on e.g. the exact sample position with respect to the focal plane. Nevertheless we chose this approach, as the more accurate method of reconstructing the background from spectra at different gate voltages was not applicable due to the presence of resonances that energetically overlap at every measured doping level. 

The shape of the differential reflection spectrum is not only determined by the reflection of the  layer, but also by its interference with the reflections from all other surfaces and interfaces within the stack and substrate. The reflection R as a function of energy E can be approximately described by the dispersive Lorentzian function


    R(E,E_0,γ,A,α)=Acos(α)γ/2/(E_0-E)^2+(γ/2)^2+Asin(α)E_0-E/(E_0-E)^2+(γ/2)^2,


where E_0, γ, and A are the resonance energy, linewidth, and amplitude, respectively. The parameter α denotes the wavelength-dependent phase shift that that arises from the interferences described above. 
In order to extract the oscillator strength f=π A, we fit the DR spectra with the total reflectivity

    R_tot(E)=∑_pR(E,E_0,p,γ_p,A_p,α_p)+f_bg(E),

where p∈{X, , ,   MP} denotes the resonance, and f_bg(E) is a heuristic quadratic function necessary to remove a residual background due to imperfect background subtraction. We use the same background function for each spectrum in any gate-dependent DR dataset. Fig. <ref> shows an example of a DR spectrum at filling factor ν = 1.25 and B_z=0 T, where all four resonances are present, with the fitted R_tot and all its components.
Due to the difficulties regarding the proper treatment of the background subtraction described above, the quantitative value for the oscillator strength extracted from the fit functions is not meaningful, and only serves to qualitatively compare oscillator strengths of resonances taken within the same measurement run.



Figure <ref> shows DR spectra and fitted reflectivity for filling factors between ν≈0.6 and ν=2, focusing on the  and  resonances.





§ CHARGE DENSITY AND MOIRÉ LENGTH CALCULATION
 

We use a simple capacitor model to convert the voltages applied to top and bottom gates to a charge carrier density in the TMD layer. We model the device as two coupled capacitors, with the TMD as the central layer. The TMD layer is always grounded (V_TMD=0). The central layer is separated from the top (bottom) gate by a  dielectric spacer with thickness d_top=50±5 nm (d_bot=36±5 nm). The thicknesses of the  gate dielectrics are estimated from their colors in bright field microscopy. We apply the voltages V_TG and V_BG to the top and bottom gate, respectively. The charge density induced on the TMD layer through the capacitive coupling is thus

    σ = ε_BNε_0(V_TG/d_t+V_BG/d_b-(1/d_t+1/d_b)V_0),

where ε_BN=3.4±0.2 is the  out-of-plane dc dielectric constant <cit.>, ε_0 is the vacuum permittivity, and V_0 is a constant that represents the finite voltage needed to surpass the charge gap of the TMD.  We determine V_0 from the integrated DR signal over the exciton /  resonance (see Fig. <ref>). As soon as charges are injected into the system, the exciton starts losing oscillator strength to the , leading to a sharp cusp in the integrated DR. 



The charge carrier density is then n=σ/e with e the elementary charge.

As described in Sec. <ref>, we determine the gate voltages at which unity filling of the moiré lattice (V_TG|_ν=1=V_BG|_ν=1=1.49 V) occur directly from the spectral features. We use Eq. <ref> to calculate the charge density at ν = 1 and find n|_ν=1=1/A_m=(1.1±0.8)·10^12 cm^-2, where A_m is the area of one moiré unit cell. The moiré length a_m is then given by a_m^2=2/√(3)A_m. From this we can calculate the corresponding twist angle of the , which is given by θ_m=arccos(1-a_BN^2/2a_m^2), with a_BN=2.5040  the lattice constant of  <cit.>.



§ PHOTOLUMINESCENCE SPECTROSCOPY
 

We show doping-dependent photoluminescence (PL) spectra at B_z=0 T in Fig. <ref>. We used a helium-neon laser at wavelength 632.8 nm (excitation power: 8.6 W before the insert) for the PL measurement. We observe a blue shift of the  at ν=1, indicating an incompressible state of electrons.





§ UMKLAPP RESONANCES AT Ν=1/3 AND Ν=2/3
 

Since the Umklapp resonance can be understood as a high momentum exciton being scattered to zero momentum by a periodic repulsive potential, we can estimate the energy splitting E_U between the Umklapp and the exciton /  by considering the band folding of the bare exciton dispersion at the edge of the periodic potential Brillouin zone. In our case, the repulsive periodic potential for the excitons stems from localized electrons in the moiré lattice. The bare exciton has both linearly and parabolically dispersing branches. As the linearly dispersing branch has a large slope, its Umklapp bands are far detuned in energy and therefore not relevant. The parabolic dispersion is given by E(𝐤)=ħ^2𝐤^2/2m_X, where 𝐤 is the momentum and m_X=1.2m_0 is the exciton mass (m_0 is the free electron mass). The splitting is therefore given by E_U=E(k_U), where k_U=|k_U| is the size of the Brillouin zone. For a triangular lattice with lattice constant a, we have k_U=2π/a. A triangular lattice with lattice constant a corresponds to an electron density of n=1/A, where A=√(3)/2a^2 is the area of the real space unit cell. We therefore have E_U(n)=4h/m_Xn.

At filling factors ν=1/3 and ν=1, we expect the electrons in the  moiré lattice to form triangular Mott–Wigner states. In Fig. <ref>(a,d,g), we show differentiated DR spectra, as well as the expected Umklapp energy E_X+E_U. For the data at B_z=7 T, we also take the Zeeman splitting E_Z of the exciton into account. We see that the ν=1/3 Umklapp resonance is at the expected energy.

At filling factor ν=2/3, the situation is more complicated, as the electrons are expected to form a hexagonal lattice. The unit cell of this hexagonal lattice has the same size as the triangular lattice formed by the electrons at ν=1/3, but has two occupied sites per unit cell. We therefore expect to see two Umklapp resonances centered around E_X|_ν=2/3+E_U|_ν=1/3. A small splitting between the two Umklapp resonances could arise due to hybridization between the two back-folded states, but we expect this splitting to be small compared to E_U. See also <cit.> for a detailed discussion of Umklapp resonances from triangular and hexagonal lattices. The Umklapp resonance we observe at ν=2/3 indeed appears at roughly half the splitting expected for a triangular lattice, confirming that the ν=2/3 Mott–Wigner state is hexagonal. We do not observe two Umklapp peaks at ν=2/3, indicating either that the two Umklapp bands do not hybridize or the splitting is smaller than the Umklapp linewidth.

We note that we do not see the ν=2/3 Umklapp in σ_- polarization at B_z=7 T. We speculate that this is due to the closeness in energy to the much stronger, blueshifting exciton resonance.





§ HOLE SIDE DIFFERENTIAL REFLECTION AT B_Z=0 T AND B_Z=7 T
 

As mentioned in the main text, we expect the moiré potential for electrons and holes generated by  to have equal shapes and depths, the only difference being an offset in the plane by half a moiré unit cell (or equivalently, a change of sign). We indeed see the same spectral features that we attribute to the moiré potential (namely, the MP and  resonances) on the hole doping side as well, see Fig. <ref>. We concentrate on the electron side for our analysis, as for low doping, the charge carrier density dependence on the gate voltage is nonlinear due to poor contact quality for holes. This nonlinear hole doping behavior has been previously observed in moiré-free graphene-contacted  devices <cit.>, therefore we do not believe it to be a consequence of the moiré potential or proximity to  layers.

We also show the DR spectra on the hole doping side at B_z=7 T in Fig. <ref>. Here, we again see the MP resonance and a single  in σ_- polarization (probing the undoped valley), as well as a cusp in the σ_+-polarized exciton. In contrast to the electron side, the  exhibits a strong redshift.







§ EFFECT OF OUT-OF-PLANE ELECTRIC FIELD
 

Some previous studies report a hysteretic electric field dependence of ferroelectric  domains due to field-induced switching of the polarization via an in-plane layer sliding transition <cit.>. We do not observe any qualitative effect of an applied out-of-plane electric field, as can be seen in Fig. <ref>, where we show DR spectra as a function of V_μ for V_E=0.5(V_TG-V_BG)=-4,0,4 V, corresponding to fields E=47,0,-47 mV/nm, respectively. We assume that we are either unable to reach the critical electric field necessary to switch the ferroelectric state due to the breakdown limit of our gate dielectric, or that the relatively small-scale  moiré pattern stabilizes the domains.





§ THEORETICAL MODEL FOR TRION BINDING IN MOIRÉ
 



 §.§ Model

 
To model the trion binding energy in the presence of the twisted  moiré we consider a rigid 1s exciton interacting with an electron that is subject to a periodic electrostatic potential. The Hamiltonian reads

    H = ^2_X/2m_X + _e^2/2m_e + U( _e) + V( _X -_e),

where _X and _e denote the momentum operators for the exciton (X) and electron (e), respectively. The exciton and electron masses are represented by m_X and m_e, and are set to 1.2m_0 and 0.7m_0 with m_0 being the bare electron mass. The terms U(_e ) and V( _X -_e) denote the moiré potential and the exciton-electron interaction potential, respectively.

Inspired by past treatments of moiré potentials in s <cit.>, we approximate the moiré potential through its lowest harmonic expansion 

    U( _e) =  ∑_j=1^6 U_j e^i _j·_e,

with U_j as the harmonic coefficients and _j the reciprocal lattice vectors of the first shell. These vectors can be derived by applying a (j-1)×π/3 rotation to _1 = (4π/√(3)a_m, 0), where a_m is the experimentally determined moiré length of 10.2 nm.

To ensure that the moiré potential is real and exhibits three-fold rotational symmetry, the harmonic coefficients must satisfy:

    U_1 = U_3 = U_5,    U_2 = U_4 = U_6,    U_1 = U_4^*.

This constraint narrows the form of the expansion coefficients to U_1 = U_0 e^i ψ, with ψ = π/2 producing a moiré potential that aligns with the expected form of twisted  near a 0^∘ twist angle. The modulation of the moiré potential Δ_m discussed in the main text is quantified by Δ_m = 6 √(3) U_0.

For the exciton-electron interaction, we adopt the effective interaction <cit.>

    V() = - W_0/(a_ X^2 + ||^2)^2

where a_ X is the 1s exciton radius of approximately 1.1 nm <cit.>. We have checked that the choice of exciton-electron interaction (e.g., Gaussian with finite range, or contact) does not affect the results presented in the main text: the minor quantitative differences are well within the uncertainty in the trion binding in the absence of moiré.



 §.§ Trion binding in moiré

To determine the trion binding energy in the presence of the moiré potential, we take advantage of the relatively weak coupling between the relative and center-of-mass (CoM) coordinates; the coordinates exactly decouple in the absence of moiré. Consequently, we reformulate the Hamiltonian presented in Eq. (<ref>) as follows:

    H = H_ rel + H_ CoM + H_ rc,

where the system is decomposed into:

    
    H_ rel   = ^2/2 m_r + V()
    
        H_ CoM   = P^2/2M + U() 
    
        H_ rc   = U(   - m_X/M) - U().


In these expressions,  and  denote the CoM and relative coordinates, respectively, with P and  representing their corresponding momentum. The masses M and m_r are defined as the total mass and the reduced mass of the exciton-electron system, respectively. The relationship between the coordinate system is summarised by:

    
    = m_X_X + m_e_e/M, 
       = _X - _e, 
       = m_e_X - m_X_e/M, 
    P   = _X + _e.


This decomposition demonstrates that if the expectation value of  is significantly smaller than the moiré length a_m, then the coupling term H_rc in Eq. (<ref>) becomes negligible. This motivates us to construct a basis using the eigenstates of the relative and CoM motions.





  §.§.§ Relative motion

In polar coordinates, the Hamiltonian of the relative motion reads 

    H_ rel = - ħ^2/2m_r( [2]r + 1/rr - 1/r^2[2]θ) + V(r).

Given the spherical symmetry of the interaction potential, angular momentum is conserved by H_ rel. The eigenstates therefore take the form ψ(r,θ) e^-i l θ/√(2 π), where l=0, ±1 , …, facilitating the discretization and efficient solution to yield the eigenstates and eigenenergies, denoted by |ψ_l n⟩ and E^r_ln, respectively. We point out that the inclusion of the moiré potential breaks the rotational symmetry, and therefore we must include multiple angular momentum channels in our basis for the trion wave function.



  §.§.§ Centre-of-mass motion

The CoM Hamiltonian can be efficiently solved using Bloch's theorem. In particular, at zero CoM quasi-momentum, we express the eigenvalue problem as

    H_ CoM|ν⟩ = E_ν|ν⟩,

where

    |ν⟩ = ∑_ϕ^(ν)_|⟩,

with the sum over all reciprocal lattice vectors. We point out that CoM quasi-momentum is a conserved quantity and we henceforth always set it to zero. We can index the reciprocal lattice vectors uniquely by the vector = (m,n) through _ = m _1 + n _6. Then the expansion coefficients satisfy the eigenvalue problem

	
    E ϕ^(ν)_   = | _|^2/2 M ϕ^(ν)_ + U_1  ϕ^(ν)_(m-1,n) + U_3 ϕ^(ν)_(m,n+1)  + U_5 ϕ^(ν)_(m+1,n-1)
            +  U_2  ϕ^(ν)_(m-1,n+1) +  U_4 ϕ^(ν)_(m+1,n) +  U_6 ϕ^(ν)_(m,n-1),


where ϕ^(ν)_≡ϕ^(ν)__.

 

  §.§.§ Eigenvalue problem for trion binding

The final ingredient required for our analysis is the matrix elements of relative-CoM coupling, which are given by

    
    ⟨ψ_nl, ν| H_ rc|ψ_n' l', ν'⟩ =  ∑_j=1^6 U_j ∑_∫ dr^2 ψ_nl^*() ψ_n'l'()  ϕ^ν_ϕ^ν'_ - _j(e^-m_X_j ·/M-1 ).


To solve for the trion's ground state, we expand the complete Hamiltonian in terms of a general two-body wave function with zero CoM quasi-momentum:

    |Ψ⟩ = ∑_nlνζ^ν_nl|ψ_nl, ν⟩.

Here, the expansion coefficients satisfy

    E ζ^ν_nl = (E^r_nl + E_ν)ζ^ν_nl + ∑_n'l' ν'⟨ψ_nl, ν| H_ rc|ψ_n' l', ν'⟩ζ^ν'_n'l'.

Given that the matrix elements of H_ rc are small, the ground state converges quickly in this basis. Equation (<ref>) allows us to determine trion binding in the presence of moiré, and to explore the effect of a finite range interaction on this binding, as discussed in the main text.

Figure <ref> shows the trion binding as a function of moiré depth. Here we also show the uncertainty in the moiré length by showing the range of trion binding energies (i.e., the blue shading). Importantly, the uncertainty from the uncertainty in the moiré length is not as large as the uncertainty in the uncertainty in the trion binding in the absence of moiré.







§ THEORETICAL MODEL FOR ELECTRONS AND EXCITONS AT TEXT
 


We now provide details on our theoretical model for the study of electrons, and their binding to an exciton, at ν=2. With two electrons per site, we expect the formation of a band insulator, albeit with strongly modified bands from strong electron correlations. Inspired by this, our model completely neglects hopping between sites, and hence we treat the electrons as localized to individual moiré unit cells; neighboring sites are accounted for via a Hartree potential. Since the electrons are confined to unit cells, we do not account for exchange interactions. Figure <ref> shows the theoretical moiré potential and the corresponding unit cell, which by construction is centered around a minimum of the potential. We begin our discussion by considering the singlet and triplet energy separation at ν=2, and then proceed to the three body problem, for two electrons and an exciton in a single unit cell.









 §.§ Singlet-Triplet energy difference

At ν=2 we have two electrons per moiré unit cell with Hamiltonian

    
    H^e_ν=2 = _e1^2/2  + U() + () + _e2^2/2 + U() + () + V_ RK( - )



where both electrons (1 and 2) enter the Hamiltonian identically. Here U() is defined in Eq. (<ref>) and () is the Hartree potential from  electrons on neighboring sites, which is given by

    () = ∫_' ∉ d^2r'   V_ RK( - ') n_e (').

Here we have introduced the electronic density n_e(), the moiré unit cell  and the Rytova-Keldysh potential

    V_ RK() = e^2/8 ε_0 r_0[ H_0 ( κ ||/r_0) - Y_0 (κ ||/r_0) ],

where H_0 and Y_0 are the Struve functions and Bessel functions of the second kind, respectively; κ≈ 4.4 is the average dielectric constant of the encapsulating materials; r_0 ≈ 3.4 nm is the screening length; e is the electric charge; and ε_0 is the vacuum permittivity <cit.>.

Since we assume that the electron wavefunctions in each moiré unit cell are identical (which defines n_e), we diagonalize the Hamiltonian in Eq. (<ref>) iteratively to reach self-consistency. Note that in performing the diagonalization we must take homogeneous Dirichlet boundary conditions such that the electrons are entirely localized in a moiré unit cell. It should also be noted that as presently shown the integral in Eq. (<ref>) is not convergent due to the long-range nature of the Rytova-Keldysh potential (which behaves asymptotically like the Coulomb potential). We regularize this divergence by replacing V_ RK() → V_ RK() e^- || / r̃, which removes the long range tail of the potential. While the choice of r̃ will always shift the absolute energies (which is not relevant to our results), for sufficiently large r̃  the wave functions and relative eigenenergies of the Hamiltonian in Eq. (<ref>) are not affected.

The eigenstates of Eq. (<ref>) have a definite parity under exchange: states that are even (odd) under exchange are associated with the spin singlet (spin triplet).
By discretizing and iteratively solving Eq. (<ref>), we find that the singlet is the ground state, but the triplet energy is only slightly higher.
Here, we emphasize that when solving for either the singlet or triplet wave function, we assume that all other sites are occupied with the same singlet or triplet wave function, respectively.
Figure <ref> shows the singlet-triplet energy difference as a function of the inverse number of grid points along each dimension of the unit cell N. The numerical data is well fit by a quadratic polynomial (i.e. c_0 + c_1/N + c_2 /N^2 with c_i the fitting parameters), which motivates us to extrapolate to 1/N=0, where the energy difference is 0.38 meV.



In order to understand this small energy difference, we decompose the singlet (Ψ^+) and triplet (Ψ^-) wave functions into single-particle orbitals. In particular, we find decompositions of form

    
    Ψ^+(,)    ≃1/√(2)(ϕ^+_0()  ϕ^+_1()+ϕ^+_1()  ϕ^+_0() )
    Ψ^-(,)    ≃1/√(2)(ϕ_0^-()ϕ_1^-()-ϕ_1^-()ϕ_0^-() ).


In both cases, the overlap of the approximation decomposition and the exact eigenstate exceeds 0.98. Figure <ref> shows the orbitals in the singlet and triplet cases, which are seen to be remarkably similar. This is a consequence of the Coulomb interaction dominating over other energy scales, such that the two electrons seek to avoid each other in the moiré unit cell. Consequently, even in the singlet case, the electrons seek out near-orthogonal orbitals, similar to those of the triplet.



The small singlet-triplet energy separation suggests that at the temperatures of the experiment (k_B T_ exp≈ 0.36 meV) the electrons form a mixture of singlets and triplets across the various moiré unit cells. Owing to this, in what follows we modify our iterative procedure by always defining the Hartree potential with the assumption that the neighboring sites host singlets. Assuming instead that the neighboring sites host triplets, or a mixture, makes no qualitative difference to the findings and only minor quantitative differences.



 §.§ Exciton-electron bound states


We now explore bound states between an exciton and two electrons. To this end, we first introduce the Hamiltonian for a free exciton

    H_ X = _ X^2/2,

where we once again enforce homogeneous Dirichlet boundary conditions, and we denote the ground state wave function and energy by ψ_ X^(0)(_ X) and E^(0)_ X, respectively.
The three-body Hamiltonian is then given by

    H^ eX_ν=2 = H^e_ν=2 +H_ X + V_σ_1( - ) + V_σ_2( - ).
 
Here, we take the exciton-electron interaction to be spin-dependent, thus phenomenologically accounting for how the spin of the electron inside the exciton affects its interaction with ↑ versus ↓ electrons. For simplicity, but without loss of generality, we take the electron inside the exciton to be spin ↓. Then, the exciton-electron interaction is given by

    
    V_↑ (r)    = -W_0/(a_ X^2 + r^2)^2
    
            V_↓ (r)    = -W_0/(a_ X^2 + r^2)^2 + P_0 exp(-r/a_ X).


For V_↑ we use the same potential that was employed in Eq. (<ref>), where W_0 is again chosen to give the correct trion binding in the absence of moiré, but now with the homogeneous Dirichlet boundary conditions. Meanwhile, for V_↓, we include a short-range repulsion, which imitates the Pauli exclusion principle. In particular, P_0 is chosen such that in the absence of moiré the trion is not bound. We find that the results are mostly insensitive to the precise value of P_0 as long as the potential supports no bound states.

Since the model in Eq. (<ref>) conserves all individual spins, we diagonalize the Hamiltonian in the individual spin basis {|↑↑⟩, |↑↓⟩, |↓↑⟩|↓↓⟩}. Resultingly, we find eigenstates and eigen-energies, which we denote by ψ^(σσ)_i(, , ) and E_i^σσ, respectively. 

We must be careful to ensure the eigenstates have the correct anti-symmetrization under electron exchange. In the case of |↑↑⟩ we only include eigenstates that satisfy ψ^(↑↑)_i(, , ) = - ψ^(↑↑)_i(, , ). Meanwhile, for differing spins the exchange operator no longer decomposes into a spin and orbital component; we instead construct anti-symmetrized eigenstates according to

    |ψ̅^(↑↓)_i⟩ = 1/√(2)( |ψ^(↑↓)_i⟩|↑↓⟩ -  |ψ^(↓↑)_i⟩|↓↑⟩),

where we have used the fact that ψ^(↑↓)_i(, , ) = ψ^(↓↑)_i(,  ,) and E^↑↓_i = E^↓↑_i.

In order to determine the oscillator strength of each bound state, we consider its overlap with the initial state: two electrons either in a singlet or triplet, and a zero-momentum exciton. In the initial state, in absence of magnetic field, the three triplet states are degenerate and symmetrically equivalent. Hence, we are free to choose our quantization axis at will, and for simplicity we choose the same basis as in our final state, where the direction is set by the electron spin in the exciton. An initial state with electrons forming the triplet |T_+⟩= |↑↑⟩ will only have overlap with states in the set {|ψ̅^(↑↑)_i⟩} (where the bar denotes anti-symmetrized states), whereas an initial state with electrons in a singlet |S⟩ = 1/√(2)(|↑↓⟩ - |↓↑⟩) or the triplet |T_0⟩ = 1/√(2)(|↑↓⟩ + |↓↑⟩) will have overlap with states in the set {|ψ̅^(↑↓)_i⟩}. We emphasize that while the exciton cannot flip the electron spin, it will still mix singlet and triplet states because the exciton becomes entangled with the electron spins.

Since the eigenstates associated with |↓↓⟩ are not bound, they are irrelevant for the attractive polaron lines and we henceforth exclude them, and the associated initial state |T_-⟩ = |↓↓⟩ from our analysis.





In general, we find that each initial state primarily has overlap with only one bound state within ∼10 meV of the ground state. In the case of |T_+⟩ there is precisely one bound state with finite overlap (with magnitude squared 0.57). Meanwhile, in the case of the singlet or |T_0⟩ there are two contributing bound states, but one dominates for each, which we show in Table <ref>. This leads us to identify each bound state by its associated initial state (which we denote by the spin-configuration, i.e., T_+, T_0 or S). Furthermore, we measure the energy of each bound state with respect to its associated initial state (where we take the energy of the exciton in the initial state as E^(0)_ X). 


Figure <ref> compares the energy of these three bound states, and the excitonic component of each bound state. We again plot the energy as a function of 1/N, and find it is well fit by a quadratic polynomial, which allows us to extrapolate to 1/N=0. Here, we find an energy splitting between the T_+ bound state and the S or T_0 bound states to be ∼ 3 meV. This splitting can be understood by looking at the exciton wave function in each of the three bound states (panels (b)–(d) in Fig. <ref>). Here we observe that in the case of T_+, the exciton can delocalize across both electrons, attracting them closer together without the short-range repulsion due to phase-space filling. This leads to the exciton effectively forming a bonding orbital with the electrons, and thereby lowering the overall energy.









apsrev4-2





 Paper Title: Slow decay rate of correlations induced by long-range extended Dzyaloshinskii-Moriya interactions.





Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute,  Chhatnag Road, Jhunsi, Prayagraj - 211019, India


§ ABSTRACT 
    We examine the impact of long-range  Dzyaloshinskii-Moriya (DM) interaction in the extended XY model on the phase diagram as well as the static and dynamical properties of quantum and classical correlation functions. It is known that in the nearest-neighbor XY model with DM interaction, the transition from the gapless chiral phase to a gapped one occurs when the strengths of the DM interaction and anisotropy coincide. We exhibit that the critical line gets modified with the range of interactions which decay according to power-law.   Specifically, instead of being gapless in the presence of a strong DM interaction,  a gapped region emerges which grows with the increase of the moderate fall-off rate (quasi-long range regime) in the presence of a transverse magnetic field. The gapless chiral phase can also be separated from a gapped one by the decay patterns of quantum mutual information and classical correlation with distant sites of the ground state which are independent of the fall-off rate in the gapless zone. We observe that the corresponding critical lines that depend on the fall-off rate can also be determined from the effective central charge involved in the scaling of entanglement entropy.  We illustrate that in a non-equilibrium setting, the relaxation dynamics of classical correlation,  the decay rate of total correlation, and the growth rate of entanglement entropy can be employed to uncover whether the evolving Hamiltonian and the Hamiltonian corresponding to the initial state are gapped or gapless. 

 Slow decay rate of correlations induced by long-range extended Dzyaloshinskii-Moriya interactions
    Tanoy Kanti Konar, Leela Ganesh Chandra Lakkaraju, Aditi Sen(De)
    
==================================================================================================









    
    



§ INTRODUCTION


Long-range order across quantum systems is desirable for a variety of quantum technological tasks, as it facilitates the sharing of correlation among distant parts of the system. Historically, such long-range order has been observed in many-body systems at quantum criticality or when the spectrum becomes gapless. In the case of the nearest-neighbor (NN) XY model with nonvanishing anisotropy parameter, it has been demonstrated that the classical correlations decay with the distance between the spins  exponentially (polynomially) when it is away from (at) the criticality.
Beyond NN models, long-range interacting (LR) systems with power-law decay in the range of interactions have been shown to exhibit polynomial decay of correlations, even when the spectrum is not gapless. These systems become an intense topic of research in the last few years, especially in the context of building analog quantum simulator<cit.> and due to their natural occurrence in several physical platforms such as the Rydberg atom arrays <cit.>, dipolar systems <cit.>, polar molecules <cit.>, trapped-ion setups <cit.>, and cold atoms in cavities <cit.>. Moreover, several established results that hold for nearest-neighbor systems came into the light of investigation in these LR systems in terms of Lieb-Robinson bound <cit.>, area law <cit.>, novel phases of matter  <cit.>,  dynamical phases <cit.> and from the perspectives of usefulness in quantum technologies like  quantum metrology <cit.> and quantum computation <cit.>. 








The nearest-neighbor Dzyaloshinskii-Moriya (DM) interaction, on the other hand,  involves an asymmetric exchange of spins caused by spin-orbit coupling <cit.>.

It has been widely explored in a variety of solid-state compounds, including Cu(C_6D_5COO)_2. 3D_2O <cit.>, Yb_4As_3<cit.>, Crl_3 <cit.> etc, exhibiting fascinating magnetic properties, like gapless chiral phase when the DM interaction strength is stronger than the anisotropy parameter in the transverse Ising <cit.>,  XY <cit.> and the gamma model <cit.>. It was also shown that the DM interaction can host a ground state in the gapless phase that contains logarithmic entanglement behavior, similar to the LR interacting systems which can enhance the fidelity in quantum teleportation <cit.>, thermal entanglement <cit.> and performance in quantum engines <cit.>.  



In the non-equilibrium domain, the DM interaction has also been examined in the context of non-equilibrium thermodynamics <cit.>, dynamical quantum phase transitions <cit.>, topological phase transition <cit.> and quantum speed limit <cit.>. 



 In this paper, we explore the long-range extended XY model in the presence of long-range extended DM interactions, which decay according to a power-law.  Specifically, we aim to analyze the responses of both long-range and DM interactions in equilibrium and non-equilibrium physics. In the former scenario, we present the phase-classification utilizing conventional order parameters, the law for the decreasing slopes of classical and quantum correlations, and the scaling of entanglement entropy (EE) with the corresponding central charge. In contrast to the NN model <cit.>, we show that given a fixed fall-off rates, there exists a gapped region in which the DM interaction is stronger than the anisotropy parameter. However, we establish that whenever the system is gapless, it is chiral. 
 Notably, in the static case, when the ground state belongs to the chiral gapless region, the classical correlation and quantum mutual information decay with constant decreasing exponent while this is not the case in the gapped regime.   In this model, we observe that the effective central charge associated with EE exhibits interesting behaviors: in the presence of LR and DM interactions, the central charge acts nonlinearly with multiple kinks that are absent from the extended Ising model without DM interactions.

  In comparison to the equilibrium scenario, the dynamical phases could not be efficiently characterized as they depend on the  initial state, evolving operator,  time and system parameters. 
   In order to distinguish between the gapless and gapped phases, we resort to a recently proposed relaxation dynamics of two-point classical correlations <cit.> and entanglement entropy of the evolved state in the transient regimes, as well as mutual information and classical correlation in the steady state domains
  (cf. <cit.> for other dynamical quantities). In our case, the initial state is  prepared by tuning the parameters of the Hamiltonian which have either gapless or gapped energy spectra while  evolution happens when the range of interactions is suddenly quenched, resulting in a gapless or gapped Hamiltonian. Specifically, in the transient  regime,  we show that the long-range DM interactions modify the exponent in the relaxation time of the two-point classical correlation between modes  and establish a generic  scaling law for the growth of entanglement entropy over time. Further, we  observe that  the behavior of mutual information and classical correlation between two arbitrary sites in the steady state differs depending on whether the Hamiltonians corresponding to the initial and dynamical states are in the same (gapped or gapless) phase or in different phases. 

  
  

  
  

 













 

The paper is organized as follows. In Sec. <ref>, we introduce the long-range extended XY model with an additional long-range DM term and find the energy spectra and eigenvectors. While 

we identify the conventional phases in Sec. <ref>, we indicate the decay of classical and quantum correlations in Sec. <ref>. The behavior of  entanglement entropy and the corresponding central charge with respect to system parameters are studied in Sec. <ref>. We then move on to the investigation of features in dynamical states, namely the relaxation dynamics of classical correlations, and the rate of entropy growth in the transient regime in Secs. <ref> and  <ref> respectively while we examine the decay of mutual information with distance between the spins  in the steady state  in Sec. <ref>. The results are summarized  in Sec. <ref>.



§ EMERGENT GAPPED PHASE WITH STRONG DM INTERACTIONS: PHASE DIAGRAM



The  critical lines  for the long-range interacting XY spin models in the presence of Coloumb-like interaction, described by a parameter α and for the nearest-neighbor transverse XY model with DM interaction  are known in the literature <cit.>. However,  the effects of long-range DM interactions has never been addressed before. In this situation, a possible query can be  – can the introduction of long-range DM interactions change the critical lines and phases of the LR Ising models? We answer this question affirmatively by first presenting the analytical prescription of this model for obtaining energy spectra and the eigenvectors, thereby leading to the modified phase-portrait of the extended long-range XY model <cit.> having long-range extended Dzyaloshinskii-Moriya interaction both in the thermodynamic limit and for finite systems.  In particular, we report the gapless chiral phase and its dependence on the fall-off rate, α. In addition, we identify appropriate order parameters that can reveal α-dependence in the modified magentic phase diagram. From the decay patterns of classical correlation including Landau parameters and quantum correlations in terms of quantum mutual information and block entanglement entropy, we again separate the gapless region from the gapped one in this model. 



 §.§ Energy spectrum of the long-range XY and DM interactions


The Hamiltonian, describing long-range extended XY model having long-range extended DM interaction in the presence of transverse magnetic field, reads as


    H   =   ∑_j=1^N∑_r=1^N/2 -J_r^'/𝒩 [1+γ/4σ_j^xℤ_r^zσ_j+r^x+1-γ/4σ_j^yℤ_r^zσ_j+r^y
          +D'/4(σ_j^xℤ_r^zσ_j+r^y-σ_j^yℤ_r^zσ_j+r^x) ]-h'/2∑_j=1^Nσ_j^z,

where ℤ_r^z = ∏_l=j+1^j+r-1σ_l^z, with ℤ_1^z=𝕀
and σ^k(k=1,2,3) is the Pauli matrix, J_r'=J/r^α with α being the strength of power-law decay of the model, γ is the anisotropy parameter, D is the strength of the DM interaction, 𝒩=∑_r=1^N/21/r^α is called the Kac-scaling factor <cit.> which ensures extensivity of the energy in the case of finite-size systems α<1 in the finite-size limit and h' is the strength of the external magnetic field. To make the analysis dimensionless, we redefine the magnetic field as h=h'/J and DM interaction strength as  D=D'/J. We consider the periodic boundary condition (PBC), i.e., σ_N+1≡σ_1.  This model can be solved analytically by mapping spins into free fermions under the following Jordan-Wigner transformation <cit.>:

    σ^x_n    =  ( c_n + c_n^†)
     ∏_m<n(1-2 c^†_m c_m) 
    σ^y_n    =i( c_n - c_n^†)
     ∏_m<n(1-2 c^†_m c_m) 
    and  σ^z_n   =1-2 c^†_n  c_n,

where c_m^†(c_m) is creation (annihilation) operator of spinless fermions and they follow fermionic commutator algebra. The corresponding free fermionic version of the Hamiltonian takes the form 

    H   =   ∑_n∑_rJ_r/2 ((1+iD)c_n^† c_n+r+γ c_n^† c_n+r^† + h.c )
          +h(c_n^† c_n-1/2),

where J_r=1/r^α. One can observe that the Hamiltonian which is quadratic in fermionic operators have long-range interaction both in hopping and superconducting terms. In addition, an extra hoping term emerges due to the DM interaction which also induces a complex phase. It is worthwhile to mention here that in the absence of DM interaction for γ=0, the model is 𝕌(1) symmetric while it is ℤ_2 symmetric with γ 0, D=0. The introduction of  DM interaction leads to a competition between these two symmetries.   In the short-range (SR) models, it is known that the conformal sector changes due to the shift of the symmetry sector <cit.>, responsible for a non-trivial phenomenon. 


Due to the presence of translational invariance in the system, momentum is a good quantum number. Hence, we perform a Fourier transform of the form, c_n=1/√(N)∑ e^-iϕ_kc_k where ϕ_k=2π k/N ∀ k∈[-N/2,N/2] and the corresponding Hamiltonian in the momentum basis can be written as H=k⊕ H_k with 

    H_k    =   ∑_k>0- [∑_r(Re(J_k^α)+DIm(J_k^α))c_k^† c_k
           +(Re(J_k^α)-DIm(J_k^α))c_-k^† c_-k
          -iγIm(J_k^α) (c_k^† c_-k^†- c_-k c_k)]
           +h(c_k^† c_k+c_-k^† c_-k)-h,

and J_k^α = ∑_r = 1^N/2J_r e^iϕ_kr. This Hamiltonian can be diagonalized by applying the Bogoliubov transformation,

    [    c_k; c_-k^† ]=[    u_k -v_k^*;    v_k  u_k^* ][    τ_k; τ_-k^† ],

where u_k=cosθ_k, v_k=isinθ_k and θ_k is called Bogoliubov angle, given by cosθ_k=-h+Re(J_k^α)-λ/√(2(λ^2-(Re(J_k^α)-h)^2)) and λ=√((h-Re(J_k^α))^2+(γ Im(J_k^α))^2). The eigenvalue corresponding to each momentum, k, is given as

    ϵ_± k=-D Im(J_k^α)  ±λ.

Note first that for D 0, although the Bogoliubov basis, (u_k, v_k)^T, are independent of DM interactions,  ϵ_kϵ_-k,  making the analysis non-trivial.

Further, when D=0, this model shows criticality at h=1 and at h=-1+2^1-α for α>1  corresponding to the both ends of the Brillouin zone, i.e., ϕ_k=0 and ϕ_k=π respectively.  In the case of α<1, the energy of the system diverges in the thermodynamic limit, hence, we consider large but finite system size <cit.>. With the introduction of DM, new gapless region emerges, which changes with α for D>γ. Such a region is known for the  nearest-neighbour XY model when D>γ <cit.>  and  gapless to gapped transition occurs at D= γ. However, we now establish that even when D>γ, long-range interactions can induce a gapped phase instead of a gapless one, which implies the modification of the critical lines in presence of LR interactions.






















 §.§ The gapless chiral phase depending on LR interactions


Apart from quantum phase transitions at zero temperature in which the energy gap vanishes, the LR Ising models possess another three distinctive regimes with respect to power-law fall-off rate α, namely, non-local (0 < α < 1), quasi-local (1 < α < 2) and local (α > 2) according to different scaling law of correlation length <cit.>. It is important to note here that in the absence of DM interaction, gap-closing never occurs by tuning the parameter α for h>0 except h_c=1. Further, we know that α≫ 2, the system with NN interactions becomes gapless for D>γ depending on h and the chiral phase emerges along with paramagnetic and ferromagnetic phases. Due to the long-range DM interactions, a competition between the range of interactions, α and the strength of DM interactions surfaces which becomes responsible for the change in condition for the gapless  to gapped transition. More specifically, for moderate values of α(<2), D>γ does not guarantee chiral gapless phase. We find the following in case of LR model. 




    For a given value of the magnetic field h (both in h≥ 0 and h<0), a gapped to gapless transition occurs when  D>γ depending upon the value of α instead of D=γ, known for the NN model.



    The proof is done by analyzing the dispersion relation in Eq. (<ref>). A system is said to be

gapped when the expression, Δ <cit.>, defined as

    Δ=max{min{ϵ_k},0}>0;  ∀ϕ_k∈ [-π, π],
 
otherwise, the spectrum is gapless (with Δ = 0). The exact point, where the gap-closing takes place, depends on the parameters of the Hamiltonian, and the corresponding momentum, k_F, called the Fermi point, can be obtained as a solution of ϵ_kk=0. However, due to the presence of long-range order, it is cumbersome to find the Fermi point analytically. By differentiating numerically, we observe that the gapless phase never occurs when D<γ while for D>γ, the gapless phase occurs although it is not ubiquitous, and changes with the variation of h,D,γ and α. In particular, when h<0, with the decrease of α, the system is gapped even when D>γ and two distinctive gapped regions emerge when 1≤α<2 and 0<α<1 which depend on the strength of D, thereby inducing (destroying) gapped (gapless) regions. However, for α>2, D > γ guarantees gapless phases and the gapless to gapped transition occurs at D=γ. On the other hand, we find, that when 1≤α<2, strong DM interactions are required to obtain the gapless phase and the desired asymmetric interaction strength for obtaining the gapless phase increases with the decrease of α (see Fig. <ref>). Note, further, that the maximum D necessary for the gapless phase does not happen exactly at α=1, but at a nearby point which can be due to finite size analysis.  



It is now evident that the interplay of α and D along with γ and h can change the critical lines of the phase diagram. Therefore, we now examine the phases in the (h,γ)- and (α, D)-planes.






α-dependent chiral phase. In the nearest-neighbor  XY model with DM interactions, the gapless phase gives rise to a chiral order that measures local current flow between the nearest neighbor sites. It can be established, by computing <cit.> the order parameter,

    𝒞ℋ= 1/4N∑_i=1^N σ^x_iσ^y_i+1-σ^y_iσ^x_i+1

between any two neighboring sites, i and i+1 in the ground state which describes the helical alignment of spins or chiral order in the z-direction. 


In the nearest-neighbor case, i.e., for the NN XY model with D > γ and suitable h values, it was shown that 𝒞ℋ > 0 when Δ = 0. 

In the LR model, we find that for a fixed h, α <2 and D > γ, whenever Δ = 0, i.e., the system is gapless, it possesses chiral order with order parameter 𝒞ℋ > 0 (as shown in Fig. <ref>). Interestingly, however, there exists region in which for D> γ, Δ>0 (as shown in the above proposition), and the corresponding chiral order parameter vanishes, i.e., 𝒞ℋ =0. 








Ferromagnetic and paramagnetic phases. We now calculate the long-range magnetic order parameter, C^xx_N/2 =  σ^x_1σ^x_N/2. When C^xx_N/2 0, the system is said to be in the ferromagnetic-x phase (FM_x) while paramagnetic when it vanishes. On one hand, we show that the FM_x phase occurs when the system is gapped and 0<h<h_c^1=1, 1<α<2, and D < γ (see Fig. <ref>). On the other hand, when h is negative, the system becomes ferromagnetic, i.e., C^xx_N/2 > 0  when h > h_c^2=-1+2^1-α with 1<α<2. When the parameters  are neither chiral nor ferromagnetic, the system is in a paramagnetic phase.




  













 §.§ Constant decaying exponent of correlations with DM interactions



From the above analysis, it is evident that the transition from the chiral gapless phase to the other magnetic gapped phase heavily depends on the fall-off rate α. Let us now investigate how the scaling of classical correlations and quantum mutual information between two spins (a measure of total correlations containing both quantum and classical correlations components) <cit.> can recognize the α-dependent phases discussed above. 

The classical correlation (CC) between two sites, separated by a distance R,  can be represented as C^xx_i,i+R = ⟨σ_i^x σ_i+R^x ⟩ while the quantum mutual information is defined as 
ℐ_i:i+R=S(ρ_i)+S(ρ_i+R)-S(ρ_i, i+R), where S(σ) = -(σlog_2σ) indicates the von-Neumann entropy of the state σ, ρ_i and ρ_i+R are the reduced density matrices of the joint state ρ_i,i+R. As we impose periodic boundary condition, the site index i can be ignored and both the classical and total correlation can be studied as C^xx_R and ℐ_R, where R = {1,2,…N/2}. In typical one-dimensional (1D) quantum spin models, the CC decays exponentially when it is away from criticality, whereas, at the criticality, the polynomial decay with R is observed. However, the extended Ising model deviates from this norm by showing different scaling laws near the critical points and away from it <cit.>. Moreover, in the long-range Kitaev chain (see Eq. (<ref>) with D = 0), quantum mutual information (QMI) persists between two distant segments of the chain, provided the system is in the non-local regime, i.e., α < 1, thereby  predicting the correlation between two distant regions of the system  <cit.>.

In our study, we expect to have a notable scaling law due to the presence of long-range and additional DM interactions which is indeed the case. Firstly, we observe that the trends of C_R^xx and ℐ_R with R is qualitatively different when the system is in the chiral phase and when it is not. Secondly, a counter-intuitive observation is that if one tunes the parameters 

 D, γ, h and α in such a way that 𝒞ℋ 0, the decay exponents of mutual information and classical correlation become constant as depicted in Fig. <ref>. Specifically, we observe that irrespective of the details of the power-law exponent and the strength of the magnetic field, the decay exponent of CC and QMI falls off as 

    ℐ_R    ∝ R^-1  and
    
        C^xx_R    ∝ R^-0.45,

which are independent of α provided the ground state belongs to the gapless chiral region (see Fig. <ref> and Table <ref>). This behavior of ℐ_R and C^xx_R highlights a more complex and highly correlated patterns between α and D within the gapless region. It indicates that although the range of interactions can control the critical lines in the (D,γ)-plane, differentiating gapless and gapped phases, the decay rate of CC and QMI in the gapless phase remains unaltered  and the chiral phase favors the sharing of both quantum and classical correlations between distant sites. More importantly, such α-dependent scaling behavior holds both for quantum and classical correlations (see Fig. <ref>).  

On the other hand,  when the system is in the non-chiral (which we refer to as achiral) phase, the ground state displays the decay exponent of both classical and quantum correlations which vary with α, i.e., depending on the internal descriptions of the model (see Fig. <ref> and Table <ref>). 

Through these observations, we elucidate the nuanced role of long-range DM interactions in modulating correlation properties across both chiral and achiral regions leading to the following proposition.


    In the chiral region of the LR XY model with DM interactions, the scaling exponents of CC and QMI are constant (independent of the fall-off rate, α) while their exponents depend on α in the achiral region.











 §.§ Amendment in scaling of entanglement entropy and central charge owing to long-range DM interactions



In a many-body system, an insight of universality classes can be obtained from the generic features of the ground state. A prominent physical quantity that serves the purpose is the block entanglement entropy  of block-size l <cit.>. In particular, after partitioning the N-party ground state into two blocks, containing l and N-l sites (with l ≪ N), we compute the block entanglement of the ground state as S_l = S(ρ_l) with ρ_l being the reduced density matrix of the N-party ground state.  The l-block entropy can be calculated from the two-point correlation functions between modes, m and n, given as

    C_mn=⟨Ψ|c_m^† c_n | Ψ⟩,    F_mn=⟨Ψ|c_m^† c_n^† |Ψ⟩,

where |Ψ⟩ is the ground state and the corresponding correlation matrix, consisting of correlation functions, can be written as

    ℂ_l=[ 𝕀-C   F; F^†   C ].

If λ_ls are the eigenvalues of ℂ_l, the von-Neumann entropy of |Ψ⟩ in the bipartition l:N-l is given as S_l=-∑_l=1^2Nλ_llog_2λ_l.

If a system is gapped which typically happens away from the criticality and when the range of interactions is relatively local, i.e., the norm of the Hamiltonian does not increase as the system size increases, S_l follows the area law, which implies S_l^NC∝ l^d-1, where d is the spatial dimension of the system and NC stands for “not at criticality" while it deviates from the area law at criticality <cit.>. Hence from the scaling behavior of S_l, one can detect the transition from a gapped to a gapless phase which is interesting since a simple scalar quantity can describe the essential properties of the Hamiltonian instead of a complete microscopic description. 

On the other hand, when the system becomes gapless, representing the critical point, e.g., in the transverse NN XY model,  S_l scales logarithmically, i.e., for a translationally invariant spin chain, at critical points, S_l^C= c/3log_2[N/πsinπ l/N]+a <cit.>, where c is the conformal or effective central charge in the conformal field theory and a is a non-universal constant. Further, it was shown that the central charge carries the signatures of an underlying symmetry. For example, c takes value 1/2 for the transverse-field Ising-like Hamiltonian at the critical point for which the Hamiltonian adheres to the ℤ_2-symmetry, while with the addition of asymmetric nearest-neighbor DM interaction, the Hamiltonian becomes  gapless region having c=1 and gapped with c=1/2. Hence EE turns out to be a powerful tool to separate a gapped phase from a gapless one.


It was also recently shown that if one considers long-range interactions instead of short-range ones <cit.>, where magnetic criticality depends on the value of α, EE depends upon α and the free-fermionic version of long-range Ising spin shows fractal entanglement apart from the volume and area law <cit.> and the scaling of entropy at critical points may not follow S_l^C. In this case, we can define an effective conformal charge, c_eff, to describe the universal properties of the system.

We are interested in finding how the scaling of S_l in the LR Ising Hamiltonian gets altered in the presence of DM interactions. Specifically, we will show the revision of a central charge c_eff, which occurs in S_l^C. Before presenting the effects of DM interaction on the central charge, we observe that at criticality, i.e., h_c^1 = 1 of the extended Ising model, c_eff monotonically increases when 1 < α < 2 and saturates to 1/2 at α≥ 2, (which is the scaling known for the NN model) as shown in Fig. <ref>(a). When α < 2, c_eff varies non-linearly with α which points out that due to the presence of long-range interaction, conformal symmetry breaks down. 

On the other hand, the introduction of long-range  DM interaction makes more modifications in c_eff –  (i) the saturation value changes to 1 when α→α_c = 2, with α_c being the point which mimics the NN transverse Ising model with NN DM interactions; (ii) for α<α_c, the conformal symmetry breaks down whereas when α > α_c,  c_eff→ 1 (see Fig. <ref>(b)). 
Our study reveals that even with the introduction of long-range asymmetric interaction, the conformal symmetry is not restored for all the values of α although c_eff indicating the conformal invariance begins for smaller values of α_c<2 which is found in the absence of DM interactions. This behavior can again be attributed to the fact that when the DM interaction is present, the system becomes gapless upon varying the power-law decaying factor α.











§ DECAY OF DYNAMICAL CORRELATION: UNCOVERING CHIRAL AND ACHIRAL PHASES



Let us now study the trends of physical quantities including quantum correlations in the time-evolved state in the transient and the steady state regimes, starting from the product state or the ground state which evolves according to the LR Hamiltonian with DM interactions. Our aim is to demonstrate that the scaling of correlations with time or with distant sites can indicate the transition from the chiral phase to the achiral ones. 



 §.§ Relaxation behavior of dynamical correlations






Let us first present the method to capture the relaxation of dynamical correlation <cit.> which is shown to comprehend distinct phases in equilibrium. The initial state and the evolution of the system in the momentum basis can be represented by the Bogoliubov–de Gennes (BdG) equation, given as


    ψ_k(0)=[u_k(0), v_k(0)]^T  and  ψ_k(t)=e^-iH_ktψ_k(0).

The corresponding dynamical correlation (DC) is defined as C_mn(t)=c_m^† c_nψ(t) which is a fermionic correlation between two modes. The relaxation of this correlation can be measured by taking the difference between correlation in arbitrary time and time in which the system reaches the steady state, given by δ C_mn(t)=C_mn(t)-C_mn(∞), describing the decay of the correlation function with time. Let the Bogoliubov angle of the initial and quenched Hamiltonian be η_k and η̃_k respectively and the difference between the angels is given as α_k=η_k-η̃_k. The relaxation of DC takes the form <cit.>

    δ C_mn(t) = 
       k∈gapped∫sin 2η̃_ksin 2α_k cos(ϵ_k+ϵ_-k)tcos[k(n-m)]= 
        
       k∉gapless∫sin 2η̃_ksin 2α_k cos(ϵ_k+ϵ_-k)tcos[k(n-m)],

where the initial state belongs to the gapped  and gapless regions in Eqs. (<ref>) and (<ref>) respectively. In our case, we have chosen m-n=1, but our result holds for any arbitrary m  and n.

When δ C_mn(t) ∼ t^-χ, χ indicates the dynamical relaxation exponent characterized by the distinctive values of the system parameters.

In the case of the nearest-neighbor XY model, it was shown that the exponent depends on the phase being commensurate or incommensurate. Specifically, in the commensurate phase, the derivative of the dispersion relation with respect to the momentum vanishes at the extreme ends of the Brillouin zone, while in the incommensurate phase, the derivative vanishes inside the Brillouin zone. If the initial state is a product state, i.e., the ground state of the model when the magnetic field goes to infinity and is then quenched with the Hamiltonian corresponding to the commensurate phase, we find χ = 3/2 while χ = 1/2 when quenched with the incommensurate phase <cit.>. 

In the NN XY model with DM interaction, if the initial state belongs to the gapless phase, the exponent changes to χ = 1 if the quenching Hamiltonian is in the commensurate phase, which indicates the presence of DM interaction. In the case of quenching by a Hamiltonian in the incommensurate phase, the exponent is either χ = 1/2 or χ = 1 depending on the parameters of the initial state <cit.>. On the other hand, considering the LR Kitaev chain, it was found that the scaling law behaves similarly to the SR ones when the initial and post-quench Hamiltonian are non-critical although the exponent changes with the parameters chosen from an equilibrium phase transition <cit.>. 

The model considered in this work possesses both LR DM interactions and LR interactions in the xy-plane and hence the interplay between α and D as mentioned in the static scenario can influence the scaling of δ C_mn(t), resulting to different χ. For investigations, possible three situations arise in the non-local regime (α < 1) - (ni) both pre- and post-quench Hamiltonian are in the chiral phase; (nii) only pre-quench Hamiltonian belongs to the chiral phase; (niii) only post-quench Hamiltonian is chosen from the chiral phase. All three cases can also be considered when α is chosen from a quasi-local regime, i.e., when α > 1 which we refer to as (qi) - (qiii). Eg., we find that χ∼ 1.5 for the case of (niii) and when (qii) χ∼ 0.3 (see Fig. <ref>). 

This shows that the exponent in DC can distinguish between the gapped and the gapless phases  of the post- and pre-quench Hamiltonians  (see Fig. <ref>). In addition, the robustness of correlation can be defined as a quantity 1/χ such that if the value of χ is small, the system does not reach the steady state correlation value for a significantly long time. This reinforces the above result that the state from the gapless phase contains a strong correlation both in terms of space and time. 



 §.§ Slow decay of total correlation and two-point correlation with LR DM interaction



We now focus on six scenarios ((ni) - (niii) and (qi) - (qiii)) to study the decay pattern of total correlation and classical correlation with R in a steady state limit, represented by ℐ_R(∞) and 𝒞^xx_R(∞) respectively. Even though the evolution is unitary, as we are studying the properties of a subsystem, a steady state can be reached in the dynamics. When the quenching Hamiltonian is in the gapped phase ((nii)), the steady state maintains both classical and quantum correlations between distant spins, and  persists even at large distance R (see Figs. <ref>(b), (c), (f) and (g)). Conversely, a gapless quenching Hamiltonian ((ni) and (niii)) results in a steady state where classical correlations are present only between nearby spins as depicted in Figs. <ref> (a), (d), (e) and (h). Notice that the decaying natures of ℐ_ℛ(∞) and C^xx_R(∞) are independent of the initial state. Remarkably, the behavior of total and classical correlations remains similar when the distance between the spins is small. 
















 §.§ Effects of long-range in dynamics of EE



In a final attempt to distinguish the gapless and gapped phases under dynamics, we investigate the
growth of entropy with time. The initial states are prepared as the ground states of gapped as well as from the gapless regions and EE, S_l(ρ(t)) ≡ S_t for a fixed l is computed after quenching the system to the gapped or gapless regions by changing the parameters α for a given γ, h and D. Let us denote the initial and final fall-off rates be α_i and α_q respectively. Firstly, we notice that the rate of growth is shown to be linear in time in the case of the XY model with NN <cit.> and long-range interacting model <cit.>, with the initial state being a product state. The trends of S_t with time depend on both α_i and α_q, thereby indicating its dependence on pre- and post-quench Hamiltonian. 

  Secondly, the rate of increment of EE with time is faster if the initial state is in a gapless region and the EE value gets saturated around time t ∼ l/2, as observed in the case of the nearest-neighbor Ising model quenched from a non-critical regime to a critical one <cit.>. However, if the initial and final states are chosen from a gapless region, the increment rate is slow and the saturation is reached faster around t=l/2 compared to a gapped case (see Fig. <ref>(d) and (e)). Also, the scaling of entropy increment until it reaches saturation can be approximated as

    S_t^±∝ a_1t/α_i ±α_q+a_2,

where a_1, and a_2 are constants (see Fig. <ref>).

When both the initial and the final Hamiltonian are gapped or one of the pre- or post-quench Hamiltonian is gapped, the saturation of EE requires much larger time than that obtained with  the Hamiltonians corresponding to the initial and final states being gapless. 








§ CONCLUSION



Quantum long-range systems are intriguing due to their non-local properties, often displaying counter-intuitive phenomena compared to a short-range Hamiltonian. These models are pervasive in physically realizable systems such as trapped ions and atomic, molecular, and optical setups.   Therefore, comprehending their universal features is essential for the advancement of quantum technologies and condensed matter physics.





We found that the long-range extended Dzyaloshinskii-Moriya (DM) interactions with the extended XY model have unique consequences on the critical behavior of the system with a variation of the fall-off rate, α, in which the range of interactions between the sites decays. Specifically,  even with a comparable or more DM interactions than the anisotropy parameter, we proved that the system remains gapped depending on α, which  varies with the direction of the transverse magnetic field although the gapless phase still possesses chiral order.  
By analyzing classical and total correlations between two arbitrary sites of the ground state in the presence of DM interactions, we observed that both  of them decay with the distance between the sites and the decay exponent is independent of α in the gapless region while it depends on the system parameters in the gapped regime and  decays faster in this case than the gapless zone.  Additionally, we exhibited   the departures from the universal trend in the block entanglement entropy in presence of long-range DM interactions which could successfully capture the interplay between long-range and DM interactions.





This study investigated how dynamical  correlations change in time when the energy spectra of the initial and the evolving Hamiltonian are gapped and gapless, driven by  α for a fixed strengths of DM interactions, and magnetic fields. 

The scaling of both classical correlations and mutual information in the  steady states reveals that the sharing of correlations is favorable when the evolving Hamiltonian is gapped.  Moreover, we found that the growth of entanglement entropy with time  is faster in the gapless regime than in the gapped phase and it saturates quickly when both the pre- and post-quench Hamiltonians are gapless.   On the one hand, these findings contribute to the understanding of fundamental aspects of quantum phase transitions, while on the other, sharability of correlations and entanglement  of the ground state as well as their dynamical patterns can be vital for establishing quantum links amongst quantum computers.



We acknowledge the support from the Interdisciplinary Cyber Physical Systems (ICPS) program of the Department of Science and Technology (DST), India, Grant No.: DST/ICPS/QuST/Theme- 1/2019/23. We  acknowledge the use of https://github.com/titaschanda/QIClibQIClib – a modern C++ library for general purpose quantum information processing and quantum computing (<https://titaschanda.github.io/QIClib>), and the cluster computing facility at the Harish-Chandra Research Institute. This research was supported in part by the INFOSYS scholarship for senior students.







 

§ CORRELATION FUNCTIONS




The two-point correlation functions are the fundamental constituents of our study. In the context of the ground state or the thermal state, the two-site correlation function between sites separated by a distance of R can usually be expressed as a determinant of an R × R Toeplitz matrix <cit.>.
However, as the Hamiltonian includes asymmetric interaction, the bipartite reduced state obtained from the N-party ground  state after tracing out N-2 parties includes the correlators like C^xy≡⟨σ^x_i σ^y_i+R⟩ = tr (σ^x ⊗σ^y ρ_R) and C^yx along with  C^ii (i=x,y,z).  Thus for the corresponding density matrix of the two distant parties of the spin chain and the time-evolved state, we need to deal directly with the Pfaffians in order to evaluate the correlations as a function of time as well as distance since there is a formation of chiral phase due to DM interaction. 

Using the Pfaffian formalism, we first write the spin correlation functions as 








    
    .
    [ C^lm_i,i+R = ⟨σ_i^l σ_i+R^m ⟩= c(l,m)pf  |  . I^lm_1,2                                                      …                                             I^lm_1,R-1                                                 J^lm_1                                                 F^lm_1                                               G^lm_1,2                                                    c .                                                      …                                               G^lm_1,r;                                                                                                             …                                                      …                                                      …                                                      …                                                      …                                                    c .                                                      …                                                      …;                                                                                                                                                         I^lm_R-2,R-1                                               J^lm_R-2                                               F^lm_R-2                                             G^lm_R-2,2                                                    c .                                                      …                                             G^lm_R-2,R;                                                                                                                                                                                                                    J^lm_R-1                                               F^lm_R-1                                             G^lm_R-1,2                                                    c .                                                      …                                             G^lm_R-1,R;                                                                                                                                                                                                                                                                               E^lm                                                 D^lm_2                                                    c .                                                      …                                                 D^lm_R;                                                                                                                                                                                                                                                                                                                                    K^lm_2                                                    c .                                                      …                                                 K^lm_R;                                                                                                                                                                                                                                                                                                                                                                                         H^lm_2,3                                                      …                                               H^lm_2,R;                                                                                                                                                                                                                                                                                                                                                                                                                                                       …                                                      …;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H^lm_R-1,R ] |,






where c(x,x)=c(y,y)=(-1)^R(R+1)/2, 

    I^xx_μ,ν      =  ⟨ A_l+μ(t) A_l+ν(t)⟩, 
    
    J^xx_μ,        =I^xx_μ,R, 
     
    H^xx_μ,ν      = ⟨ B_l+μ-1(t) B_l+ν-1(t)⟩, 
    
    K^xx_ν      = H^xx_1,ν, 
     
    G^xx_μ,ν      = ⟨ A_l+μ(t) B_l+ν-1(t)⟩, 
     
    F^xx_μ      =G^xx_μ,1,  
    
    E^xx      =G^xx_R,1,  
    
    D^xx_ν      =G^xx_R,ν,



    I^yy_μ,ν      =  ⟨ A_l+μ-1(t) A_l+ν-1(t)⟩, 
    
    J^yy_μ      =I^yy_μ,R, 
     
    H^yy_μ,ν      = ⟨ B_l+μ(t) B_l+ν(t)⟩, 
    
    K^yy_ν      = H^yy_1,ν, 
     
    G^yy_μ,ν      = ⟨ A_l+μ-1(t) B_l+ν(t)⟩, 
     
    F^yy_μ      =G^yy_μ,1,  
    
    E^yy      =G^yy_R,1,  
    
    D^yy_ν      =G^yy_R,ν,

with s(x,y)=s(y,x)= -i (-1)^R(R-1)/2, we have

    I^xy_μ,ν      =  ⟨ A_l+μ(t) A_l+ν(t)⟩, 
    
    G^xy_μ,ν      = ⟨ A_l+μ(t) B_l+ν(t)⟩, 
    
    J^xy_μ      = G^xy_μ,0, 
     
    F^xy_μ      =G^xy_μ,1,  
    
    H^xy_μ,ν      = ⟨ B_l+μ(t) B_l+ν(t)⟩, 
    
    E^xy      =H^xy_0,1,  
    
    D^xy_ν      = H^xy_0,ν, 
      
    K^xy_ν      =H^xy_1,ν,

and

    I^yx_μ,ν      =  ⟨ A_l+μ-1(t) A_l+ν-1(t)⟩, 
    
    G^yx_μ,ν      = ⟨ A_l+μ-1(t) B_l+ν-1(t)⟩, 
    
    J^yx_μ      = I^yx_μ,R, 
     
    F^yx_μ      =I^yx_μ,R+1,  
    
    E^yx      =I^yx_r,r+1,  
    
    D^yx_ν      = G^yx_R,ν,  
    
    K^yx_ν      =G^yx_R+1,ν, 
     
    H^yx_μ,ν      = ⟨ B_l+μ-1(t) B_l+ν-1(t)⟩.


Each of the elements in the Pfaffian can be constructed from the expectation values of one of the operators, A_l A_l+R, B_l B_l+R and A_l B_l+R with

A_i=c_i^†+c_i,   B_i=c_i^†-c_i.





When recasted in the same Fourier basis as taken while diagonalization, the operators for the k^th momentum have the matrix form

    (A_l A_l+R)_k =  [[          cos(kR)        -isin(kR)                0                0;        -isin(kR)          cos(kR)                0                0;                0                0 cos(kR)-isin(kR)                0;                0                0                0 cos(kR)+isin(kR) ]],


    (B_l B_l+R)_k =   [[          -cos(kR)           sin(kR)                 0                 0;          -sin(kR)          -cos(kR)                 0                 0;                 0                 0 -cos(kR)+isin(kR)                 0;                 0                 0                 0 -cos(kR)-isin(kR) ]],



    (A_l B_l+R)_k =   [[   cos(kR)  isin(kR)         0         0; -isin(kR)  -cos(kR)         0         0;         0         0         0         0;         0         0         0         0 ]].
 
The single-site transverse magnetization in the same Fourier basis is given by

    σ_z^k = [[ -1  0  0  0;  0  1  0  0;  0  0  0  0;  0  0  0  0 ]].






The corresponding expectation value of the operators for each value of k, say O^k, with respect to the time-evolved  state is computed as ⟨ O ⟩ = ∑_k=1^N/2Tr(ρ_β^k(t) O^k).

Once we find all the single-site magnetizations m_j^α ∀ α={x,y,z} at j=i and j=i+R and all possible two-site correlation functions C_i,i+R^l,m ∀  l,m={x,y,z}  from the Pfaffians described above, we can construct the two-site reduced density matrix between sites i and I+R as we know any two party density matrix for the given Hamiltonian is given as


    ρ_ij=1/4 (𝕀_4 + m^z(σ^z_j+σ^z_i) + ∑_k,l C^kl_ijσ^k_i ⊗σ^l_j),

where k,l∈{x,y}.








 Paper Title: Persistent Topological Negativity in a High-Temperature Mixed-State.






Department of Physics, University of California, Santa Barbara, CA 93106, USA	


Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106


Department of Physics, University of California, Santa Barbara, CA 93106, USA	

§ ABSTRACT 
We study the entanglement structure of the Greenberger–Horne–Zeilinger (GHZ) state as it thermalizes under a strongly-symmetric quantum channel describing the Metropolis-Hastings dynamics for the d-dimensional classical Ising model at inverse temperature β. This channel outputs the classical Gibbs state when acting on a product state in the computational basis.  When applying this channel to a GHZ state in spatial dimension d>1, the resulting mixed state changes character at the Ising phase transition temperature from being long-range entangled to short-range-entangled as temperature increases. Nevertheless, we show that the topological entanglement negativity of a large region is insensitive to this transition and takes the same value as that of the pure GHZ state at any finite temperature β>0. We establish this result by devising a local operations and classical communication (LOCC) “decoder" that provides matching lower and upper bounds on the negativity in the thermodynamic limit which may be of independent interest. This perspective connects the negativity to an error-correction problem on the (d-1)-dimensional bipartitioning surface and explains the persistent negativity in certain correlated noise models found in previous studies. Numerical results confirm our analysis. 


 Persistent Topological Negativity in a High-Temperature Mixed-State
    Sagar Vijay
    February 2024
===================================================================



Introduction: The nature and fate of quantum entanglement in mixed-states of quantum many-body systems – arising in a description of thermal quantum matter  or after a pure state experiences local decoherence – have been recently investigated in order to understand the universal quantum correlations that can persist in open quantum systems <cit.>. Patterns of long-range-entanglement (LRE) in mixed quantum matter are intricately related to quantum many-body phases which can act as quantum error-correcting codes <cit.>.  Thermal states of quantum many-body systems are also now known to be efficiently classically-preparable at sufficiently high temperatures <cit.>.

The entanglement negativity provides a well-studied diagnostic of bipartite entanglement in a mixed-state, which vanishes if the state can be prepared by from a product state by local operations and classical communication (LOCC) across the entanglement bipartition <cit.>.  While the negativity of a region A grows with the area of its boundary in a generic Gibbs state <cit.>, contributions to the negativity which are independent of coarse-grained details of the bipartitioning surface signal the presence of LRE in a mixed-state and can probe the presence of mixed-state topological quantum order <cit.>.  Diagnostics of non-local mixed-state entanglement, such as the topological entanglement negativity <cit.>, are challenging to study analytically, and the subtle behavior of mixed-state LRE under a finite-depth local quantum channel (FDLC) remains to be fully understood <cit.>.  

In this work, we investigate the fate of mixed-state quantum entanglement under a Metropolis-Hastings <cit.> quantum channel describing the equilibration of a thermal classical system, specifically that of a d-dimensional Ising model at inverse temperature β. When acting on a pure state in the computational basis, this channel simply prepares a classical thermal state of the d-dimensional Ising model. 

The nature of the mixed quantum many-body state produced by the action of this channel on a generic input state is more striking. When the initial state is chosen to be a Greenberger–Horne–Zeilinger (GHZ) state, a symmetric, long-range-entangled ground-state of the ferromagnetic  phase, we show that the resulting quantum many-body mixed state at any finite temperature β > 0 has a negativity which is asymptotically exactly that of the pure GHZ state in the limit that the boundary of the sub-region becomes large. In spatial dimensions d>1, the negativity of a large, contiguous region in this mixed state ρ_β is thus a constant, independent of geometric details of the bipartitioning surface, and insensitive to the thermal phase transition, as summarized in Eq. (<ref>). Nevertheless, the entanglement properties of the mixed-state do change across this transition; we show that ρ_β cannot be expressed as a convex sum of short-range-entangled (SRE) pure states in the ordered phase but is preparable using an FDLC acting on a product initial state in the high-temperature, disordered phase.  This observed behavior is in stark contrast to the negativity of the Gibbs state of a local, quantum many-body Hamiltonian which necessarily vanishes above a finite temperature <cit.>.  

This result is established by using the fact that the entanglement negativity is an entanglement monotone <cit.>, so that the negativity of a subregion A of any state ρ, denoted (ρ), cannot increase under an LOCC operation  across the bipartitioning surface

    ([ρ])≤(ρ).

This property lends a new perspective on mixed-state negativity, which is of possibly broader interest outside of the specific mixed-states that we study. We explicitly construct an LOCC operation that recovers the GHZ state at any finite temperature and for any bipartition in the asymptotic limit that the boundary A becomes large.  We show that for a contiguous bipartition, this channel is the decoder for a (d-1)-dimensional repetition code, which the two parties in the LOCC protocol must use to determine the precise unitary circuit that must be applied to recover the GHZ state.  The well-known success of the (d-1)-dimensional repetition code up to a maximal error strength in dimensions d > 1 is intimately related to the recovery of the GHZ state, and thus the constant entanglement negativity, at any finite temperature. 

We comment on broader utility of this perspective for understanding mixed-state negativity, by demonstrating that this can provide a simple understanding of why the topological entanglement negativity can remain a constant in correlated noise models (e.g. in <cit.>).


Setup: Consider N qubits on a d-dimensional hypercubic lattice, in a state described by the density matrix

    ρ_β≡1/_β∑_σe^-β H|ψ_σ⟩⟨ψ_σ|.

Here, 

    H ≡ -J∑_⟨ i,j⟩Z_iZ_j

is the energy of a classical Ising model on the d-dimensional hypercubic lattice with ferromagnetic (J>0) nearest-neighbor interactions, while _β≡(e^-β H) is the corresponding partition function at inverse temperature β.  Furthermore, the state
|ψ_σ⟩≡ (|σ⟩ + U|σ⟩)/√(2), 
where |σ⟩≡|σ_1,…,σ_N⟩ is a product state in the Pauli Z basis, and U≡∏_jX_j is a unitary transformation which generates the Ising symmetry of Eq. (<ref>).

The mixed-state ρ_β is naturally obtained by starting with a GHZ state |ψ_+⟩ = (|↑⋯⟩ + |↓⋯⟩)/√(2), which is an equal-amplitude superposition of the macroscopically-distinct all-up and all-down qubit configurations, and then applying a channel Φ_β which describes a Metropolis-Hastings algorithm <cit.> in which the energy H is measured, and then a unitary operation X_j is applied on a random qubit with probability min{1, exp(-β Δ E_j)}, where Δ E_j≡ E_new - E_old is the energy difference between the new and old qubit configurations. 

Repeated measurements and feedback, starting from any product state in the Pauli Z basis, will eventually produce a Gibbs ensemble of the d-dimensional classical Ising model at inverse temperature β <cit.>.   We specifically define Φ_β to be  repeated application of the Metropolis-Hastings algorithm so as to produce a thermal steady-state when acting on the all-up state ρ_0≡|↑⋯⟩⟨↑⋯|, i.e.

    Φ_β[ρ_0] = 1/_β∑_σe^-β H|σ⟩⟨σ|.

Each operation that comprises the channel Φ_β manifestly commutes with the Ising symmetry transformation U.  Therefore, U is a strong symmetry of this channel <cit.>, i.e.    UΦ_β[ρ]=Φ_β[Uρ] for any state ρ, so that

    Φ_β[|ψ_+⟩⟨ψ_+|] = 1+U/√(2)Φ_β[ρ_0]1+U/√(2) = ρ_β.


Mixed-State Entanglement:
The state ρ_β witnesses a thermal phase transition with increasing temperature in spatial dimensions d>1 between a ferromagnetically-ordered and a disordered state. At any temperature, the state ρ_β has a strong Ising symmetry Uρ_β = ρ_β.  However, the ferromagnetically-ordered phase is also characterized by long-range-order in local operators which are charged under this symmetry (ρ_βZ_iZ_j) |i-j|→∞0.  These conditions are sufficient to establish  <cit.> that this mixed-state cannot be described as an ensemble of SRE pure-states.  

We may also show that in the disordered phase, ρ_β can be prepared by an FDLC, and is thus an SRE mixed-state [The ability to prepare ρ_β by an FDLC implies that this mixed-state describes an ensemble of SRE pure state.].  In the Supplemental Material <cit.>, we formally show that ρ_β = ℰ_ZZ[|ϕ(β)⟩⟨ϕ(β)|], 
where ℰ_ZZ is a quantum channel that describes maximal dephasing of Z_iZ_j along all nearest-neighbor bonds.  Furthermore, we show <cit.> that the wavefunction |ϕ(β)⟩≡_β^-1/2∑_μe^-β H/2|μ⟩, 
where |μ⟩ denotes a product state in the Pauli X basis, is the exact ground-state of the  Hamiltonian

    H_0(β) = ∑_j[∏_i∈NN(j)e^-β J Z_iZ_j -X_j],

which exhibits a phase transition at a critical value of β=β_c(d) which is in the d-dimensional Ising universality class.  When β<β_c(d),  |ϕ(β)⟩ is the Ising-symmetric ground-state in the disordered phase and can thus be prepared from a product state using a finite-depth local unitary circuit.  Thus in this range of temperatures, ρ_β can be prepared by an FDLC by composing this circuit with the quantum channel ℰ_ZZ.


Conditional Mutual Information:
We now investigate entanglement properties of ρ_β and their relation to the Ising phase transition. The conditional mutual information (CMI) is a measure of tripartite correlations in a state and has been used to study various aspects of entanglement in quantum systems<cit.>, including probing mixed state quantum phases of matter<cit.>. 
Given subsystems A, B, and C, the CMI is defined as

    I(A:C|B)
           =S(AB)+S(BC)-S(B)-S(ABC).

Consider the partitioning shown in Fig. <ref> with r>0, where C is taken to be the complement of AB. Note that for any proper subset A of qubits, ρ_β,A=_A̅(ρ_β)=ρ_β,A^Cl, where ρ_β^Cl=e^-β H/_β is the classical Gibbs state [This in particular shows that for mutual information we have I(A:C)=I^Cl(A:C) provided that r>0, where I^Cl is computed with respect to ρ_β^Cl.]. This shows that the first three terms in Eq. (<ref>) take the same value when computed for ρ_β as for ρ_β^Cl. Moreover, it is straightforward to use  Eq.(<ref>) to show S(ρ_β)=S(ρ_β^Cl)-log(2). Therefore I(A:B|C)=log(2)+I^Cl(A:C|B),
where I^Cl(A:C|B) is the CMI computed for ρ_β^Cl. On the other hand, since B is assumed to be non-empty, there is no interaction term in H with support on both A and C and hence I^Cl(A:C|B)=0 <cit.>, and

    I(A:B|C)=log(2),

independent of r, temperature β and the system size. It is worh mentioning that in contrast, for thermal Gibbs state of local quantum Hamiltonians, CMI decays exponentially with r <cit.>. 


Entanglement Negativity:
We now investigate the entanglement negativity of a subregion A of the mixed-state (<ref>), (ρ_β) ≡ (||ρ_β^T_A||_1 - 1)/2.  We will show that for a region A, 

    (ρ_β) = {[ 1/2 - O(exp(-|∂ A|))                  β>0;                     ;                    0                  β=0 ].

where |∂ A| is the number of qubits on the boundary of A. The second result is easily established. At infinite temperature, ρ_β = 0 = (1 + U)/2^N, so that ρ_β^T_A = ρ_β and (ρ_β=0)=0. 

The first result is determined as follows. The negativity is a convex function <cit.> so that 

    (ρ_β) ≤1/_β∑_σe^β J∑_⟨ i,j⟩σ_iσ_j(|ψ_σ⟩⟨ψ_σ|) = 1/2

for any region A, since |ψ_σ⟩ is related to the GHZ state |ψ_+⟩ by single qubit gates, so that (|ψ_σ⟩⟨ψ_σ|) = (|ψ_+⟩⟨ψ_+|).  The latter is exactly 1/2 for any region A. The negativity is also an entanglement monotone <cit.>, see Eq. (<ref>).  Here, we show that given the state ρ_β for any β > 0 and any region A, there is an LOCC operation  with respect to A and its complement A̅ that recovers the GHZ state with a fidelity that approaches unity exponentially quickly in the size of the boundary of A.  Combining this with (<ref>), allows us to establish Eq. (<ref>).  



We first construct the LOCC channel  for a contiguous bipartition, before generalizing this construction for a non-contiguous subregion A.  The state ρ_β describes the ensemble of pure-states |ψ_σ⟩ drawn with probability e^β J∑_⟨ i,j⟩σ_iσ_j/_β. Given a state from this ensemble, we measure the operators Z_iZ_j along all nearest-neighbor bonds lying exclusively within A or .  The resulting pattern of domain walls – an example of which is shown in the left panel of  Fig. <ref> in two spatial dimensions – is consistent with having drawn the state  |ψ_σ⟩ or the state  ∏_j∈ AX_j|ψ_σ⟩ and thus leaves one classical bit's worth of ambiguity in determining the precise state.  Equivalently, this ambiguity corresponds to the two possible ways that the observed domain walls can connect across the bipartition, as shown in Fig. <ref>.   

The preparation of the GHZ state through LOCC thus maps onto a decoding problem in a classical (d-1)-dimensional repetition code living on the interface between A and A.  The classical bit of information is the relative alignment of any two qubits across the bipartition in the Pauli Z basis (e.g. in Fig. <ref>, the classical bit can be taken to be τ_1≡σ_1σ_1).  The measured “syndromes" of the repetition code (e.g. τ_1τ_2 = (σ_1σ_2)(σ_1σ_2)) and the Boltzmann weights for the d-dimensional Ising model are used to perform maximum-likelihood decoding to guess the correct state.  Specifically, let w_1 and w_2 denote the Boltzmann weights corresponding to states (i) |ψ_σ⟩ and (ii) ∏_j∈ AX_j|ψ_σ⟩, respectively. We will guess that the state we have is (i) with probability p ≡ w_1/(w_1+w_2) and (ii) with probability 1-p.  We will subsequently apply the appropriate single-site unitary gates to turn this into the GHZ state.
The LOCC channel is thus 


  * Measure Z_iZ_j along all nearest-neighbor bonds lying exclusively within A or .

  * Let |σ⟩ be a product state in the Pauli Z basis which is consistent with the measurement outcomes. Apply the unitary U_1 = ∏_jX_j^(1-σ_j)/2 with probability p, or the unitary U_2 = ∏_j∈ AX_j∏_jX_j^(1-σ_j)/2 with probability 1-p. 


At any finite temperature, the relative probability p/(1-p) = w_1/w_2 scales exponentially in the size of the boundary of A, due to the fact that the energy of an Ising domain wall is extensive in its length.  Thus, the decoder for the repetition code should be successful at any finite temperature, and fail exactly at infinite temperature when w_1 = w_2 = 1/2.  This is consistent with the known behavior of the repetition code with single-qubit bit-flip errors with probability q, for which perfect decoding is possible for any q≤ 1/2. 

We may generalize this channel for the case where A is a non-contiguous region.  In this case, step 1 (above) is replaced by measurements of Z_iZ_j across all pairs of sites entirely within A and entirely within A̅.  As before, only one classical bit's worth of ambiguity thus remains after these measurements, in determining the precise state.  The quantum circuit that takes this state onto the GHZ state is then determined by a decoder for a repetition code with |∂ A| sites.  Thus, decoding the correct quantum circuit should again be successful at any β > 0, as |∂ A| becomes large, regardless of the geometry of the bipartition.  

Consistent with these expectations, we show <cit.> that for any bipartition, the fidelity of the recovered state R(ρ_β) with the GHZ state is exactly

    ℱ((ρ_β),|ψ_+⟩⟨ψ_+|) = 1/2[1 - ⟨tanh(β H_∂)⟩]

where H_∂ denotes terms in the Ising Hamiltonian acting exclusively along bonds that cross the bipartition between A and A and the expectation value is taken in the thermal state for the classical Ising model.  At any finite temperature, the energy H_∂ is negative, and extensively large in the boundary of A, so that ⟨tanh(β H_∂)⟩→ 1 as A becomes large.

Numerical Results: 
To numerically confirm this result, we obtain an alternate expression for the entanglement negativity of any bi-partition <cit.> 

    𝒩_A[ρ_β]=1/4|1-e^2β H_∂[σ]|.

One may now use classical Monte Carlo simulations to compute the thermal average in Eq.(<ref>).
However, care must be taken with sampling the spin configurations, since the expression inside the absolute value could be exponentially large for configurations with an exponentially small Boltzmann weight, in such a way that the overall contribution to the expectation value is O(1). 




The aforementioned issue can be avoided by using a technique known as importance sampling. Let q[σ] denote the probability distribution defined as q[σ_A,σ_]=1/2p[σ_A,σ_]+1/2p[σ_A,_], where p[σ]=e^β J∑_⟨ i,j⟩σ_iσ_j/𝒵_β. To sample from q[σ], one can simply sample from p[σ] and then with 1/2 probability, flip the spins in . We may thus write Eq.(<ref>) as,

    𝒩_A[ρ_β]
           =1/4∑_σ q[σ]p[σ]/q[σ]|1-e^2β H_∂[σ]|
       =1/2|tanh(β H_∂[σ])|_q.

where the final expectation value is taken with respect to the distribution q. The advantage of Eq.(<ref>) compared to Eq.(<ref>) is that the expression inside the expectation value is now bounded by unity everywhere, so that contributions from rare configurations are no longer a concern. 

In the following we present the numerical results for a 2D system of qubits placed on a L× L square lattice with periodic boundary conditions. 
Fig. <ref> shows the negativity 𝒩_A(ρ_β) when A is taken to be the L×L/2 cylinder. As is clear from Fig. <ref>, in the thermodynamic limit, the negativity is equal to 1/2 even above the Ising critical temperature (marked by the dashed line in Fig.<ref>), as expected from the expression in (<ref>). The fact that the negativity of subregions with extensive boundary is 1/2 irrespective of temperature, means that if one defines the topological negativity as the constant contribution to the negativity as in Ref. <cit.>, one would find that the resulting mixed state has 𝒩_topo=-1/2 even above the critical temperature, despite the fact that for T>T_c – as shown above – ρ_β is an SRE mixed-state. <cit.> (cf. Ref. <cit.>). 

Although the thermal phase transition at T_c has no effect in the negativity of subregions with extensive boundaries, the negativity of finite subregions (while continuous) becomes singular at T=T_c. Fig. <ref> shows the derivative of the negativity of a single site with respect to temperature which shows a singularity at T=T_c, which is consistent with the singularity of the heat capacity at the d=2 classical Ising phase transition, as to be expected from Eq. (<ref>). 




Discussion and Outlook: Using LOCC to recover states with a known pattern of LRE may provide another perspective that can be used to investigate mixed-state LRE under decoherence. For correlated decoherence, this is apparent in specific examples.  The two-dimensional toric code dephased by local operators Z_j+δX_j at each site j, with the vector δ given in Fig. <ref>, was studied in Ref. <cit.>.  Quantum state trajectories under this channel only contain patterns of ℤ_2 charge and flux which are separated by δ.  Therefore, for any dephasing strength, it is possible to use LOCC across the bipartition in Fig. <ref> to deterministically recover the toric code ground-state. The ℤ_2 charge and flux lying on the bipartitioning surface, which cannot be directly measured in the LOCC protocol, can be perfectly inferred from stabilizer measurements in A and A due to the noise correlations.  The negativity for this bipartition and for any dephasing strength <cit.> is thus exactly that of the toric-code ground-state <cit.>. Another example for which the LOCC perspective presented here could be insightful is the study of entanglement structure in the toric code with single-site Y decoherence <cit.>. In this case, while the noise model itself is not correlated, the LOCC protocol can benefit from the correlations that are present in the syndromes due to the extensively many strong symmetries of this channel <cit.>. The detailed study of this model and other correlated noise models is left to a future work. 

We identify other directions implied by our results.  First, it is interesting to consider the fate of mixed-state entanglement under thermalization by genuinely quantum channels which are strongly-symmetric.
More concretely, we may consider thermalizing the GHZ state via a strongly symmetric implementation of the quantum Metropolis-Hastings dynamics<cit.> for the d-dimensional transverse field quantum Ising model. The resulting mixed state would be the projection of the quantum Gibbs state onto the symmetric sector. A similar LOCC protocol to recover the GHZ state can be used to relate the entanglement negativity to the performance of a (d-1)-dimensional repetition code under a coherent correlated noise model. Energetic considerations can be used to argue about the success of the LOCC protocol in turning the resulting mixed state into the GHZ state. This lower bounds the negativity of the resulting mixed-state and may suggest that it 
would have a non-trivial negativity at high temperatures, though this remains to be confirmed. 

Second, while the persistence of topological negativity at high temperatures in our work can be understood as a consequence of the repetition code possessing maximum error threshold, it is interesting to search for settings where a similar line of argument would lead to a different code with a finite threshold. Such a system, would exhibit a finite temperature phase transition in negativity, which may not necessarily coincide with the standard order-disorder thermal phase transition. Furthermore, while we focused on the Metropolis-Hastings channel for the classical Ising model, a possibly fruitful future direction to pursue is to consider other classical systems with different symmetries and study the fate of entanglement in an initially symmetric state (e.g. continues symmetries, higher-form symmetries or non-Abelian symmetries <cit.>). 


Lastly, given that the topological negativity is unable to detect the phase transition in the complexity of the mixed-state ρ_β (from LRE to SRE), it is interesting to look for other quantities that would be able to capture the complexity of the quantum phase transition in our system as well as in more general settings. 

Acknowledgments:
We thank Yimu Bao, Matthew Fisher, Tarun Grover, Tibor Rakovszky, Subhayan Sahu, Shengqi Sang and Yaodong Li for useful discussions. This research was supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). Use was made of computational facilities purchased with funds from the National Science Foundation (CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; NSF DMR 2308708) at UC Santa Barbara. SV is partly supported as an Alfred P. Sloan Research Fellow.








    Supplemental Material





§ ENTANGLEMENT OF Ρ_Β

Here, we show that the state (<ref>) is short-range-entangled (SRE) throughout the disordered phase at sufficiently high temperatures, and in any spatial dimension.  First, observe that this state may be written as

    ρ_β = ∑_μp_μ|ϕ_μ⟩⟨ϕ_μ|

where |ϕ_μ⟩≡ρ_β^1/2|μ⟩/ √(p_μ) and p_μ≡⟨μ|ρ_β|μ⟩, while |μ⟩ is a product state in the Pauli X basis.  Because of the strong symmetry of this state, (ρ_βU) = 1, where U ≡∏_jX_j.  As a result, each pure state appearing in (<ref>) must satisfy ∏_jX_j|ϕ_μ⟩ = |ϕ_μ⟩.  

Furthermore, observe that

    ρ_β^1/2 = 1/√(_β)∑_σe^-β H/2|ψ_σ⟩⟨ψ_σ|

It is evident from this expression that [Z_iZ_j,ρ_β^1/2]=0 for any i, j.  As a result, any state |ϕ_μ⟩ appearing in (<ref>) – for which U|ϕ_μ⟩ = |ϕ_μ⟩ – can be obtained by applying single-site unitary gates on the reference state |ϕ_+⟩ = ρ_β^1/2|+⋯ +⟩/√(p_+), where |+⋯ +⟩ is a product state in which X_j=+1 for each qubit:

    |ϕ_μ⟩ = ∏_jZ_j^1-μ_j/2|ϕ_+⟩

Finally, we note that p_μ = ⟨μ|ρ_β|μ⟩ = ⟨+⋯ +|∏_jZ_j^1-μ_j/2ρ_β∏_jZ_j^1-μ_j/2|+⋯ +⟩ = ⟨+⋯ +|ρ_β|+⋯ +⟩ = p_+.  As a result, p_μ = 1/2^N-1 where N is the total number of qubits.  To summarize, we may write (<ref>) as

    ρ_β   = 1/2^N-1∑_μ  s.t.  ∏_jμ_j= +1∏_jZ_j^1-μ_j/2|ϕ_+⟩⟨ϕ_+|∏_jZ_j^1-μ_j/2

We now re-cast the summation by introducing Ising degrees of freedom σ_ij along bonds of the hypercubic lattice, and letting μ_j≡ A_j[σ] ≡∏_i∈ NN(j)σ_ij, where the product is taken over sites that are nearest-neighbor to j.  With periodic boundary conditions, ∏_jA_j[σ] = +1, and as a result, the constraint on the summation ∏_jμ_j = +1 is naturally satisfied by this replacement.  With this replacement, we may write

    ρ_β   = 1/2^N_b∑_σ∏_jZ_j^1-A_j[σ]/2|ϕ_+⟩⟨ϕ_+|∏_jZ_j^1-A_j[σ]/2

Here, N_b = d· N is the total number of bonds on the d-dimensional hypercubic lattice.  The pre-factor ensures the normalization of the density matrix[Another way to derive this pre-factor is to observe that for a given configuration of μ_j satisfying the constraint ∏_jμ_j = 1, there are 2^N_b/2^N-1 different configurations of σ_ij such that μ_j = A_j[σ]. ]. Finally, using the fact that Z_j^1-A_j[σ]/2 = ∏_i∈NN(j)Z_j^1-σ_ij/2, we may write ∏_jZ_j^1-A_j[σ]/2 = ∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2, where the product in the second expression is taken over distinct, nearest-neighbor bonds.  As a result,

    ρ_β   = 1/2^N_b∑_σ∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2|ϕ_+⟩⟨ϕ_+|∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2

We conclude that ρ_β may be obtained by dephasing Z_iZ_j on each bond of the lattice, with maximal strength, a manifestly finite-depth, local quantum channel.

Finally, we may argue that the state |ϕ_+⟩ is SRE when the system is in a disordered phase.  We do this by first writing

    |ϕ_+⟩ = √(2^N-1)ρ_β^1/2|+⋯ +⟩ = √(2^N-1/_β)∑_σe^-β H/2|ψ_σ⟩⟨ψ_σ|+⋯ +⟩ = 1/√(_β)∑_σe^-β H/2|σ⟩

We now observe that (<ref>) is the ground-state of the Hamiltonian

    H = -∑_jQ_j

where

    Q_j≡ -X_j + ∏_i∈NN(j)e^-β J Z_iZ_j


 


§ LOCC OPERATIONS AND RECOVERING THE GHZ STATE


Here, we will show the existence of an LOCC protocol   with respect to bi-partition A and , which transforms state ρ_β back to the GHZ state in the thermodynamic limit. 
The LOCC protocol consists of two steps:


    
  * Measure Z_iZ_j on each link that is entirely in A or in  subregions, finding the outcome μ_ij=± 1. This does not include the links across the boundary, ∂ A.

    
  * For any given set of measurement outcomes μ={μ_i,j}, there exist two possible states which have domain wall configuration consistent with μ. 
     We pick one of the two states based on a coin toss weighted with their respective Boltzmann weight in ρ_β (Eq.(<ref>)) and
      apply a product of Pauli X_i operators to eliminate the domain walls in that assumed configuration.

In the following, we will derive Eq.(<ref>) for the fidelity between the GHZ state, |ψ_+⟩, and the output state (ρ_β).


 §.§ Fidelity


Let {M_μ} be the projective measurements that measure whether there is a domain wall or not on the links which are entirely
in region A or A̅,


    M_μ≡∏_<i,j>∈ A, A̅1+μ_i,jZ_iZ_j/2,

where μ_ij is the measurement outcome of measuring Z_iZ_j and the product only includes next nearest sites which are both in A or both in .
Note that,

    M_μ|ψ_σ⟩ = ∏_<i,j>∈ A, A̅δ_μ_i,j,σ_iσ_j|ψ_σ⟩.
 

A set of measurement outcomes μ specifies a certain domain wall configuration in A and , with some domain walls ending on the boundary ∂ A. Let 1∈ A and 1̅∈ be two adjacent sites across the boundary and define τ=± 1 to be a marker for whether or not there is a domain wall across the edge connecting 1 and 1̅, i.e. τ≡σ_1 σ_1̅ (see Fig. <ref>). Note that μ and τ together uniquely specify the global domain wall configuration, because there should be even number of domain walls across each plaquette of the square lattice. In particular if we label the sites along the boundary as shown in Fig.<ref>, we have,

    σ_i σ_ i̅=τ∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1.

Let X_μ,τ be the appropriate product of single qubit Pauli-Xs that corresponds to the domain wall configuration specified by μ and τ. In particular, if M_μ|ψ_σ⟩=|ψ_σ⟩, we have

    X_μ,τ|ψ_σ⟩≡|ψ_+⟩   if σ_1σ_1̅=τ
    ∏_j∈ AX_j|ψ_+⟩   if σ_1σ_1̅≠τ.

Given the measurement outcomes μ, the decoder  applies X_μ,τ with τ=± 1 chosen randomly with probability P_μ,τ. The probability P_μ,τ is chosen to be proportional to the Boltzmann weight of the spin configuration specified by μ and τ and according to Eq.(<ref>) can be written as,

    P_μ,τ = e^β J τ∑_i≥ 1∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1/∑_τ'=± 1 e^β J τ' ∑_i≥ 1∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1.

Putting everything together, we can compute the average “decoding" fidelity,


    F = ψ_+(ρ_β)ψ_+= ∑_σ,μ,τe^-β H[σ]/𝒵_βP_μ,τ⟨ψ_+|X_μ,τM_μ|ψ_σ⟩^2.


Using Eqs.(<ref>) and (<ref>) this simplifies to


    F = ∑_σ,μ,τe^-β H[σ]/𝒵_β P_μ,τ∏_<i,j>∈ A, A̅δ_μ_i,j,σ_iσ_jδ_σ_1σ_ 1̅,τ.

Plugging the expression for P_μ,τ into Eq. (<ref>), we get


    F =   ∑_σ,τe^-β H[σ]/𝒵_βe^β Jτ∑_i≥1σ_1σ_iσ_ 1̅σ_ i̅/∑_τ'=±1 e^β Jτ'∑_i≥1σ_1σ_iσ_ 1̅σ_ i̅δ_σ_1σ_ 1̅,τ
    
         =   ∑_σe^-β H[σ]/𝒵_βe^β J∑_i≥1σ_iσ_ i̅/∑_τ'=±1 e^βτ'σ_1 σ_ 1̅ J∑_i≥1σ_i σ_ i̅
    
         =   1/2∑_σe^-β H[σ]/𝒵_β(1-tanh(β J∑_i≥1σ_i σ_ i̅))
    
         =   1/2[1-⟨tanh(β H_∂)⟩]

where H_∂ is the boundary Hamiltonian between A and A̅. 



 §.§ Mapping to the d-1 repetition code


The problem of choosing which domain wall configuration to assume can be understood as a decoding problem for a (d-1)-dimensional classical repetition code. To see this, note that the decoding task in a classical error correcting code is basically figuring out what error has happened based on a given set of syndromes. In particular the decoding task in the classical repetition code on N bits can be described as follows. Consider an error vector e∈{1,-1}^N such that e_i=-1 if a bit flip has happened on the i'th bit and e_i=1 otherwise. 
The goal of the decoder is to figure out e, knowing only the syndrome values, i.e. e_i e_j for adjacent i and j on the appropriate graph, e.g. the d-dimensional square lattice in the standard d-dimensional repetition code. Moreover, the decoder assumes an underlying noise model, i.e. a random distribution from which the vector e is drawn. For example, in the independent and identically distributed (iid) noise model, it assumes each component e_i is -1 with probability p and 1 with probability 1-p, independent of other components. 

In our setting finding the domain wall configuration is clearly the same task, with e_i defined to be e_i≡σ_i σ_ i̅. Note that we can not know the value of e_i, but we know the value of e_i e_j=μ_i,j μ_i̅,j̅ for adjacent i and j. At low temperatures, the domain walls are rare so most e_is would be 1, and as one increases the temperature more e_i would be -1. The underlying noise model is given by P_μ,τ as described in the previous section. This noise model is not iid, but rather a correlated noise model. However, as we argue in the next section, the repetition code is able to guess e correctly with probability 1 in the thermodynamic limit unless β=0, at which point each e_i is ± 1 compeletly at random. 



§ THRESHOLD FOR THE REPETITION CODE

Here we argue that the repetition code decoding under the noise distribution P_μ,τ is successful with probability 1 for any β>0 in the thermodynamic limit. First we review the argument for maximum error correction threshold in the presence of the iid noise and then we argue that the same should remain true for more generic noise models. We use quantum notation for convenience although the problem we are considering is inherently classical. 

Consider an initial state |Ψ⟩ undergoing some errors, leaving it in the mixed state

    ρ = ∑_e p_e e|Ψ⟩⟨Ψ|e

where e is a product of X bit flips and p_e is the probability of having that error. We rewrite the sum to distinguish between the errors that leave ρ with fewer than N/2 bit flips and errors with higher weights:

    ρ = ∑_e: |e|<N/2p_e e|Ψ⟩⟨Ψ|e + ∑_e:|e|≥ N/2p_e e|Ψ⟩⟨Ψ|e.

The error correction channel, , measures the syndromes, and between two possible error configuration consistent with a syndrome, applies the one with the smaller weight. Therefore it always successfully corrects the errors when there are less than N/2 bit flips, hence,

    (ρ)=   ∑_e: |e|<N/2 p_e (e|Ψ⟩⟨Ψ|e) + ∑_e:|e|≥ N/2p_e  (e|Ψ⟩⟨Ψ|e)
    
        =   ∑_e: |e|<N/2 p_e|Ψ⟩⟨Ψ| + ∑_e:|e|≥ N/2p_e (e|Ψ⟩⟨Ψ|e).


The fidelity of this post-error correction state with the original state is lower bounded as,


    F =   ⟨Ψ|(ρ)|Ψ⟩
    
        =   ∑_e: |e|<N/2 p_e +  ∑_e:|e|≥ N/2p_e ⟨Ψ|(e|Ψ⟩⟨Ψ|e)|Ψ⟩≥∑_e: |e|<N/2 p_e


where the lower bound is because the second term in the expression above is non-negative. Now we will show that this lower bound is equal to one in the thermodynamic limit. First, we rearrange the sum to be over |e|=k, the number of bit flips error, and because  p_k is a binomial distribution,
p_k = Nkp^k(1-p)^N-p.

In the limit of large N, we can approximate the binomial distribution as a Gaussian distribution with mean Np and standard deviation √(Np(1-p)) 
via the central limit theorem. Since N/2 is N/2-Np/√(Np(1-p))=O( √(N)) standard deviations away from the peak of the Gaussian, we see that 

lim_N→∞∑_k=0^N/2-1 p_k = 1

when p<1/2. Since fidelity is upper-bounded by one, the fidelity is indeed equal to one in the thermodynamic limit. Thus, the repetition code will work successfully when p<1/2. 

For the more general case where e is drawn based on the P_μ,τ distribution, the bound in Eq.(<ref>) is still valid. Therefore, to show perfect recovery we only need to show that with probability one, less than half of the edges contain a domain wall. Intuitively, this is true because for β>0, not having a domain wall on an edge is energetically favorable. To make it rigorous, one can define indicator variables I_i=(1-e_i)/2 in terms of which the number of domain walls can be written as k=∑_iI_i. When β >0 it is more probable for I_i to be 0 rather than 1 and hence the expectation value of k/|∂ A| is less than 1/2. Noting that lim_|i-j|→∞Cov(I_i,I_j)=0, the weak law of large numbers for weakly dependent variables<cit.> shows that k/|∂ A| concentrates around its mean, which is less than 1/2 when β>0 in the thermodynamic limit. 



§ NEGATIVITY SPECTRUM

Here we calculate the entanglement negativity between a sub-region A and its complement A̅, in the state ρ_β. 
Let  denote the spin configuration that is related to σ by flipping every spin. Accordingly, the state |ψ_σ⟩ can be written as |ψ_σ⟩=|σ⟩+|⟩/√(√(2)) and Eq. (<ref>) can be rewritten as,


    ρ_β = 1/2𝒵_β∑_σ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅| +|σ⟩⟨σ̅| +|σ̅⟩⟨σ|),

where H[σ]=-J ∑_i,jσ_i σ_j is the classical Ising energy of the spin configuration σ. Taking the partial transpose of ρ_β with respect to sub-region A yields,

    ρ_β^T_A   = 1/2𝒵_β∑_σ[ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅|) +e^-β H[σ_A,σ_A](|σ_A,σ_A⟩⟨σ_A, σ̅_A| +|σ_A, σ̅_A⟩⟨σ_A,σ_A|)]
       =1/2𝒵_β∑_σ[ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅|) +e^-β H[σ_A,_A](|σ⟩⟨σ| +|σ⟩⟨σ|)]


where σ_A and σ_A̅ denote the spin configuration σ restericted to regions A and A̅ respectively. Thus, the eigenvalues and eigenstates of ρ_A^T_A are

    λ_σ;±=e^-β H[σ]± e^-β H[σ_A,σ̅_A]/_β, |ψ_σ^(±)⟩≡|σ⟩±|σ⟩/√(2).



We can now calculate the entanglement negativity  from the sum of all the negative eigenvalues. Firstly,
it is only the λ_σ;- eigenvalues which can be negative. 
We can also see that λ_σ;-=λ_σ̅;- from the ℤ_2 symmetry of the Hamiltonian and they correspond to the same eigenvector, up to some phase. 
Moreover, 
note that λ_σ;-=-λ_σ_A,σ̅_A̅;- so for a given spin configuration, either λ_σ;- or λ_σ_A,σ̅_A̅;- would be negative. Thus, we account for the two incidences of double counting with an additional 1/4 and get


    𝒩_A[ρ_β] = 1/4_β∑_σe^-β H[σ] - e^-β H[σ_A,σ_A].


Factoring out e^-β H[σ] we get


    𝒩_A[ρ_β]    = 1/4_β∑_σ e^-β H[σ]1 - e^-β( H[σ_A,σ_A]- H[σ])
       =1/4⟨1 - e^2β H_∂[σ])⟩.






 Paper Title: Nonmagnetic Phase Transition in La$_3$TiSb$_5$ Observed in $^{121}$Sb Nuclear Quadrupole Resonance.




    
    manago@riko.shimane-u.ac.jp
    
    Department of Applied Physics, Shimane University, Matsue 690-8504, Japan
    
    ICSR, Shimane University, Matsue 690-8504, Japan
    
    Department of Applied Physics, Shimane University, Matsue 690-8504, Japan
    
    Department of Physics, Kobe University, Kobe 657-8501, Japan

§ ABSTRACT 
    We measured the electrical resistivity and the ^121Sb nuclear quadrupole resonance (NQR) spectra
    of La_3TiSb_5, which is a nonmagnetic reference system for a locally noncentrosymmetric
    antiferromagnet Ce_3TiSb_5, in order to determine its physical properties.
    The resistivity exhibited a hump anomaly at 120 K, suggesting a phase transition.
    The NQR spectra split below 110 K, whereas the nuclear spin-lattice relaxation exhibited no anomalies.
    These results can be explained by the incommensurate charge-density-wave (CDW) transition.
    This finding suggests the importance of quasi-one-dimensional characteristics and CDW instability
    in these systems.

Nonmagnetic Phase Transition in La_3TiSb_5 Observed in ^121Sb Nuclear Quadrupole Resonance
    Hisatomo Harima
    August 5, 2024
==========================================================================================




Locally noncentrosymmetric systems have received considerable attention because various unusual phenomena
can be realized, for example, exotic superconductivity<cit.>
and cross-correlation responses<cit.>.
Cross-correlation responses are expected to occur when odd-parity multipoles emerge
in conjunction with the magnetic transitions<cit.>.
The magnetoelectric effect, that is, current-induced magnetization, is an example of an unusual response
and is confirmed for two antiferromagnetic metals
UNi_4B<cit.> and Ce_3TiBi_5 <cit.>.
The origin of the magnetoelectric effect in these systems remains under active discussion.
Understanding the magnetic structure is crucial to elucidate its mechanism.

Antiferromagnet Ce_3TiBi_5 belongs to the series of hexagonal RE_3TiX_5 systems
(RE: rare-earth element; X = Sb, Bi)
and these materials crystallize in a hexagonal P6_3/mcm (No. 193, D_6h^3) space group symmetry<cit.>.
Rare-earth atoms form a zigzag chain along the c axis possessing no inversion symmetry.
Ce_3TiBi_5 exhibits a cycloid order with moments lying in the plane of the zigzag chain
below T_N=5.0 K<cit.>.
By contrast, Ce_3TiSb_5 exhibits an antiferromagnetic transition at T_N=5.2 K
with ordered moments along the c axis with long-period modulation<cit.>.
Another remarkable feature of RE_3TiX_5 systems is the quasi-one-dimensional chain along the c axis.
This is common in the nonmagnetic reference system La_3TiSb_5,
and Sb(1) and the Ti atoms are aligned along the c axis, as shown in Fig. <ref>.
This could lead to the instability of the Fermi surface,
resulting in charge-density-wave (CDW) or spin-density-wave transitions.
Among the RE_3TiX_5 systems, La_3TiSb_5 exhibits an electrical resistivity anomaly
of unknown origin at approximately 120 K<cit.>.
Some phase transition is expected in La_3TiSb_5.

In this letter, we report the results of the electrical resistivity and ^121Sb nuclear quadrupole resonance (NQR) measurements
on La_3TiSb_5 to reveal the anomalies at lower temperatures.
The resistivity exhibited a hump anomaly at 120 K.
The NQR spectrum began to split below the transition temperature of ∼ 110 K.
This split was explained using a simple model that assumed spatially modulated electric field gradient (EFG)
because of the incommensurate CDW transition.
No clear anomaly was observed in the nuclear spin-lattice relaxation rate 1/T_1.
This finding highlights the importance of the quasi-one-dimensional character of these systems.




The single-crystalline La_3TiSb_5 sample was synthesized using the Sn-flux method,
as in Ref. J.Phys.Cond.Matt.29.145601.
La (99.9%), Ti (99.5%), Sb (99.99%), and Sn (99.999%) in molar ratios
of La:Ti:Sb:Sn = 2–3:1:5:20–50
were placed in an alumina crucible and sealed in a quartz tube.
The ingredients were heated to 1000°C, maintained at this temperature for 10 h,
and subsequently cooled to 500°C at ∼ -1.5°C/h.
Needle-shaped crystals were obtained.
The excess Sn flux was removed by centrifugation.
The sample was cleaned with dilute hydrochloric acid to remove residual Sn flux.
Electrical resistivity measurements were performed on the single-crystal samples using
the standard four-terminal method.
The sample was crushed into a powder for NQR measurements to enhance the signal intensity.
In addition, the powdered sample was sealed in epoxy (Stycast 1266) in a rod shape to prevent oxidization.
The standard spin-echo method was used for NQR measurements.
The frequency-swept spectra were obtained by summing multiple Fourier-transformed spin-echo signals
at various frequencies.





Figure <ref> shows the electrical resistivity of La_3TiSb_5 samples along the c axis
in the range of 2–290 K.
A hump anomaly was observed at ∼ 120 K for the best-quality sample,
which agrees with a previous study<cit.>.
This suggests a reduction in the density of states (DOS) at the Fermi energy level.
The samples used in this study did not exhibit zero resistivity because of the superconductivity of the Sn flux below
4 K<cit.>, although some samples exhibited a slight decrease at this temperature.
The sample quality varied to some extent, and the residual resistivity ratio (RRR) ρ(300 K)/ρ(2 K)
was between 3 and 8.
Furthermore, higher RRR samples exhibited higher and sharper anomalies.
Another sample from the same batch as the electrical resistivity was used for the NQR measurements.






A nucleus with I≥ 1 possesses a nuclear quadrupole moment Q,
and the nuclear spin Hamiltonian without a magnetic field is

    ℋ_Q = hν_Q/6{[3I_z^2 - I(I+1)] + 1/2η(I_+^2 + I_-^2)
        },

where ν_Q denotes the nuclear quadrupole frequency and η denotes the asymmetric parameter.
These parameters are expressed by the local EFG V_ij at the nuclear site:
ν_Q=3eQV_zz/[2I(2I-1)h] and η = (V_xx - V_yy)/V_zz when
the principal axes of the EFG are selected along the x, y, and z axes, such that
| V_zz|≥| V_yy|≥| V_xx|.
Two isotopes of Sb exist: ^121Sb (I=5/2) and ^123Sb (I=7/2).
For ^121Sb, two NQR transition lines
ν_1 and ν_2 are expected for each site.
The three-fold symmetry at the Sb(1) site ensures η=0, which leads to
simple resonant frequencies ν_1=ν_Q and ν_2=2ν_Q
for the ± 1/2 ↔± 3/2 and ± 3/2 ↔± 5/2 transitions.
In contrast, η is nonzero for the Sb(2) site because of the lower symmetry,
resulting in an η-dependent ratio between ν_1 and ν_2.
The ^123Sb gives three lines per site, and the resonant frequencies are related to those of ^121Sb
because these nuclei are located at the same crystal site.

Figure <ref> shows the ^121Sb and ^123Sb NQR spectra at the selected temperatures.
The experimental ν_Q value for ^121Sb was determined from the ± 1/2 ↔± 3/2
transition line for Sb(1) at ^121ν_1(1)=^121ν_Q(1) (≃ 65.3 MHz at 300 K).
Sb(2) parameters were evaluated based on the frequency of
ν_1(2) and ν_2(2) lines (∼ 18 and 31 MHz at 5–7 K, respectively) using numerical diagonalization
of the Hamiltonian in Eq. (<ref>).
Table <ref> lists the quadrupole parameters of Sb(1) and Sb(2) for the ^121Sb nucleus
obtained through band calculation and the NQR experiment.
The band calculation result is in good agreement with the experimental values.
The ratio of ν_Q between ^121Sb and ^123Sb nuclei was
^121ν_Q(1)/^123ν_Q(1)=1.64 for Sb(1) site at 300 K, being
consistent with the expected value from the quadrupole moments<cit.>:
^121{Q/[2I(2I-1)]}/^123{Q/[2I(2I-1)]}=1.65.
The signal intensity of ^123Sb is weaker than that of ^121Sb
due to the smaller gyromagnetic ratio and natural abundance, although it is unexpectedly weaker
compared to the typical relative intensity with that of ^121Sb.
Hereafter, the NQR result for the ^121Sb nucleus is shown.



Figure <ref> shows the NQR spectra at (a) Sb(2) ^121ν_2(2) and
(b) Sb(1) ^121ν_1(1) between 7–300 K.
The NQR spectra at ^121ν_1(1) broadened below 100 K and split into two peaks at lower temperatures.
This demonstrates a phase transition with symmetry reduction.
In contrast, the spectra at the ^121ν_2(2) line showed a small anomaly at approximately 100 K.
No clear splits were observed.




The site dependence of the NQR split can be explained by the
difference in the local structure between two sites.
Sb(1) site forms a quasi-one-dimensional chain along the c axis possessing the CDW instability,
whereas Sb(2) forms a zigzag chain.
The nuclear relaxation shown later suggests the nonmagnetic transition.
Therefore,
the split peak in Fig. <ref>(b) was analyzed using a spatially modulated quadrupole frequency
caused by the CDW transition

    ν_Q(x)=ν_0 + Wsin qx,

where W and q denote the amplitude and wavenumber of the modulation, respectively.
The nuclear position along the q vector is denoted by x.
If the wavelength 2π/q cannot be expressed as a rational multiple of the lattice constant,
qx is uniformly distributed between zero and 2π, which results in a continuous distribution of
ν_Q(x) between ν_0-W and ν_0+W.
Then, the NQR intensity I(f) at frequency f is given by I(f) df ∝ dθ, where
θ = qx and f=ν_0 + Wsinθ.
The NQR spectral shape is described as follows<cit.>:

    I(f) = 
                   A[W^2 - (f-ν_0)^2]^-1/2    (| f-ν_0| < W),
    
                   0     (| f-ν_0|≥ W),

where A denotes a constant.
This yields a double-peak spectrum at f=ν_0± W with a finite intensity between
two peaks.
The asymmetric parameter η is fixed to zero for simplicity
in this model.
This can be determined by comparing frequencies between ^121ν_1(1) and ^121ν_2(1).

The fitting results obtained using Eq. (<ref>) are represented by the dashed line in
Fig. <ref>(b) for spectra at 100 K or below.
For a better fit to the experimental data, the above function was broadened using a Gaussian function
and is multiplied by a linearly varying weight function.
This simple model reproduces the experimental Sb(1) NQR spectrum at temperatures below 100 K.
The analyzed split width W is shown in Fig. <ref>(a).
This temperature dependence appeared to be an order parameter with a second-order phase transition and
was analyzed using the following approximate BCS-type gap function<cit.>:

    W(T)=W(0)tanh[a(T_A/T-1)^1/2],

where W(0), a, and T_A denote fitting parameters.
Although the selection of this model is arbitrary, it approximately reproduces the experimental result,
as indicated by the dashed lines in Fig. <ref>(a).
The transition temperature was evaluated to be T_A≃ 110 K.
This value was close to 120 K, where an electrical resistivity anomaly was observed
in Ref. Chem.Mater.14.4867 and in the present study (Fig. <ref>).
The relative split width regarding the quadrupole frequency is
W(0)/ν_0≃ 0.073 for T → 0.
In contrast, the increase in the spectral width in ν_2(2) line is less than 0.15 MHz below T_A
estimated from the change in the half width at half maxima, yielding W(0)/ν_0≤ 0.005.
Because the NQR anomaly is more pronounced in Sb(1), which forms a quasi-one-dimensional chain,
this result supports the CDW transition originating from Fermi surface nesting.
The NQR result, combined with the electrical resistivity, was consistent with the
incommensurate CDW transition.

The anomaly at T_A was also observed in the temperature dependence of ^121ν_1(1)
or ^121ν_2(2) for Sb(1) and Sb(2), as shown in Fig. <ref>(b) and (c).
The frequencies gradually increased as the temperature decreased, which is ascribed to the lattice
shrinkage.
This can be explained by empirical temperature variations in
the NQR frequency<cit.>
ν_Q(T)=ν_Q(0)(1-α T^3/2) above T_A,
and the fitted result is indicated by the dashed lines.
The deviation from this curve is pronounced in ^121ν_1(1)
below T_A, and a small kink is observed in ^121ν_2(2) at T_A.
These results suggest a lattice constant change accompanying the CDW ordering.




Figure <ref> shows the nuclear spin-lattice relaxation rate divided by temperature  1/T_1T
for Sb(1) and Sb(2).
The relaxation rate probes the dynamics of the magnetic and electric origins.
The nuclear relaxation curve M(t) was analyzed using

    M(∞)-M(t)/M(∞)-M(0)=R(t),

where R(t) denotes a normalized relaxation curve consisting of multiple exponential functions.
In the analysis, only the magnetic fluctuations were taken
into consideration for the nuclear relaxation process, and the quadrupole relaxation
mechanism was neglected.
^121ν_1(1) line corresponds to a ± 1/2 ↔± 3/2 transition and R(t) is described as

    R(t)=3/28exp(-3t/T_1) + 25/28exp(-10t/T_1).

For ^121ν_2(2), a finite η altered the relaxation curve from that of the
± 3/2 ↔± 5/2 transition, and thus, we adopt the following
function<cit.> obtained from numerical calculations for η=0.38:

    R(t)=0.3875exp(-3t/T_1) + 0.6125exp(-9.0875t/T_1).

The relaxation curves exhibited a single component in two sites.
1/T_1 at the Sb(1) site was measured at the lower peaks below T_A.
Overall, 1/T_1T is nearly constant over the entire temperature range at these sites, indicating that
La_3TiSb_5 is a normal metal with small magnetic fluctuations.
Because magnetic fluctuations are absent in Sb(1), which exhibits an NQR line split,
the ordered phase is nonmagnetic.
This value is somewhat larger for Sb(2) than for Sb(1) below T_A,
which may be due to the change in the magnetic fluctuations
or hyperfine coupling constant caused by the ordering.

The absence of an anomaly in 1/T_1T may be unexpected, given the drastic NQR line split and resistivity hump.
The 1/T_1T behavior in the CDW state varies for different materials.
No detectable changes have been reported in the CDW state of 2H-NbSe_2<cit.> as in La_3TiSb_5,
whereas an enhancement of 1/T_1 T was observed in BaTi_2Sb_2O<cit.>
and SrPt_2As_2<cit.> toward T_CDW.
Site-dependent 1/T_1 T has been reported for Ta_4Pd_3Te_16<cit.>
and CsV_3Sb_5<cit.>.
The small change in ^121Sb 1/T_1T of La_3TiSb_5 suggests that 1/T_1T
is mainly caused by the electrons not directly involved in the CDW transition,
and that the reduction of DOS accompanying the transition
does not strongly affect 1/T_1T.
It is interesting to obtain 1/T_1T at the La site to clarify the site dependence
of the CDW anomaly.

Although the simple CDW model expressed in Eq. (<ref>) reproduced the experimental results,
the NQR spectrum deviates slightly from the expected spectral shape in Eq. (<ref>):
the lower-frequency peaks were stronger, and an edge structure was observed at approximately 66 MHz at lower temperatures,
as shown in Fig. <ref>(b).
Therefore, the actual CDW modulation may be more complicated than expected.
The complex spectrum may be related to the presence of Sb(1) and Ti chains with different symmetries.

Sb(1) lacks local inversion symmetry, whereas Ti is located at the inversion center.
These factors can cause successive CDW transitions similar to those of
LaAgSb_2<cit.> and SrPt_2As_2<cit.>.
Further studies, including x-ray or neutron diffraction experiments, may reveal the details of
the CDW structure in La_3TiSb_5.

Finally, we discuss the isostructural systems.
CDW instability may also exist in Ce_3AX_5 systems (A = Ti, Zr; X = Sb, Bi).
However, only antiferromagnetic transitions have been reported
thus far<cit.>.
In Ce_3TiSb_5 and Ce_3TiBi_5, the magnetic structures are somewhat different.
Ce_3TiSb_5 exhibits a long-period modulation of the ordered moment along the easy axis
(in plane) with double transitions<cit.>.
In contrast, the cycloid structure has been reported for Ce_3TiBi_5, where the moment can be along the hard (c) axis<cit.>.
This difference may be partly due to the various mechanisms involved in the
magnetic structures of these systems.
The CDW instability inferred in this study can also be classified into these mechanisms.
Future experimental and theoretical studies will unravel the interplay between magnetism and CDW instability.

In summary, electrical resistivity and ^121Sb NQR measurements were performed on
La_3TiSb_5, and the NQR spectrum of Sb(1) split below T_A≃ 110 K.
This corresponds to a hump anomaly in the electrical resistivity.
The broad double-peak spectrum can be explained by the incommensurate CDW structure.
The nuclear spin-lattice relaxation rate shows no anomalies across the transition,
indicating the nonmagnetic ordering.
The CDW instability, owing to its quasi-one-dimensional character, can affect the magnetism of the systems
where the magnetoelectric effect is expected.


    The authors thank M. Yogi and T. Mutou for their insightful discussions.
    The authors also thank N. Masada and Y. Sano for their support in the experiments.
    This work was supported by JSPS KAKENHI Grant Numbers JP21K03447 and JP24K17012.







 Paper Title: Anharmonic quantum muon effects in the kagome antiferromagnet Zn-Barlowite.








Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland 
Jožef Stefan Institute, Jamova c. 39, SI-1000 Ljubljana, Slovenia

Faculty of Mathematics and Physics, University of Ljubljana, Jamova c. 19, SI-1000 Ljubljana, Slovenia

 



§ ABSTRACT 

Muon spin spectroscopy (μSR) is a powerful local probe technique e.g. used for the investigation of exotic frustrated magnetism. Ab initio simulations using Density Functional Theory with the muon treated as a point-like defect (DFT+μ) are commonly employed to determine the interstitial lattice positions where the muon comes to rest after implantation. These muon stopping sites are critical for accurately interpreting μSR data. For example, for the quantum spin liquid candidate Zn-Barlowite, DFT+μ simulations identify two types of muon stopping sites: a higher-energy site where the muon is located between a fluorine and a bromine atom and three similar sites near an OH group.
However, our study shows that the μSR spectra of Zn-Barlowite cannot be adequately described using muon sites determined by the conventional DFT+μ approach. Instead, accurate reproduction of the μSR data requires treating the muon as a spatially extended quantum particle with a skewed wavefunction due to the anharmonicity of the surrounding electrostatic potential. The quantum nature of the muon significantly affects its lattice position and, consequently, the observed μSR spectra. Our findings highlight the potential of μSR to study the localization of quantum particles, using the muon as the probe and particle under investigation. The light mass of the muon amplifies quantum effects, enhancing the sensitivity of our measurements and enabling a detailed comparison between experimental data and theoretical calculations. These results can be directly applied to the theoretical calculations of hydrogen localization, where quantum effects, though smaller, may still be relevant in real materials.






Anharmonic quantum muon effects in the kagome antiferromagnet Zn-Barlowite 
    F. Hotz^1 *, M. Gomilšek^2,3 *, T.  Arh^2,3, A. Zorko^2,3  and H. Luetkens^1
    August 5, 2024
================================================================================

The inherent sensitivity of muon spectroscopy (μSR) to small internal magnetic fields over a broad frequency range has made it indispensable in studies of high-T_c superconductors, topological magnets, and quantum spin liquids <cit.>. However when interpreting μSR data, knowledge of the positions (stopping sites) and behavior of implanted muons in studied materials is often crucial. Recent developments in ab initio density functional theory (DFT) calculations of muon stopping sites in a classical DFT+μ approach have partially solved this problem under the critical but hard-to-justify assumption that it suffices to treat stopped muons as classical point particles at rest <cit.>. However, similarly to quantum effects of nuclei (where even a proton is ∼9× heavier than a muon), which are important in many material classes, from superconductors to hydrogen-storage materials <cit.>, strong quantum effects of muons have also been observed in a range of materials <cit.>. These include quantum muon wavefunction spread (uncertainty in position), quantum entanglement of muon–nuclear positions, quantum tunneling, and large zero-point energy (ZPE, ∼ 0.7 eV) of muons at their stopping sites. These effects can strongly influence the results of μSR measurements <cit.> and should, in many cases, be considered.
This is especially important in the search for notoriously elusive quantum spin liquid (QSL) states with exotic fractional excitations, where μSR has turned out to be a uniquely well-suited method for investigating their behavior <cit.>. Some of the most promising experimental realizations of QSLs are spin systems with a highly frustrated kagome spin-lattice in the family of paratacamites, which includes  Herbersmithite <cit.>, Zn-brochantite <cit.>, Zn-Barlowite, Cu_4-xZn_x(OH)_6FBr <cit.>. Many of the main conclusions about the existence and properties of a QSL state in these materials are based on μSR, including a recent study on Zn-Barlowite <cit.>.


In this work, we combine μSR measurements and detailed quantum ab inito calculations of muons in Zn-Barlowite to demonstrate that muons must be described as quantum particles in this system to correctly capture both their average positions at a given stopping site (shifted from the classical DFT+μ prediction due to strong anharmonic effects affecting the muon wavefunction) as well as the stability of muon stopping sites due to large, site-dependent muon ZPE contributions to their total energy. We argue that many of these effects are generic both to QSL candidate <cit.> and other materials and must be taken into account when drawing conclusions from μSR data. 



We start by performing classical DFT+μ calculations on Zn-Barlowite, where we find four distinct muon-stopping sites. The muon forms a μ–OH complex in the three lower energy muon sites [Fig. 1(a–c)], while in the highest-energy site, it forms a nearly-linear F–μ–Br complex [Fig. <ref> d)]. Peculiarly, the occupancy of the latter site, which was confirmed by experiment <cit.>, seems in question, as it is 1 eV higher than the μ–OH sites, violating the heuristic that only sites with energies less than ∼0.5 eV above the lowest-energy site are expected to be occupied <cit.>. The first and the most accessible approximation to improve on the classical point-particle description of DFT+μ is the quantum harmonic approximation, valid for small displacements of the muon from its classical position <cit.>. Neglecting quantum muon–nuclear entanglement, the muon ZPE at a for each site is estimated by summing the contributions ħω_j / 2 of the three main Γ-point normal phonon modes ω_j of the muon calculated by DFT <cit.>. 
This leads to ZPE and total muon site energies listed in Table <ref> (see Appendix <ref>). We see that the stability of the highest-energy F–μ–Br site is improved by taking the quantum ZPE into account. However, as the highest-frequency normal mode at the μ–OH sites is along the μ–O direction, one could expect that the harmonic approximation breaks down due to anharmonic effects, with the energy penalty higher for displacements towards the oxygen than away from it. 


 Paper Title: Information Scrambling at Quantum Hall Interfaces and Their Analog to Black Hole Event Horizon.









Ken K. W. Ma Hometown: Orlando, Florida 
 
kwokwai_ma@alumni.brown.edu
Kun Yang () Department of Physics, Florida State University and the National High Magnetic Field Laboratory, 

 Address: 1800 E Paul Dirac Dr, Tallahassee, Florida 32310 

kunyang@magnet.fsu.edu





Information Scrambling at Quantum Hall Interfaces and Their Analog to Black Hole Event Horizon
    Ken K. W. Ma and Kun Yang
    August 5, 2024
==============================================================================================



The black hole information paradox has been hotly debated for the last few decades without a full resolution. This makes it desirable to find analogues of this paradox in simple and experimentally accessible systems, whose resolutions may shed light on this longstanding and fundamental problem. Here, we review and resolve the apparent “information paradox" in two different interfaces separating Abelian and non-Abelian quantum Hall states. In both cases, the information carried by the pseudospin degree of freedom of the Abelian anyons get scrambled when they cross the interface and enter the non-Abelian quantum Hall liquid. Nevertheless, it is found that the scrambling mechanism depends on the nature of the interface. The corresponding analogues of different concepts in black hole physics such as event horizon, black hole interior, Hawking radiation, and Page curve will also be discussed.





§ INTRODUCTION


The existence of black holes has received strong support from recent observations <cit.>. Instead of being a region which nothing can escape from, Hawking predicted that a black hole emits radiation and evaporates slowly <cit.>. He also concluded that the radiation carries no information except mass, angular momentum, and charge of the black hole <cit.>. This result points to possible loss of information in black holes. On one hand, it is consistent with the no-hair theorem <cit.>. On the other hand, quantum mechanics forbids information loss due to the unitarity of time evolution. This apparent contradiction leads to the black hole information paradox <cit.>. It is believed that the resolution of this paradox may provide important clues on how to combine quantum mechanics and general relativity.

Various approaches have been proposed to resolve this black hole information paradox. Among them, the holographic principle <cit.> supports the preservation of unitarity and information. In particular, information can be encoded holographically on surfaces, such as the event horizon. This belief is substantiated by the discovery of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence <cit.>. Recently, the firewall scenario <cit.> was proposed to resolve the conflict between black hole complementarity <cit.> and monogamy of entanglement <cit.>. If this conjecture is correct, the firewall at the event horizon (or black hole's boundary) may also break the entanglement between the outgoing and the in-falling particles. Thus, the boundary can be as important as, or even more important than, the interior of a black hole.

Suppose black hole evaporation is a unitary process. Then how is information hidden in the black hole released from Hawking radiation? Page argued that the release of information starts slowly at the beginning, but becomes faster in the later stage of the evaporation <cit.>. If the system was initially in a pure state, entropy of the radiation (coming from its entanglement with the remainder of black hole) would first increase from zero but eventually decrease back to zero when the black hole evaporates completely, thus recovering the pure state nature of the system and all the (quantum) information it carries. This feature is now known as the Page curve. Based on quantum information theory, the thought experiment by Hayden and Preskill (Hayden-Preskill protocol) has provided further insight on retrieving information from Hawking radiation <cit.>. Suppose the black hole has already passed its Page time and become maximally entangled with its previously emitted Hawking radiation. If the internal dynamics of black hole can be described by an instantaneous random unitary transformation, then any additional information entering the black hole can be recovered from Hawking radiation almost immediately (a very short time compared to the lifetime of the black hole) <cit.>. The protocol has postulated the existence of information scrambling <cit.>, which describes the dispersal of local information into entanglement and correlation throughout the entire system. Thus, the original information is stored nonlocally and cannot be recovered via local measurement. This concept has attracted considerable attention in the context of many-body dynamics <cit.> and quantum neural network <cit.>, which has been verified in recent experiments <cit.>. In addition, recent studies have recovered the Page curve for AdS black holes <cit.>. However, a full resolution of the paradox remains an open problem <cit.>. It is thus desirable to mimic the information paradox in simple and experimentally accessible systems, that allows for a complete understanding of this process.

Somewhat similar to the holographic principle, the bulk-edge correspondence relates the topologically protected edge modes and bulk topological orders in fractional quantum Hall (FQH) systems <cit.>. This allows us to learn about the bulk by probing the edge of the system <cit.>. Comparatively speaking, interfaces between a pair of FQH states are explored much less <cit.>. In fact, the physics and dynamics of interfaces are much richer than simple edges <cit.>. For example, as we demonstrate below, certain interfaces allow quasiparticle tunneling between two different FQH states, even if the quasiparticles are of very different nature. If they have different internal degrees of freedom, (local) information carried by them needs to be transmuted (or scrambled in a specific way) to prevent information loss. This motivates us to explore analogues of black hole information paradox in quantum Hall interfaces.

In this chapter, we review and resolve a condensed matter analogue of information paradox in two different quantum Hall (QH) interfaces <cit.>. We first consider the interface between Halperin-331 <cit.> and Pfaffian (Moore-Read) <cit.> QH states. The latter is a famous non-Abelian state which hosts non-Abelian anyons that may be useful in topological quantum computation <cit.>. Meanwhile, both 331 and Pfaffian states may be realized in bilayer systems or wide quantum wells at filling factor ν=1/2 <cit.>. Due to the competition between interlayer tunneling and intralayer Coulomb interaction, a phase transition between the 331 and Pfaffian states was predicted <cit.>. This suggests the possibility of creating a 331-Pfaffian interface by controlling the tunneling strengths in different regions of the bilayer system<cit.>. Interestingly, the Abelian 331 state has a pseudospin (layer) degree of freedom, which is absent in the Pfaffian state. If the original information carried by the pseudospin degree of freedom becomes irrecoverable after quasiparticles cross the interface and enter the Pfaffian liquid, it leads to an “information paradox". We demonstrate that the information is scrambled and stored nonlocally in the Pfaffian liquid and the interface. We also mimic black hole evaporation in the same system, and find it satisfies the Page curve naturally. In other words, the original pseudospin information is recovered and the “information paradox" in our model is resolved. Following the similar logic, we consider the “information paradox" in another QH interface formed by Halperin 330 state and Read Rezayi state at level four (abbreviated as RR_4 state) <cit.>. On one hand, the setting closely resembles the 331-Pfaffian interface by having a pseudospin degree of freedom in the 330 state that is absent in the RR_4 state. Both states may be realized in bilayer systems at ν=2/3 <cit.>. On the other hand, we will show that the nature of 330-RR_4 interface is different from the 331-Pfaffian interface. This leads to different mechanisms for scrambling the original pseudospin degree of freedom carried by the Abelian anyons in the two QH interfaces. Here, we must emphasize that we are not aiming at a resolution of the original information paradox in astrophysical black holes. This is clearly unachievable by proposing a simple analogy. Instead, we want to simulate some important concepts in resolving the original paradox in a simple and accessible manner.



§ BASICS OF QUANTUM HALL EFFECT


To set the stage for later discussion, we first review briefly some basic concepts in quantum Hall (QH) physics <cit.>. Electrons moving in two dimensions (x-y plane) and a perpendicular magnetic field (in the z direction) have their energy levels being quantized in Landau levels. A very important parameter characterizing the system is the Landau level filling factor ν, which is the ratio between the number of electrons and the number of magnetic flux quanta enclosed by the system. Of particular interest is the case ν<1, where only the lowest Landau level is partially filled by electrons at low temperature, known as the fractional QH (FQH) regime. Since the kinetic energy of these electrons is quenched due to Landau quantization, the interaction between them dominates the properties of the system. Various FQH states, which possess numerous fascinating properties, are realized in this strongly correlated electronic system. The exotic properties of FQH states are associated with the topological order they possess <cit.>. Most prominent among them is the existence of low-energy excitations (quasiparticles) that have fractional charges and obey fractional statistics (between bosonic and fermionic statistics) <cit.>. A famous example is the Laughlin state at ν=1/3 <cit.>, in which quasiparticle with a fractional charge e/3 <cit.> and a fractional statistics 2π/3 can exist <cit.>. Note that both fractional charge <cit.> and fractional statistics were observed experimentally <cit.>. Such exotic quasiparticles are called anyons, and the possible types of anyons are associated with the specific topological order.



 §.§ Bulk-edge correspondence and conformal field theory


The bulk-edge correspondence, another consequence of the topological order, relates the edge structure and the bulk topological order in FQH systems. In particular, it predicts the existence of gapless edge modes described by conformal field theories (CFTs), and there is a one-to-one correspondences between the bulk topological order and edge CFT <cit.>. In our previous example, the edge of the Laughlin state at ν=1/3 has a single chiral bosonic edge mode ϕ, which can be described by the Lagrangian density,

    ℒ_1/3
    =-3/4π∂_xϕ(∂_t-v∂_x)ϕ.

Here, v is the speed of the edge mode, and ϕ is a (chiral) bosonic field. In general, the edge of a FQH liquid can have more than one edge mode. For Abelian FQH states, the corresponding edge theory is described by <cit.>

    ℒ_ edge
    =-1/4π∑_i,jK_ij∂_tϕ_i∂_xϕ_j
    -1/4π∑_i,jV_ij∂_xϕ_i∂_xϕ_j.

Importantly, the K matrix encodes all information of the topological order. For a FQH state in a bilayer system, it may (but not always) be described by a two-component topological order which has a 2× 2 K matrix. In this situation, two different edge modes exist. Furthermore, the possible type of edge modes is not limited to bosonic mode. Other types of modes such as Majorana fermion modes exist when the topological orders are non-Abelian <cit.>.

With the knowledge of the edge structure in hand, different low-energy excitations in the FQH system can be described or created by suitable CFT operators <cit.>. For example, a charge-e/3 quasiparticle and an electron in the ν=1/3 Laughlin state are created by the operators :exp(iϕ): and :exp(3iϕ):, respectively. Here, :𝒱: denotes the normal ordering of the vertex operator 𝒱. When there is no confusion, this normal ordering notation will be dropped in the later discussion. For a FQH state being described by a multicomponent topological order, there are multiple types of anyons (described by different CFT operators) that have the same electric charge. In other words, the anyons have an additional degree of freedom. This point will become clear when we discuss our setup.



§ INFORMATION PARADOX IN 331-PFAFFIAN INTERFACE



Both Halperin-331 and Pfaffian quantum Hall liquids have the same Landau level filling factor ν=1/2, which can be realized in a bilayer system. For the 331 liquid, it is described by a two-component topological order with the K matrix <cit.>,

    K=[ 3 1; 1 3 ].

The two different edge modes are denoted as ϕ_↑ and ϕ_↓. The two most relevant operators creating an electron are exp(3iϕ_↑+iϕ_↓) and exp(iϕ_↑+3iϕ_↓). On the other hand, the Pfaffian liquid is described by a single-component non-Abelian order with K=2. Its edge has a bosonic mode ϕ and a Majorana fermion mode ψ <cit.>. The corresponding electron operator is ψexp(2iϕ). Since the Halperin-331 and Pfaffian edges have opposite chiralities at the interface, the interface is described by the Lagrangian density <cit.>,

    ℒ
    =   -1/4π∑_i,jK_ij∂_tϕ_i∂_xϕ_j
    +2/4π∂_tϕ_l∂_xϕ_l
    -iψ_l∂_tψ_l
    
       -ℋ(ϕ, ψ).

Here, the indices i,j=↑, ↓ denote the layer or the pseudospin.
All ϕ_↑, ϕ_↓, and ϕ_l are charge modes. The first two are right-moving along the edge of the 331 liquid, whereas the last one is left-moving along the edge of the Pfaffian liquid. The Pfaffian liquid also has a left-moving neutral Majorana fermion mode ψ_l along the edge. The edge structures of both quantum Hall states are illustrated in Fig. <ref>(a). As shown by one of us, a relevant random electron tunneling between the Pfaffian and 331 edges can lead to a phase transition at the interface <cit.>.

Now, we follow Ref. <cit.> and briefly summarize how different modes get localized at the interface. This also allows us to introduce useful notations for later discussion. One can define a charge mode ϕ_r=ϕ_↑+ϕ_↓ and a neutral spin mode ϕ_n=ϕ_↑-ϕ_↓ in the 331 liquid. Using this new set of modes, the topological term of the 331 edge becomes

    ℒ_331
    =-2/4π∂_tϕ_r∂_xϕ_r
    -1/4π∂_tϕ_n∂_xϕ_n.

The overall charge density at the interface is given by

    ρ(x)
    =1/2π∂_x(ϕ_↑+ϕ_↓+ϕ_l)
    =1/2π∂_xϕ_c.

The random electron tunneling between the Pfaffian and 331 edges is described by

    H_ T
    =   ∫ξ(x)ψ_l
    (e^3iϕ_↑+iϕ_↓+2iϕ_l
    +e^iϕ_↑+3iϕ_↓+2iϕ_l) dx
    +H.c.
    
    =   ∫ |ξ(x)|ψ_l(x) ψ_r(x)cos[2ϕ_c(x)+φ(x)] dx.

Here, ξ(x) denotes the random tunneling amplitude. In the second line, |ξ(x)| and
φ(x) are the magnitude and the phase of ξ(x), respectively. We have also fermionized exp[iϕ_n(x)]=ψ_r(x)+iψ_R(x). The resulting edge modes are shown in Fig. <ref>(b). If H_ T is relevant in the renormalization group sense, then both charge modes, ψ_l, and ψ_r are localized at the interface. After the localization, only a single right-moving Majorana fermion mode remains gapless and propagates freely at long distance, as shown in Fig. <ref>(c). Notice that this gapless mode is neither an original edge mode of the 331 nor the Pfaffian state.



Since Pfaffian and 331 states are quantum Hall states formed by superconducting pairing between composite fermions <cit.>, both of these two states have quasiparticles with the smallest possible charge of e/4 <cit.>. However, there is a fundamental difference between these quasiparticles. For the 331 state, there are two different types of Abelian e/4 quasiparticles created by the vertex operators, e^iϕ_↑ and e^iϕ_↓. One may view them as quasiparticles with different pseudospins. When we formulate the information paradox in the following discussion, this pseudospin will be regarded as the degree of freedom of the Abelian quasiparticles. For the Pfaffian state, there is only one type of e/4 quasiparticle created by σ e^-iϕ_l/2 <cit.>. Here, σ is the twist field with a scaling dimension 1/16 in the chiral Ising CFT <cit.>. We summarize the three primary fields and their properties in Table <ref>. In particular, σ satisfies the fusion rule σ×σ=ψ+I. Note that we have omitted the subscript l for the Majorana field to make the discussion of Ising CFT general. Its proper meaning should be clear from context. The fusion rule indicates that the quasiparticle is non-Abelian. An interesting question is what happens if a quasiparticle is dragged from the 331 liquid in to the Pfaffian liquid? It seems that the pseudospin information would be lost. In this sense, we can define the interface between the two different QH liquids as the “event horizon with a firewall" in our setup. This definition or analogy makes sense since the interface plays the role of a one-way surface of information in our setup, and the “destruction" of pseudospin information at the interface resembles a firewall conjectured in Ref. <cit.>. In addition, the Pfaffian liquid can be viewed as the interior of a “black hole". Suppose this analogous black hole can evaporate (discussed in Sec. <ref>) and the original pseudospin information cannot be recovered at the end of the evaporation. Then, the lost of information contradicts to the fact that quasiparticle tunneling is a unitary process. Thus, we have identified an apparent “information paradox".





 §.§ 331-Pfaffian interface from anyon condensation



Before resolving the paradox, we reformulate the above discussion in the framework of anyon condensation <cit.>. This technique has been commonly applied to study possible transitions between topologically ordered phases. In the context of quantum Hall physics, it was used to study the interface between Pfaffian and non-Abelian spin-singlet (NASS) quantum Hall states <cit.>. In this section, we first use anyon condensation to deduce the CFT description of the 331-Pfaffian interface. In the next section, we apply the same technique to resolve the paradox. Along the way, we adopt a pedagogical approach and aim at relating the rather abstract technique to the more physical picture in Sec. <ref>. It will allow us to highlight the advantages of applying anyon condensation in studying quantum Hall interfaces.

From Sec. <ref>, we know that the edges of the Halperin-331 and Pfaffian liquids are described by CFTs with central charges 2 and 3/2, respectively. These two edges are counterpropagating at the interface. Hence, we expect the resulting CFT describing the 331-Pfaffian interface has a net central charge of 2-3/2=1/2. To deduce exactly what the CFT is, it is first necessary to separate the charge and neutral sectors for both Halperin-331 and Pfaffian liquids. It is because a charge mode cannot be gapped out by coupling to a neutral mode in a usual situation. Equivalently, we do not consider the possibility of condensing charge bosons, which will break the U(1) gauge symmetry. The separation was already achieved in Sec. <ref>. In particular, the combination of charge modes ϕ_c=ϕ_r+ϕ_l was shown to be gapped out (more precisely, localized) by H_ T <cit.>. Therefore, we can focus our discussion on the neutral sectors.

As stated previously, the neutral sector of the Pfaffian state is described by a chiral Ising CFT. For the Halperin-331 state, its neutral sector is governed by the spin mode ϕ_n, which is described by the U(1)_4 CFT. Different primary fields in this Abelian CFT are summarized in Table <ref>. Note that any two vertex operators in the form
e^iαϕ_n/2 and e^i(α+4ℤ)ϕ_n/2 are identified.



The structure (remaining gapless modes) of the 331-Pfaffian interface is solely determined by anyon condensation in the neutral sectors. This condensation occurs in the
U(1)_4 ×Ising CFT. We emphasize again that the bar denotes conjugation of the Ising CFT due to the opposite chiralities between the 331 and Pfaffian edges at the interface.  Compared to the original CFT, anyons in the conjugate CFT have the same fusion rules, but complex conjugated topological spins and braiding phases. Alternatively, one may interpret the condensation as a coset construction <cit.>. We label a generic anyon as (e^iαϕ_n/2, t̅). Here, the parameter
α=0,1,2,3 determines the corresponding primary fields in the U(1)_4 CFT. Meanwhile, t̅={I̅, ψ̅, σ̅} denotes the primary fields in the conjugate Ising CFT. In the present case, there is only one condensable boson,
B=(e^iϕ_n, ψ̅). The condensation of B leads to confinement of some of the anyons in the condensed phase. An anyon remains unconfined if and only if it has a trivial mutual statistics with B. This condition ensures that an unconfined anyon has a consistent topological spin in the condensed phase. Furthermore, two anyons are identified when they differ from each other by a multiple of B. Using operator product expansion, it is straightforward to deduce the six (or three after identification) deconfined anyons in the condensed phase. They are listed in Table <ref>. Their corresponding topological sectors, namely Ĩ, ψ̃, and σ̃ are defined according to their conformal spins. From the table, we conclude that the 331-Pfaffian interface is described by a chiral Ising CFT. Note that this Ising CFT has an opposite chirality to the one describing the Pfaffian edge at the interface.



Now, one may wonder why going through such abstract and seemingly redundant procedures to find out the CFT describing the interface. Doesn't the net central charge c=1/2 directly indicate that it should be an Ising CFT? There are two reasons for analyzing this simple system by anyon condensation. First of all, it is fortunate that for the 331-Pfaffian interface, the mechanism and consequences of anyon condensation can be visualized in a very transparent and physical manner, but this is a very special case. In Eq. (<ref>), the electron tunneling between counterpropagating edges at the interface couples ψ_l and e^iϕ_n. This leads to a mass term and eventually gaps out the counterpropagating ψ_l and ψ_r. Only ψ_R remains gapless at the interface. This was demonstrated by fermionizing e^iϕ_n=ψ_r+iψ_R. This type of arguments does not always generalize to more complicated interfaces. On the other hand, the condensation of B systematically captures the gaping process and leads to a correct CFT description of the 331-Pfaffian interface. More importantly, anyon condensation relates every primary field in the original and condensed phases. These relations cannot be obtained from the argument in Sec. <ref>.



 §.§ Transmutation of pseudospin information
 


We now discuss the transmutation of pseudospin information when Abelian quasiparticles cross the interface. Let us first comment on the charge sectors. As one will see, they basically play no role in the resolution of the paradox. Since charge-e/4 quasiparticles are allowed in both Halperin-331 and Pfaffian liquids, dragging quasiparticles across the interface does not require any absorption of net charge at the interface. Furthermore, the gapping of ϕ_c indicates that the dragging will not create any low-energy charge excitation at the interface <cit.>.

Thus, we focus on the neutral sectors. An Abelian quasiparticle in the Halperin-331 liquid has its pseudospin degree of freedom carried solely by the neutral mode ϕ_n. This is observed by writing

    e^iϕ_↑   =e^iϕ_r/2e^iϕ_n/2,
    
    
    e^iϕ_↓   =e^iϕ_r/2e^-iϕ_n/2.

Hence, the vertex operators 𝒱_1 and 𝒱_3 encode the spin-up and spin-down states of an Abelian charge-e/4 quasiparticle, respectively. These two operators are not defined in the Pfaffian liquid. To understand the transmutation of quasiparticles when they cross the interface, we need to represent the four primary fields in the U(1)_4 CFT as different products between two Ising CFTs. One of them describes the interface, whereas the other describes the Pfaffian order. Both Ising CFTs now have the same chirality to match the central charges, 1=1/2+1/2. From Table <ref>, one can obtain the inverted expressions:

    𝒱_0
       ≡ I_1
    =I_1/2×Ĩ+ψ×ψ̃,
    
    𝒱_1
       ≡ e^iϕ_n/2
    =σ×σ̃,
    
    𝒱_2
       ≡ e^iϕ_n
    =ψ×Ĩ+I_1/2×ψ̃,
    
    𝒱_3
       ≡ e^-iϕ_n/2
    =σ×σ̃.

The tilded and untilded fields are in the CFTs describing the interface and the Pfaffian liquid, respectively. Also, the subscripts 1 and 1/2 in the identity fields denote the central charges of the corresponding CFTs. When there is no confusion, these subscripts will be skipped.

Eq. (<ref>) suggests that the U(1)_4 CFT is obtained from condensing the boson b=(ψ,ψ̃) <cit.>. This result is consistent with the orbifold construction <cit.>. After the condensation, one of the unconfined particles is (σ, σ̃). We should state clearly that these two twist fields describe excitations (anyons) at different regions of the system, so it is meaningless to consider their fusion. In other words, the present situation is different from the case of a Pfaffian liquid, in which σ in the bulk and σ at the edge created from vacuum must fuse to I for conserving fermion parity. Importantly,

    (σ, σ̃)×(σ, σ̃)
    =(ψ, ψ̃)+(I, Ĩ)+(ψ, Ĩ)+(I, ψ̃).

The first two terms on the right hand side show that two orthogonal copies of vacuum exist, so (σ, σ̃) needs to split into two inequivalent types of anyons in the resulting U(1)_4 CFT <cit.>. We denote them as (σ, σ̃)_1 and (σ, σ̃)_2. Both of them have conformal spins 1/8, which are identified as the vertex operators 𝒱_1 and 𝒱_3 in the U(1)_4 CFT  (see Table <ref>). The fusion rules are consistent by imposing the conditions
(σ, σ̃)_1×(σ, σ̃)_1=
(σ, σ̃)_2×(σ, σ̃)_2
=𝒱_2
and (σ, σ̃)_1×(σ, σ̃)_2=𝒱_0. Following Ref. <cit.>, we interpret the above result as an incoming pseudospin-up quasiparticle transmutes into a neutral anyon σ̃ at the interface, and another anyon σ free to move in the Pfaffian liquid. To be more precise, the last anyon actually carries charge e/4, but we skip displaying its charge sector e^-iϕ_l/2 explicitly. The same conclusion holds for an incoming quasiparticle with pseudospin down. We illustrate the results in Fig. <ref>.





  §.§.§ Matching of Hilbert spaces and analogue of information scrambling



It is obvious that the total quantum dimension of (σ, σ̃)_1 and (σ, σ̃)_2 is two. It matches the two-dimensional Hilbert space spanned by the pseudospin degree of freedom of an Abelian charge-e/4 quasiparticle. This matching is guaranteed mathematically by the commutativity between fusion and restriction in anyon condensation <cit.>. Interestingly, the information of pseudospin is being stored nonlocally at the interface and in the interior of Pfaffian liquid. There is no local measurement that can distinguish between (σ, σ̃)_1 and (σ, σ̃)_2. Hence, it is impossible to recover the original information from any local measurement. This feature resembles a quantum information scrambling, which can be defined as the spreading of local information into many-body entanglement and correlation in the whole system <cit.>.

The situation becomes more interesting when we keep dragging more Abelian quasiparticles across the interface. Suppose N-1 charge-e/4 quasiparticles were already dragged. We assume the corresponding N-1 anyons σ̃ created at the interface are well separated from each other, so that no fusion occurs between them. We also pose the same assumption for the N-1 anyons created in the Pfaffian liquid. Consider dragging an additional charge-e/4 quasiparticle across the interface. This process increases both numbers of σ̃ and σ by one. As a result, there are N neutral anyons
σ̃ at the interface, and N non-Abelian anyons in the interior of the Pfaffian liquid. The dimension of the corresponding topological Hilbert space is then increased by a factor of two, which is consistent with the one bit of information carried by the additional Abelian quasiparticle from the Halperin-331 liquid. We illustrate the example of N=6 in Figs. <ref>(a) and <ref>(b). Now, we relax the confining potential, and allow the anyons to move and braid <cit.>. The braiding can further scramble the original information <cit.>. The N anyons at the interface are indistinguishable, so are those N anyons in the Pfaffian liquid. Meanwhile, the information carried by pseudospins of the original N Abelian quasiparticles is still preserved. Both Hilbert spaces for indistinguishable anyons at the interface and indistinguishable anyons in the Pfaffian liquid have dimensions 2^N/2. It is intriguing that the interface and the Pfaffian liquid store the same amount of information. This does not hold in the Pfaffian-NASS interface <cit.>.

The above discussion also suggests another important feature. In addition to being stored nonlocally, the original pseudospin information is actually “hidden" in the fusion channels of the non-Abelian anyons. Hence, the scrambled information is protected topologically and will not be destroyed by any local perturbation. This property is essential in topological quantum computation (TQC) <cit.>.



 §.§ Upper bound of information storage and holographic principle



Our previous discussion assumed that local anyons in the system can be well separated to prevent fusion. This assumption leads to a natural question. How much information can be stored nonlocally with the topological protection that has been described?

Recall that the minimum separation between two anyons is in the order of the magnetic length ℓ_B, so that they are well defined individually and do not fuse. From this, one may naively think that the maximum amount of information can be stored is N_A∼ A/πℓ_B^2, where A denotes the area of the Pfaffian liquid. This argument is valid if the information is carried solely by anyons in the Pfaffian liquid. However, this is not true in the present case. We have assumed both Pfaffian liquid and 331-Pfaffian interface were initially in the ground state with no excitations. As we discussed, the nonlocal storage of pseudospin information of the Abelian quasiparticles from the Halperin-331 liquid requires both anyons at the interface and in the Pfaffian liquid.

For an interface with a length (perimeter) L, it can only accommodate N_L∼ L/ℓ_B neutral anyons σ̃. Since the radius R of a circular quantum Hall droplet satisfies R≫ℓ_B, one has N_L≪ N_A. When the number of σ̃ gets close to or exceeds N_L, different σ̃ anyons start to fuse. The resulting particles will be either a fermion or a boson that can propagate back to the Halperin-331 liquid. More explicitly, one has

    (I,σ̃)×(I,σ̃)
    =(I,Ĩ)+(I,ψ̃).

For the first fusion outcome, the two σ̃ anyons at the interface can fuse to a neutral boson with its spin part described by 𝒱_0 [see Eq. (<ref>)]. This neutral boson can then split to a pair of quasihole and quasiparticle with opposite charges but the same pseudospin, and propagate in the 331 liquid. For the second fusion outcome, the neutral fermion can split to a pair of quasihole and quasiparticle with opposite pseudospins propagating in the 331 liquid. Consequently, some of the hidden information is released and being accessible by local probes. Therefore, the “black hole” is no longer completely black. Note that the released information is not protected topologically and can suffer from quantum decoherence. The discussion shows that the length of the interface sets an upper bound of storing information nonlocally and topologically via (σ,σ̃) pairs. Furthermore, the magnetic length ℓ_B=√(1/eB) (in the unit of h/2π=c=1) plays the role of Planck length in the present system. Here, B denotes the magnetic field.

The above observation actually resembles the argument from holographic principle in black holes. Based on this principle, the maximum amount of information can be stored in a black hole is not determined by its volume, but bounded by its area <cit.>. This is because the Bekenstein entropy of the black hole is proportional to its area <cit.>, which limits the number of degree of freedoms the black hole can have. In contrast to the quantum Hall interface, the black hole can always store and “hide" more information by increasing its area. Since the length of the quantum Hall interface is assumed to be fixed, the analogous (a weaker version of) holographic principle there implies that Hawking radiation in the form of Abelian quasiparticles and quasiholes will be released when the bound N_L∼ L/ℓ_B is reached. Roughly speaking, any additional incoming information is thus reflected by the interface (event horizon).



§ INFORMATION PARADOX IN 330-RR_4 INTERFACE


In the previous section, we demonstrated that an original Abelian anyon from the Halperin-331 QH liquid would transmute into a pair of anyons, when it crosses the interface and enters the non-Abelian Pfaffian QH liquid. More specifically, one of the resulting anyons is neutral and being created at the interface, and the remaining one is created in the Pfaffian liquid. By proximitizing different pairs of QH liquids, a wide variety of interfaces can be formed. The corresponding quasiparticle transmutation (if any) will depend on the properties of the interface. Hence, different ways of scrambling the original information may be realized in different QH interfaces. Recall that the 331-Pfaffian interface has a chiral central charge of c=2-3/2=1/2. This indicates that there must be a gapless chiral Majorana fermion mode on the interface <cit.>. As a result, some of the scrambled information can be stored at the interface. Meanwhile, an interface with a zero chiral central charge can be created by proximitizing a pair of different QH states with identical central charges for their edges. Interfaces of this kind have been studied extensively, and importantly, can be gapped or gapless. When both QH phases are Abelian, they are described by the K-matrix formalism <cit.>. This enables one to study the gapness of the interface through the concepts of Lagrangian subgroups <cit.> or the null vector criteria <cit.>. For an interface involving non-Abelian QH state(s), its gapness and the corresponding gapped phases can be explored by the anyon condensation approach <cit.>. A rigorous mathematical formalism was developed in <cit.>. We note that gapped interfaces between different non-Abelian topologically ordered states is still an ongoing research topic.

Motivated by the above discussion, we consider in this section the information paradox and the corresponding quasiparticle transmutation in the interface between the Abelian Halperin-330 state <cit.> and the non-Abelian Read-Rezayi state at level four <cit.>. Both states or phases may describe the fractional QH (FQH) state in a bilayer system at the total Landau-level filling factor ν=2/3. Whether these phases are favorable or not in a realistic sample depends on the actual microscopic details of the system. Here, we assume both states and the interface between them can be realized, and study the possible consequences. Previous theoretical work suggested that a continuous phase transition between the Halperin-330 state and the RR_4 state might be triggered by tuning the interlayer tunneling strength in a bilayer FQH system <cit.>. This transition and similar phase transitions in bilayer systems at other filling factors can be studied systematically through the anyon condensation approach <cit.>. In particular, the condensation of a specific type of anyon in the RR_4 state leads to the Halperin-330 state. Based on the “folding trick" <cit.>, it is expected that a gapped boundary between these two FQH states can form. Nevertheless, the precise gapping mechanism and the possibility of having different phases for the gapped interface have not been addressed. Suppose the interface can be gapped, the scrambled pseudospin information may be entirely stored by anyons in the non-Abelian RR_4 FQH liquid with no information stored at the interface. This feature is completely different from the one in the aforementioned 331-Pfaffian interface <cit.>.



 §.§ Review of Halperin-330 and Read-Rezayi states



The Halperin-330 state is a two-component Abelian topological order, which is characterized by the matrix <cit.>,

    K=[ 3 0; 0 3 ],

and the associated t=(1,1)^T. Clearly, one has ν=t^T K^-1t=2/3. In a more explicit form, the corresponding topological term for the Halperin-330 edge is

    L_0
    =-3/4π(∂_tϕ_↑∂_xϕ_↑
    +∂_tϕ_↓∂_xϕ_↓).

Here, the subscripts ↑, ↓ denote the layer or the pseudospin index. Roughly speaking, each layer of the Halperin-330 state is a simple Laughlin state at 
ν=1/3  <cit.>. Since ϕ_↑ and ϕ_↓ have the same chirality, the edge is maximally chiral and has a central charge c=2. 

The two most relevant vertex operators for electrons are exp(3iϕ_↑) and exp(3iϕ_↓). These two different operators create respectively an electron in the upper and the lower layer. We call them as electrons with pseudospin up and pseudospin down. Both electrons operators have scaling dimensions 3/2, indicating that they are indeed fermionic as required. The most fundamental anyon that the Halperin-330 state can host has charge e/3. Depending on its pseudospin, it is created by the operator exp(iϕ_↑) or exp(iϕ_↓). The OPE between any one of them and each of the electron operators is singlevalued. For example, one has the following OPEs:

    lim_z→ w
    e^3iϕ_↑(z)× e^iϕ_↑(w)   ∼ (z-w)e^4iϕ_↑(w),
    
    lim_z→ w
    e^3iϕ_↓(z)× e^iϕ_↑(w)   ∼ e^i[3ϕ_↓(w)+ϕ_↑(w)].

Notice that |α|=1 is the smallest possible nonzero value for the generic operator exp(iαϕ_↑) to have singlevalued OPEs with both electron operators. This fact verifies that the e/3 anyon is the smallest-charge anyon that can be host by the Halperin-330 state.



  §.§.§ Parafermion conformal field theory and Read-Rezayi states


Different from the Halperin-330 state, the Read-Rezayi states are a series of non-Abelian topological orders introduced from the CFT approach <cit.>. Each RR state consists of two different types of CFTs. First, it has a compactified U(1) holomorphic boson ϕ. On the edge of the system, ϕ corresponds to the gapless charge mode described by

    L_ϕ
    =-1/4πν∂_xϕ(∂_t+v∂_x)ϕ.

This mode fixes the charge density ρ(x)=∂_xϕ/2π and the quantized electrical Hall conductance σ_xy=ν e^2/h. The possible values of the filling factor ν depend on the second type of CFT in the RR state, which is the chiral ℤ_k parafermion CFT with the central charge <cit.>,

    c=2(k-1)/k+2.
 

The above parafermion CFT can be obtained from the SU(2)_k/U(1) coset construction <cit.>. Here, k∈ℕ is the level of the corresponding Wess-Zumino-Witten model. Following the notations in Ref. <cit.> (which are more convenient for the general discussion), each Virasoro primary field in the parafermion CFT is labeled as Φ^ℓ_m with ℓ+m≡ 0 (mod  2). The index ℓ=0,1,⋯, k. By imposing the field identification, Φ^ℓ_m=Φ^ℓ_m+2k=Φ^k-ℓ_m-k, the index m can be restricted to the range -ℓ<m≤ℓ, and ℓ>0. Hence, there are in total k(k+1)/2 Virasoro primary fields. Their scaling dimensions are given by

    h^ℓ_m=ℓ(ℓ+2)/4(k+2)-m^2/4k.

Since the CFT is chiral or holomorphic, the scaling dimension of the field is also its conformal spin. The fusion rule between different fields is

    Φ^ℓ_1_m_1×Φ^ℓ_2_m_2
    =∑^min(ℓ_1+ℓ_2, 2k-ℓ_1-ℓ_2)_ℓ=|ℓ_1-ℓ_2|Φ^ℓ_m_1+m_2.

When k≥ 2, the ℤ_k parafermion CFT is non-Abelian. A famous example is the Ising CFT at k=2, which has a non-Abelian anyon σ corresponding to the field Φ^1_1. The Ising CFT is relevant in the description of FQH state at ν=5/2 <cit.>. The parafermion CFTs and their related FQH states have attracted considerable amount of attention as the non-Abelian anyons there are useful in topological quantum computation <cit.>. 

An anyon (including the electron) in the RR state is created by the operator, 
ηexp(iωϕ). Here, η is a Virasoro primary field in the parafermion CFT. It is customary to choose η=Φ^k_k-2 for constructing the electron operator. The conformal spin for Φ^k_k-2e^iϕ/ν is

    h=k-1/k+1/2ν.

By requiring ν>0 and h being an half-integer, the possible filling factor is fixed at ν=k/(Mk+2), where M is a positive odd integer. For k=4 and M=1, the Read-Rezayi state may describe a FQH state of electrons at ν=2/3. This matches the filling factor of the Halperin-330 state. In particular, the Lagrangian density describing the chiral Bose mode ϕ_l on the RR_4 edge is

    L_ϕ_l=-3/8π∂_xϕ_l(-∂_t+v∂_x)ϕ_l.

Since we will eventually study the interface between two FQH liquids, we define the chirality of ϕ_l as opposite to the chiralities of the two Bose modes on the Halperin-330 edge. The subscript l indicates that ϕ_l is a left-moving mode. As a reminder, the complete edge theory for the RR_4 state also involves the neutral parafermionic sector (see the particular discussion in Ref. <cit.>). 

For later reference, the ten different Virasoro primary fields in the ℤ_4 CFT and their fusion rules are listed in Table <ref>. Note that we have switched to another set of notations (somehow more convenient for this particular discussion) for the primary fields. The identification with the notations in the previous discussion:

    I   =Φ^4_4, ψ_1=Φ^4_2, ψ_2=Φ^4_0, ψ_3=Φ^4_-2, 
    σ_+=Φ^3_3,
    
    σ_-   =Φ^1_1, ϵ=Φ^2_0, ρ=Φ^2_2, 
    χ_-=Φ^3_1, χ_+=Φ^3_-1.

After determining the electron operator, all other operators for quasiparticles can be deduced in a “brute-force" manner by requiring the OPE between ηexp(iωϕ) and the electron operator is single-valued. This restricts the possible values of ω in the vertex operator for each separate η as listed in Table <ref>. From the result, the charge of the corresponding quasiparticles 2ω/3 (in unit of e) and the whole spectrum of anyons that the Read-Rezayi state can host is determined. Notice that the quantum dimension of the anyon a is determined by the largest eigenvalue of the fusion matrix N_a with matrix elements (N_a)_bc=N^c_ab. Here, N^c_ab is defined from the fusion between the anyons a and b, and the resulting anyon c:  

    a× b=∑_c N^c_ab c.

The maximum eigenvalue is positive and nondegenerate, as guaranteed by the Perron-Frobenius theorem. For an Abelian anyon, the quantum dimension is one. Otherwise, the anyon is non-Abelian.







 §.§ Mathematical details of anyon condensation



Here, we discuss the interface between the Halperin-330 RR_4 states (abbreviated as 330-RR_4 interface) from the anyon condensation perspective. Readers who are uninterested in the mathematical details may skip this subsection. The corresponding physical picture for the fully gapped interface and mechanism of scrambling the pseudospin degree of freedom in the system are reviewed in Secs. <ref> and <ref>.

Since both edges of the QH states have the same central charge but with opposite chiralities at the interface, the interface has a zero chiral central charge. In principle, this kind of interface can be (but not guaranteed to be) fully gapped. For example, the edge of the ℤ_2 toric code (i.e., the boundary between it and the vacuum) remains gapless if the chosen boundary does not break the translational symmetry <cit.>. However, the edge in a generic situation (with the so-called smooth or rough boundary) is gapped due to the condensation of either one of the self-bosons, e or m on the edge <cit.>. The discussion suggests that there are two different possible gapped boundaries. When the gapped boundary is obtained by condensing e, then m is confined and becomes a boundary excitation, or vice versa. Can the 330-RR_4 interface be fully gapped? If the answer is affirmative, there will be no gapless excitations on the interface. Then, it may be possible to transmute the original information carried by certain types of the Abelian anyons completely into topological information stored by anyons in the non-Abelian RR_4 liquid.

In the following discussion, we first employ the anyon condensation approach to explore the gapness of the 330-RR_4 interface. We use 𝒜 and ℬ to denote the sets of anyons for the Halperin-330 state and RR_4 state, respectively. The possible phases of the interface are determined by the resulting phases from condensing different possible anyons (if it occurs) in 𝒜×ℬ̅. Here, ℬ̅ indicates the conjugation of ℬ. In the present case, we have

    𝒜   ={e^iαϕ_↑e^iβϕ_↓},
    
    ℬ̅   ={η e^iω_ηϕ_l}.

Here, α, β are integers, and η denotes the Virasoro primary field in the ℤ_4 parafermion CFT. For each η, the possible values of ω_η are given by the last row in Table <ref> with m∈ℤ. Since both electrons in the Halperin-330 state and RR_4 state are fermions that have trivial mutual statistics with every anyon in 𝒜×ℬ̅, 𝒜×ℬ̅ is a fermionic topological order. Specifically, we introduce the symbol Ψ_e=e^3iϕ_↑ to denote the electron with pseudospin up for the Halperin-330 state. It is a trivial fermion in 𝒜×ℬ̅.



  §.§.§ Determination of Lagrangian subset


The interface can be fully gapped only if there exists a Lagrangian subset ℒ⊂𝒜×ℬ̅, in which the condensed anyons are the only deconfined anyons in the condensed phase. To identify some possible ℒ, we follow the strategy in Ref. <cit.> by first condensing Abelian anyons. After that, we will analyze the resulting phase and check for the necessity of condensing more anyons in the system to obtain a Lagrangian subset. Finally, we discuss the underlying reason that gives rise to the anyon condensation. It is vital to clarify that one can actually condense non-Abelian anyons, but the strategy below greatly simplifies the analysis. When the Lagrangian subset ℒ only consists of Abelian bosons, then the explicit conditions for having a fully gapped interface are:


  * trivial monodromy for all a∈ℒ: M^aa̅_I=1;

  * trivial mutual statistics for all a, b∈ℒ: M^ab_a× b=1;

  * confinement for all a∉ℒ: there exists at least one b∈ℒ that has a non-trivial braiding phase with a.

The condition (i) ensures that a is a boson. If a is its own antiparticle (i.e., a=a̅ and a×a̅=I), then a should have an integer conformal spin. When a is not its own antiparticle, then it further requires that b=a× a also has an integer conformal spin <cit.>. For condition (ii), it implies that the anyon c=a× b also needs to be a boson, otherwise it needs to be condensed as well <cit.>. Note that one can also condense fermions (by fusing it with the trivial fermion) in a fermionic topological order <cit.>, but this turns out unnecessary in our discussion.



  §.§.§ Separation of charge and neutral sectors


From Table <ref>, it is clear that the four Abelian anyons in the ℤ_4 parafermion CFT are I, ψ_1, ψ_2, and ψ_3. To deduce the set of condensable Abelian bosons, one can in principle analyze all anyons with η={I, ψ_1, ψ_2, ψ_3} and generic values for α, β, and ω_η in the vertex operators. However, this general treatment turns out to be unphysical. As we will discuss in Sec. <ref>, the gapped interface originates from the gapping or localization of counterpropagating edge modes due to electron and quasiparticle tunneling process. Therefore, the charge modes and neutral modes should obtain expectation values independently. This suggests us to separate the charge and neutral sectors in the problem, which is achieved by introducing a new set of modes for the Halperin-330 edge:

    ϕ_r=ϕ_↑+ϕ_↓,
    
    ϕ_n=ϕ_↑-ϕ_↓.

They correspond to the overall charge mode and neutral spin mode for the Halperin-330 edge, respectively. Both modes are right-moving. Using the new set of modes, the topological term in Eq. (<ref>) becomes

    L_0=-3/8π(∂_tϕ_r∂_xϕ_r
    +∂_tϕ_n∂_xϕ_n).
 
The t vector becomes t=(1,0)^T, which verifies that ϕ_n is a neutral mode. Furthermore, we have

    𝒜×ℬ̅
    ={e^iQϕ_r e^iSϕ_n}×{η e^iω_ηϕ_l}.

Here, Q=(α+β)/2 and S=(α-β)/2. Since α∈ℤ and β∈ℤ, both Q and S can only take half-integer or integer values. The corresponding anyon in 𝒜 has charge 2Q/3 (in the unit of e). Furthermore, the conformal spins for the two vertex operators exp(iQϕ_r) and 
exp(iSϕ_n) are

    s_Q=h_Q=Q^2/3 , 
    s_S=h_S=S^2/3.

Each of them will change by an integer value whenever Q or S is changed by 3ℤ. Physically, this corresponds to the fusion between the anyon with a trivial boson in the Halperin-330 state. Thus, we can identify a pair of anyons with 
(Q, S) and (Q+3ℤ, S+3ℤ). More explicitly, one can rescale the fields φ_r=ϕ_r/2 and φ_n=ϕ_n/2 and rewrite Eq. (<ref>) as

    L_0=-6/4π(∂_tφ_r∂_xφ_r
    +∂_tφ_n∂_xφ_n).
 
The possible vertex operators take the form exp(ipφ_r) and 
exp(iqφ_n), where both p and q can only take integer values now. Thus, both φ_r and φ_n (and hence the original fields, ϕ_r and ϕ_n) are compactified bosons in the U(1)_6 CFT which has exp(6iφ) as the trivial boson. For later discussion, the six primary fields involving the neutral spin mode in the U(1)_6 CFT  are summarized in Table <ref>. Moreover, a pair of anyons with (Q,S) and [Q+3(ℤ+1/2), S+3(ℤ+1/2)] differ by the fusion of an odd multiples of electrons in the Halperin-330 state. 





  §.§.§ Condensation of Abelian bosons


Following the above discussion, we look for possible values of Q, ω_η, and S such that both anyons with operators exp(iQϕ_r)exp(iω_ηϕ_l) and ηexp(iSϕ_n) are bosonic. The conformal spin for exp(iQϕ_r)exp(iω_ηϕ_l) is

    s(Q,ω_η)=Q^2-ω_η^2/3.

Therefore, a possible solution for s(Q,ω_η)=0 (or alternatively, the null vector for the K matrix describing the charge sector) is (Q,ω_η)=(1,1). Hence, we first condense the following four bosons:

    b_0
       =e^im_0(ϕ_r+ϕ_l),
    
    
    b_1
       =ψ_1 e^3iϕ_n/2e^i(m_1+1/2)(ϕ_r+ϕ_l),
    
    
    b_2
       =ψ_2 e^im_2(ϕ_r+ϕ_l),
    
    
    b_3
       =ψ_3 e^3iϕ_n/2e^i(m_3+1/2)(ϕ_r+ϕ_l).

Here, all m_i∈ℤ. It is easy to check that all of the above bosons satisfy the conditions (i) and (ii) in the previous discussion. This verifies that all of them can be condensed simultaneously <cit.>. It is worthwhile for clarifying that being a boson does not mean that it must be condensable. For example, the condensation of non-Abelian bosons can be obstructed by a no-go theorem <cit.>. Also, a self-dual boson having Frobenius-Schur indicator ϰ=-1 is not condensable <cit.>. Nevertheless, none of these limitations applies here.

Since a deconfined anyon must have trivial mutual statistics with all the condensed bosons, we first put m_2=0 to eliminate a large set of possibilities. The corresponding monodromy is given by

    M^ηψ_2_η×ψ_2
    =exp[-(2π i)(h_η×ψ_2-h_η-h_ψ_2)].

Hence, all anyons with η={σ_+, σ_-, χ_+, χ_-} are confined. This result was used by Barkeshli and Wen in showing that the phase transition from the RR_4 state to the Halperin-330 state could be described by the anyon condensation of ψ_2 <cit.>. By setting m_2≠ 0, the monodromy between Y=ηexp(iSϕ_n)exp(iQϕ_r)exp(iω_ηϕ_l) and b_2 is given by

    M^Yb_2_Y× b_2
    =exp[4m_2 π i/3(Q-ω_η)] , 
    m_2∈ℤ.

Here, η={I, ψ_1, ψ_2, ψ_3, ϵ, ρ}. Thus, the set of possible deconfined anyons is further reduced to
e^iω_η(ϕ_l+ϕ_r)
e^3iγϕ_r/2
e^iSϕ_n×{I, ψ_1, ψ_2, ψ_3, ϵ, ρ}, with γ∈ℤ. This set of anyons have trivial mutual statistics with b_0 also. By requiring them to have trivial mutual statistics with b_1 and b_3, it further reduces the possible set of deconfined anyons to four apparently different classes. For the first class, we have

    𝒯_1
       =e^im(ϕ_l+ϕ_r)
    e^3i(p+1/2)ϕ_r
    e^i(q+1/2)ϕ_n×{I, ψ_2, ϵ}
       ∼ e^im(ϕ_l+ϕ_r)[e^3i(ϕ_r+ϕ_n)/2]
    {𝒱_0, 𝒱_2, 𝒱_4}×{I, ψ_2, ϵ}
       ∼ e^im(ϕ_l+ϕ_r)Ψ_e
    {𝒱_0, 𝒱_2, 𝒱_4}×{I, ψ_2, ϵ}
       ∼Ψ_e
    {𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}.

All m, p, and q in the above calculation are integers. It is recalled that 
Ψ_e=exp(3iϕ_↑)=exp[3i(ϕ_r+ϕ_n)/2] is the electron in the Halperin-330 state. In the second line, we used the fact that ϕ_n is a compactified boson in the U(1)_6 CFT, and labeled the corresponding vertex operators by the symbols in Table <ref>. Moreover, we used the fact that exp(3ipϕ_r) is a trivial boson to make the identification denoted by ∼. In the last line, the identification is made by fusing the set of anyons in the third line with the condensed boson b_2 with m_2=-m. 

Using similar procedures, we determine the second class of deconfined anyons as

    𝒯_2
       =e^im(ϕ_l+ϕ_r)
    e^3ipϕ_r
    e^iqϕ_n×{I, ψ_2, ϵ}
       ∼{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}.

By setting η={ψ_1, ψ_3, ρ}, we have the remaining two classes of deconfined anyons,

    𝒯_3
       =e^i(m+1/2)(ϕ_l+ϕ_r)
    e^3ipϕ_r
    e^i(q+1/2)ϕ_n×{ψ_1, ψ_3, ρ}
       ∼ e ^i(ϕ_l+ϕ_r)/2{𝒱_1, 𝒱_3, 𝒱_5}×{ψ_1, ψ_3, ρ},

and

    𝒯_4
       =e^i(m+1/2)(ϕ_l+ϕ_r)
    e^3i(p+1/2)ϕ_r
    e^iqϕ_n×{ψ_1, ψ_3, ρ}
       ∼Ψ_e  e ^i(ϕ_l+ϕ_r)/2{𝒱_1, 𝒱_3, 𝒱_5}×{ψ_1, ψ_3, ρ}.

Meanwhile, they are actually equivalent to 𝒯_2 and 𝒯_1, respectively. This is observed by fusing 𝒯_3 and 𝒯_4 with either b_1 or b_3 to obtain

    𝒯_3    ∼{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}∼𝒯_2,
    
    𝒯_4    ∼Ψ_e{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}∼𝒯_1.

Therefore, the above four classes of deconfined anyons can be combined into

    𝒯   ={1, Ψ_e}×{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}
       ={1, Ψ_e}×𝒯_B.

Since 𝒱_2 and 𝒱_4 cannot be obtained from any fusion between the bosons in the set {b_0, b_1, b_2, b_3}, this set is not a Lagrangian subset. In the second line of Eq. (<ref>), 
𝒯_B is a bosonic order with 𝒱_0 I being the trivial boson. The separable form in Eq. (<ref>) is guaranteed for any Abelian fermionic order <cit.>. The Abelianity of 𝒯 becomes transparent after splitting the non-Abelian anyons in 𝒯_B. Notice that (Ψ_e)^2=exp(3iϕ_r)exp(3iϕ_n)∼ 1. 



  §.§.§ Splitting of non-Abelian anyons and the ℤ_3 toric code


Since some of the anyons in the original phase have been identified as the trivial vacuum after the anyon condensation, the unconfined non-Abelian anyons in 𝒯 may need to split. Consider the fusion:

    ϵ×ϵ
    =I+ ψ_2+ϵ∼ 2I + ϵ.

Due to the identification ψ_2∼ I, the vacuum appears twice. Consequently,
ϵ needs to split into two Abelian anyons, ϵ=ϵ_1+ϵ_2. This matches the quantum dimension as 2=1+1. Furthermore, the splitting and the fusion rules are consistent only if

    ϵ_1×ϵ_1=ϵ_2,
    
       ϵ_2×ϵ_2=ϵ_1,
    
       ϵ_1×ϵ_2
    =ϵ_2×ϵ_1=I.

Then, the fusion rule ρ×ρ=I+ψ_2+ϵ implies that the non-Abelian anyon ρ also needs to split, ρ=ρ_1+ρ_2. After the splitting, a possible (but not unique) consistent set of fusion rules are

    ρ_1×ρ_1=ϵ_1 , ρ_2×ρ_2=ϵ_2 , ρ_1×ρ_2=I,
    
       ϵ_1×ψ_1=ρ_2
     , ϵ_2×ψ_1=ρ_1,
    
       ρ_1×ψ_1=ϵ_2
     , ρ_2×ψ_1=ϵ_1,
    
       ρ_1×ϵ_1=ψ_1,
     , ρ_1×ϵ_2=ρ_2,
    
       ρ_2×ϵ_1=ρ_1,
     , ρ_2×ϵ_2=ψ_3.

Note that ψ_3∼ψ_1 in the condensed phase since they differ by a fusion with ψ_2 (i.e, b_2 with m_2=0). Therefore, the fusion rules involving ψ_3 are the same as those involving ψ_1. By making the change(s), ϵ_1↔ϵ_2, or/and ρ_1↔ρ_2, one can still obtain a consistent set of fusion rules. This is reasonable as it is impossible to uniquely define or fix the anyons resulting from the splitting. On the other hand, they must be inequivalent. Without loss of generality, we will stick with the set of fusion rules in Eqs. (<ref>) – (<ref>) in the following discussion.

The above discussion shows that 𝒯_B actually consists of nine different Abelian anyons, so 𝒯 is an Abelian fermionic topological order. Then, what is the topological order 𝒯? By labeling the nine anyons in 𝒯_B as shown in Table <ref>, it is observed that these anyons take the form of e^p m^q, where p, q∈ℤ. Moreover, one has e^3=m^3=𝕀. Hence,𝒯_B is the ℤ_3 toric code. This topological order is a generalization of the more famous ℤ_2 toric code that only has four anyons, {1, e, m, f≡ em} <cit.>. Since it is the topological order built from (mathematically, the modular tensor category constructed over) the quantum double model with the finite group ℤ_3 <cit.>, the ℤ_3 toric code is also denoted as 𝔇(ℤ_3) <cit.>. It was suggested that this special topological order can be realized by proximitizing a bilayer system of electrons and holes in separate Laughlin states with respective filling factors ± 1/3 to a superconductor <cit.>. It was also pointed out that the more general ℤ_p toric code appears as the symmetry-enriched neutral sector of non-diagonal quantum Hall states <cit.>. Here, we have shown that the interface between the Halperin-330 and RR_4 states with suitable anyon condensation also leads to the ℤ_3 toric code. The physical origin that triggers such an anyon condensation will be discussed in Sec. <ref>. On the application side, it was suggested that the ℤ_3 toric code could be used in implementing universal topological quantum computation <cit.>. To summarize, the condensation of b_0, b_1, b_2, and b_3 leads to the condensed phase with deconfined anyons,

    𝒯={1, Ψ_e}×𝔇(ℤ_3).

Note that 𝒯 does not describe a fully gapped interface. This is because 
𝔇(ℤ_3) is not a topologically trivial order. Alternatively, the edge of 𝔇(ℤ_3) is gappable but the edge remains gapless unless a further anyon condensation occurs, which we are going to discuss below.






  §.§.§ Two different phases of gapped interfaces


As we have shown, the set {b_0, b_1, b_2, b_3} is not a Lagrangian subset, and the condensation of this set of bosons does not lead to a fully gapped interface. On the other hand, a fully gapped interface can be achieved by further condensing some bosonic particles in 𝔇(ℤ_3). By recycling the results in Refs. <cit.>, one can immediately conclude that there are two different types of fully gapped interfaces between the Halperin-330 FQH state and the RR_4 FQH state.

The first possible kind of fully gapped interface is obtained by a further condensation of e and e^2 in 𝔇(ℤ_3). It is straightforward to verify that both of them satisfy conditions (i) and (ii) in the previous discussion. This implies that they can be condensed simultaneously. Moreover, their condensation leads to the confinement of the remaining anyons (except the identity 𝕀) in 
𝔇(ℤ_3). Therefore, we determine the first Lagrangian subgroup <cit.> for 𝒜×ℬ̅ as

    ℒ_e
    ={b_0, b_1, b_2, b_3}×{𝕀, e, e^2}.

The corresponding fully gapped interface is usually known as the e-boundary. Notice that the symbol × actually means picking an element from each set on the right hand side of Eq (<ref>) and then fuse them (see Eq. (<ref>) for example). The product structure of ℒ_e ensures that it is a maximal set of condensable bosons.

Another possible type of fully gapped interface is obtained by condensing m and m^2 in 𝔇(ℤ_3) instead. This leads to the second Lagrangian subgroup,

    ℒ_m
    ={b_0, b_1, b_2, b_3}×{𝕀, m, m^2}.

The corresponding fully gapped interface is known as the m-boundary. For both e- and m-boundaries, the remaining deconfined anyons outside the Lagrangian subgroups are ℱ_0={1, Ψ_e}. This is the trivial fermionic topological order, which indicates that the condensation of anyons in ℒ_e or ℒ_m leads to a fully gapped interface <cit.>.



 §.§ Physical picture of the gapped interface
  


While anyon condensation has provided a systematic and mathematical approach in studying the gapness of the interface, it will be also desirable to understand the gapping of the interface in a more physical picture. We claim that the anyon condensation may originate from the electron and quasiparticle tunneling processes at the interface in two different steps.

At the beginning, the Halperin-330 and Read Rezayi states are two topologically distinct phases. Therefore, only electrons can tunnel across the two different FQH liquids. In general, the electron tunneling process is described by the following Lagrangian density,

    L_el, tun
    =    ξ_1,a(x)(ψ_1 e^3iϕ_l/2e^3iϕ_↑)^a
    
       +ξ_2,a(x)(ψ_1 e^3iϕ_l/2e^3iϕ_↓)^a
    +H.c.
    
    =    ξ_1,a(x)[ψ_1 e^3iϕ_n/2 e^3i(ϕ_l+ϕ_r)/2]^a
    
       +ξ_2,a(x)[ψ_1 e^-3iϕ_n/2 e^3i(ϕ_l+ϕ_r)/2]^a
    +H.c.

Note that H.c. stands for the Hermitian conjugation. We include the exponent a>1 to describe multi-electron tunneling processes. As the electron operators ψ_1 e^3iϕ_l/2 and e^3iϕ_σ (where σ=↑, ↓) enter the same number of times, L_el, tun conserves the total electric charge. Furthermore, the tunneling amplitudes ξ_1,a(x) and ξ_2,a(x) are random functions in x. This is because the electron tunneling generally does not conserve momentum, and disorder needs to be involved. Due to the random nature of the tunneling, some of the processes are actually irrelevant in the renormalization group sense <cit.>. In this situation, we will need to assume the tunneling strength W_a, defined as ξ_a(x)ξ_a(x')=W_aδ(x-x'), is sufficiently large so that the charge modes 3a[ϕ_l(x)+ϕ_r(x)]/2 can still be pinned. Then, this combination of counterpropagating charge modes obtains a nonzero expectation value, and indicates the localization (analogous to mode gapping in nonrandom tunneling) of charge modes. For the neutral sector, it is expceted that the combination ψ_1 e^± 3iϕ_n/2 will also be gapped by ℒ_el, tun in the strong coupling regime. By defining (ψ_1)^2∼ψ_2, (ψ_1)^3∼ψ_3, and (ψ_1)^4∼ I, the tunneling processes described by ℒ_el, tun lead to the condensation of b_0, b_1, b_2, and b_3 in Eqs. (<ref>) – (<ref>). 

After the above condensation or charge-mode localization, the interface remains gapless. Meanwhile, quasiparticles with charges e/3 and 2e/3 can tunnel across this “new" gapless interface. Let us specifically consider the tunneling of charge e/3 anyon with the topological sector ρ in the Read-Rezayi liquid. This process can be described by the following Lagrangian density,

    L_e/3, tun
    
    =    ζ_1(x)(ρ e^iϕ_l/2e^iϕ_↑)
    +ζ_2(x)(ρ e^iϕ_l/2e^iϕ_↓)
    +H.c.
    
    =    ζ_1(x)[ρ𝒱_1 e^i(ϕ_l+ϕ_r)/2]
    +ζ_2(x)[ρ𝒱_5 e^i(ϕ_l+ϕ_r)/2]
    +H.c.

Since the charge modes have been localized, we focus on the neutral sector. Now, both combinations ρ𝒱_1 and ρ𝒱_5 have zero conformal spins (see Tables <ref> and <ref> for reference), which indicates that they are bosonic. In principle, they may be condensed or gapped as well. Consider the set of anyons generated from any fusion between ρ𝒱_1 and 
ρ𝒱_5, we have

    {ρ𝒱_1, ρ𝒱_5}×{ρ𝒱_1, ρ𝒱_5}
    ={I, ψ_2, ϵ}×{𝒱_0, 𝒱_2, 𝒱_4}

A further fusion with {ρ𝒱_1, ρ𝒱_5} gives

    {ψ_1, ψ_3, ρ}×{𝒱_1, 𝒱_3, 𝒱_5}.

Hence, the fusion between ρ𝒱_1 and ρ𝒱_5 generates the list of anyons (more precisely, their neutral sectors) in ℒ_e and ℒ_m. In this sense, the condensation described by the Lagrangian subgroup 
ℒ_e or ℒ_m may be understood as originating from the gapping of modes due to e/3 quasiparticle tunneling at the interface. However, this is only possible if we can treat the whole system as a single topological phase. Otherwise, only electrons can tunnel across two topologically distinct phases (for example, between a FQH liquid and a normal metal) <cit.>.



 §.§ Transmutation of pseudospin information



Previously, we found that it is possible (at least in principle) to form a gapped interface between the Halperin-330 state and the Read-Rezayi state at level four. Although both states may describe the FQH state in a bilayer system at total filling factor 2/3, they host different sets of anyons. Specifically, the Halperin-330 state is a two-component state, in which the anyons possess the layer or pseudospin degree of freedom that is absent in the RR_4 state. Meanwhile, the RR_4 state supports non-Abelian anyons that have more complicated fusion rules than the usual Abelian quasiparticles. This observation leads to a natural question: What happens when an Abelian quasiparticle from the Halperin-330 liquid crosses the gapped interface and enters the non-Abelian RR_4 FQH liquid? In the opposite direction, what happens when we drag a non-Abelian quasiparticle from the RR_4 liquid to the Halperin-330 liquid? The answers to these questions lead to the ideas of topological quantum information scrambling and Andreev-like reflection of non-Abelian anyons. 

An anyon a originally in the topological phase 𝒜 can pass through the interface (described by the condensed phase of 𝒜×ℬ̅) and transmutes into an anyon b in ℬ if and only if ab is a deconfined anyon in the condensed phase. This idea was introduced to study the transmutation between anyons in the interface between the Pfaffian and non-Abelian spin-singlet FQH states <cit.>. Now, we employ the same kind of argument to study the transmutation of Abelian anyons in the Halperin-330 FQH liquid when they cross the interface and enter the RR_4 liquid. 



  §.§.§ Transmutation of Abelian charge e/3 anyon


First, an Abelian charge e/3 anyon in the Halperin-330 state can have pseudospin up or pseudospin down. We denote them as (e/3, ↑) and (e/3, ↓), respectively. Their associated CFT operators are

    (e/3, ↑)≡exp(iϕ_↑)
    =𝒱_1exp(iϕ_r/2),
    
    
    (e/3, ↓)≡exp(iϕ_↓)
    =𝒱_5exp(iϕ_r/2).

When they cross the fully gapped 330-RR_4 interface, their transmutation depends on the type of the gapped interface (i.e., whether it is an e- or m-boundary). 

Suppose the fully gapped interface is described by the e-boundary. Then, the deconfined anyons are those in ℒ_e in Eq. (<ref>). In particular, we have the following pair of deconfined anyons:

    e^2× b_1
       = [ρ_1 exp(iϕ_l/2)] [𝒱_1exp(iϕ_r/2)],
    
    
    e× b_1
       = [ρ_2 exp(iϕ_l/2)] [𝒱_5exp(iϕ_r/2)].

These are the only two deconfined anyons that involve (e/3,↑) and (e/3,↓). Since e^2× b_1 and e× b_1 are condensed bosons in 
ℒ_e, they are identified as the vacuum sector in the condensed phase. Based on the interpretation in Ref. <cit.>, (e/3, ↑) and (e/3, ↓) will transmute into ρ_1 exp(iϕ_l/2) and ρ_2 exp(iϕ_l/2) in the RR_4 FQH state, respectively. It is important to clarify that these transmuted anyons are now in ℬ, but not ℬ̅. Since ρ_1≠ρ_2, the resulting Abelian anyons are different.



Consider the case when the fully gapped interface is described by the m-boundary that has the Lagrangian subset ℒ_m in Eq. (<ref>). Now, we have the following pair of deconfined anyons:

    m× b_1
       = [ρ_2 exp(iϕ_l/2)] [𝒱_1exp(iϕ_r/2)],
    
    
    m^2× b_1
       = [ρ_1 exp(iϕ_l/2)] [𝒱_5exp(iϕ_r/2)].

Thus, (e/3, ↑) and (e/3, ↓) will transmute respectively into 
ρ_2 exp(iϕ_l/2) and ρ_1 exp(iϕ_l/2). Notice that the result is different from the one in the e-boundary. We summarize the above results in Fig. <ref>.





  §.§.§ Transmutation of Abelian charge 2e/3 anyon


Similarly, there will be a transmutation of the pseudospin information carried by a charge 2e/3 anyon when it crosses the interface and enters the RR_4 FQH liquid. There are three different pseudospin states for the charge 2e/3 anyon in the Halperin-330 FQH liquid. They are spin up, spin zero, and spin down. Physically, they can be viewed as the combination of two spin-up, one spin-up and one spin-down, and two spin-down e/3 anyons. The corresponding CFT operators are

    (2e/3, ↑)
       ≡exp(2iϕ_↑)
    =𝒱_2exp(iϕ_r),
    
    
    (2e/3, 0)
       ≡exp[i(ϕ_↑+ϕ_↓)]
    =𝒱_0exp(iϕ_r).
    
    
    (2e/3, ↓)
       ≡exp(2iϕ_↓)
    =𝒱_4exp(iϕ_r).


Following the previous discussion, we first discuss their transmutation when the fully gapped interface is described by the e-boundary. In this case, one has the following three deconfined anyons,

    e× b_0
       = [ϵ_1 exp(iϕ_l)] [𝒱_2exp(iϕ_r)],
    
    𝕀× b_0
       = [I exp(iϕ_l)] [𝒱_0exp(iϕ_r)],
    
    
    e^2× b_0
       = [ϵ_2 exp(iϕ_l)] [𝒱_4exp(iϕ_r)],

Thus, (2e/3, ↑) transmutes into ϵ_1 exp(iϕ_l); (2e/3, 0) becomes I exp(iϕ_l); (2e/3, ↓) becomes ϵ_2 exp(iϕ_l). For the case of having the m-boundary, one has

    m^2× b_0
       = [ϵ_2 exp(iϕ_l)] [𝒱_2exp(iϕ_r)],
    
    
    m× b_0
       = [ϵ_1 exp(iϕ_l)] [𝒱_4exp(iϕ_r)].

In this scenario, (2e/3, ↑) transmutes into ϵ_2 exp(iϕ_l), whereas (2e/3, ↓) becomes ϵ_1 exp(iϕ_l). The results are illustrated in Fig. <ref>.



The original pseudospin information of a charge e/3 or a charge 2e/3 anyon from the Halperin-330 state is entirely encoded in the neutral spin mode ϕ_n. Clearly, this is a local information which can be accessed via local measurement. More specifically, we know in which layer the Abelian anyon was created. Depending on both the original pseudospin state and the boundary type of the fully gapped interface, the pseudospin information will be transmuted into ρ_1 or ρ_2 (for the charge e/3 anyon), or ϵ_1, ϵ_2, or I (for the charge 2e/3 anyon). Although one knows that the split anyons, say ρ_1 and ρ_2, are different, it is impossible to distinguish between them via any local measurement. In fact, their inequivalence is only manifested topologically in their braiding and fusion rules with other anyons (including themselves) as summarized in Eqs. (<ref>) – (<ref>). In this sense, we claim that the original local pseudospin information is completely scrambled into a form of nonlocal information being stored and protected topologically by the anyons in the RR_4 FQH liquid. No information is stored on the interface. It is reasonable as the interface is fully gapped, and does not support any low-energy gapless excitations there. This feature is different from the situation of having a gapless QH interface. As a concrete example, our recent work <cit.> showed that the pseudospin information for an Abelian charge e/4 anyon from the Halperin-331 state will be scrambled and stored nonlocally by a pair of vortices (one on the interface, and another one in the Pfaffian FQH liquid) when the Abelian anyon crosses the gapless 331-Pfaffian interface. The comparison here demonstrates the dependence of the scrambling mechanism on the gapness of the QH interface.



 §.§ Some remarks


One may realize that the transmuted particle from the charge 2e/3 anyon can be obtained directly by fusing a pair of transmuted particles from the charge e/3 anyons. This is not a coincidence. It is guaranteed from the commutativity between restriction and fusion in anyon condensation <cit.>. This important property also ensures that the total quantum dimension of the split anyons is identical to the quantum dimension of the original non-Abelian anyon. Moreover, the total conformal spin for the transmuted anyon(s) matches the one for the original anyon. In the present case, we have a fully gapepd interface, and the vertex operators describing the charge sector of the deconfined anyons always have zero conformal spins. Thus, the conformal spin for the parafermionic field η from the ℤ_4 CFT must match with the conformal spin for the vertex operator describing the neutral spin mode ϕ_n. This holds in all our results, and further highlights the advantage of separating the charge and neutral sectors as we did in our analysis.

In principle, one can drag more quasiparticles from the Halperin-330 liquid into the RR_4 liquid. Suppose the resulting anyons are well separated from each other (i.e., with a separation much larger than the magnetic length). Then, the braiding between the transmuted anyons will lead to a further scrambling of the original pseudospin information <cit.>. In order to recover the original pseudospin information, one basically needs to “pull" all the anyons back to the Halperin-330 liquid. However, the recovered information will be in a highly entangled form, which again resembles the definition of quantum information scrambling.



§ SIMULATION OF BLACK HOLE EVAPORATION
 

In order to resolve the information paradox in our model, it is necessary to mimic a black hole evaporation in the quantum Hall interface. To simplify our discussion, we only consider the 331-Pfaffian interface here. As we will show, the process recovers the original information carried by the pseudospin degree of freedom naturally. To simplify the discussion, we assume only charge-e/4 Abelian quasiparticles were dragged across the interface before the “evaporation". In general, one can also drag quasiholes and quasiparticles with other charges. Under the above assumption, we argued in Sec. <ref> that neutral anyons σ̃ and non-Abelian charge-e/4 quasiparticles carrying σ are created. This is illustrated in Figs. <ref>(a) and <ref> (b).



The “black hole evaporation" is simulated by shrinking the Pfaffian region. Experimentally, it may be achieved by reducing the interlayer tunneling in the bilayer system <cit.>. When a non-Abelian quasiparticle reaches the shrinking interface, it is released back to the Halperin-331 liquid. This process plays the role of Hawking radiation in the present setup. There are two different mechanisms for the conversion from a non-Abelian quasiparticle into an Abelian quasiparticle. First, the former may encounter an existing σ̃ at the interface. In this case, they recombine and transmute back to an Abelian quasiparticle [see Eqs. (<ref>) and (<ref>)]. This special scenario is shown in Fig. <ref>(c). In order for this recombination to occur, it requires a highly delicate control of the shrinking process. Thus, it is unlikely to recombine all the existing σ̃ and σ (with charge sector skipped) in this way. On the other hand, it is likely that a non-Abelian anyon reaches the interface at a position with no neutral anyon σ̃. Being already outside the Pfaffian liquid, the non-Abelian anyon still needs to transmute into an Abelian quasiparticle. In this case, a pair of σ̃ needs to be created at the interface. One of them combines with the non-Abelian anyon to covert into an Abelian quasiparticle in the Halperin-331 liquid. The remaining one is left at the interface. This mechanism is shown in Fig. <ref>(d). Since the additional pair of σ̃ are created from the vacuum, they must have their fusion channel in the trivial topological sector. Hence, they do not carry additional information. All above processes are unitary, so information should be preserved.



 §.§ Recovery of Page curve


Now, we show that the above “black hole evaporation" recovers all original information and satisfies the Page curve. Here, the first subsystem consists of anyons remaining in the Pfaffian liquid and the interface. Another subsystem consists of Abelian quasiparticles in the Halperin-331 liquid. For simplicity, we call these two subsystems as (I) and (II), respectively. Since we treat the Pfaffian liquid as the black hole and the interface as an event horizon, reduced density matrices at different stages are obtained by partial tracing out (I). At the beginning of the evaporation, (II) is in a vacuum state with no quasiparticles, so its entropy is zero. When the Pfaffian liquid starts to shrink, the entropy of Abelian quasiparticles originating from their entanglement with (I) increases. However, the increase in entanglement entropy will not continue forever. By keep shrinking the Pfaffian region, the number of non-Abelian anyons and the dimension of the corresponding Hilbert space decrease. Hence, the dimensions of Hilbert spaces of (I) and (II) will become comparable and eventually equal to each other. The entanglement entropy reaches its maximum at this moment <cit.>, which is known as the Page time. The Page time depends on the actual shrinking process. After passing the Page time, the entanglement entropy starts to decrease. 

In the most idealistic (yet most unlikely) situation which one can recombine all N non-Abelian anyons with the originally existing σ̃ (exist before the evaporation) at the interface, the Page time occurs when N/2 quasiparticles are released to the 331 region. This feature does not hold in a generic situation. One can actually deduce the average entanglement entropy of (II) in the most idealistic case. We assume the initial state of the total system (before shrinking the Pfaffian region) is a random pure state |Ψ⟩ in the 2^N-dimensional Hilbert space. The entanglement entropy is averaged with respect to the unitary invariant Haar measure on the space of unitary vectors |Ψ⟩ in the 2^N-dimensional Hilbert space <cit.>. Suppose j non-Abelian anyons have been dragged out from the Pfaffian region and transmuted back to Abelian quasiparticles in the Halperin-331 liquid. The corresponding Hilbert space dimensions of (I) and (II) are given by n=2^N-j and m=2^j, respectively. When m≤ n, the conjecture by Page (later proved by Sen <cit.>) suggests that the average entanglement entropy of (II) takes the form <cit.>,

    ⟨ S_ (II)⟩≡ S_m,n
    =(∑_k=n+1^mn1/k)-m-1/2n.

For m>n, one obtains ⟨ S_ (II)⟩ by interchanging m and n in Eq. (<ref>). By plotting ⟨ S_ (II)⟩ versus lnm, one concludes that it is identical to the one in Fig. 1 of Ref. <cit.>.

From our previous discussion, it is very likely that the number of σ̃ anyons increases during the “black hole evaporation". This leads to two consequences. First, it is possible that all non-Abelian anyons in the Pfaffian liquid have been released to the 331 liquid, but some σ̃ anyons still remain at the interface. These anyons are entangled with the Abelian anyons in the Halperin-331 region, so the entanglement entropy is still nonzero. Second, the bound N_L∼ L/ℓ_B can be satisfied easily during the shrinking process. The discussion in Sec. <ref> showed that pairs of Abelian quasiparticles and quasiholes with opposite charges will be released to the 331 region. After eliminating the Pfaffian region completely, the total number of anyons in the 331 region needs not be equal to the number in the original configuration (before sending the anyons in the Pfaffian liquid). Only in the most idealistic situation that we mentioned previously, these two numbers are equal as shown in Fig. <ref>(a) and (e). In general, the final state of the system will have a superposition of different total numbers of anyons in the 331 liquid. It is reasonable since the total charge in the system is still conserved. This idea is illustrated in Fig. <ref>(f). In particular, Fig. <ref>(f) denotes a superposition state of 6+M charge-e/4 quasiparticles and M charge--e/4 quasiholes, where M is a non-negative integer. This kind of superposition state is actually a closer analogue of the actual Hawking radiation emitted from a black hole, which consists of different types of particles or excitations.

Independent of the actual shrinking process, the system must return to a pure state when the Pfaffian region is eliminated completely. Then, the entanglement entropy goes back to zero and resembles the Page curve. The original pseudospin information is recovered but in a highly entangled form. Thus, the paradox in our model is resolved. Our above discussion suggests that the Page curve in the present system should be more complicated than the one in Fig. 1 of Ref. <cit.>.



§ SUMMARY AND DISCUSSION


To conclude, we have identified and resolved an “information paradox" in the 331-Pfaffian and the 330-RR_4 quantum Hall interface respectively <cit.>. In both cases, the paradox originates from an apparent inability to recover the original pseudospin information carried by Abelian quasiparticles after they cross the interface and enter the non-Abelian single-component QH liquid. 

Employing the technique of anyon condensation, we found that each incoming Abelian anyon needs to be transmuted into new types of anyons when the former crosses the interface. Moreover, the mechanism of transmutation and its corresponding scrambling of pseudospin information depend on the nature of quantum Hall interfaces. For the 331-Pfaffian interface, an incoming Abelian anyon is transmuted into a pair of non-Abelian anyons. One of them is created in the Pfaffian liquid, whereas the other is created on the interface. This is expected because the interface remains gapless and being described by a conformal field theory with central charge c=1/2. On the other hand, the 330-RR_4 interface can be fully gapped. In this case, the incoming Abelian anyon is completely transmuted into a non-Abelian anyon in the Read-Rezayi quantum Hall liquid. Abelian anyons with opposite pseudospins would transmute into non-Abelian anyons satisfying different fusion channels.

Regardless of the transmutation mechanism, the original pseudospin information is stored nonlocally in the system. As a result it cannot be recovered by any local measurement. We believe this is a fair analogy to an object falling into a real black hole, in the sense that while the information it carries is not lost, it does become inaccessible to an (outside) observer. This resembles the idea of quantum information scrambling, which is consistent with the modern viewpoint that black holes are fast (perhaps the fastest) information scramblers <cit.>. The matching between the dimensions of Hilbert spaces for the Abelian quasiparticle and non-Abelian anyons further verifies the preservation of information.

In addition, we considered the case when more quasiparticles are dragged across the interface. Focusing on the 331-Pfaffian interface, we argued that the maximum amount of information the system can store in a topologically protected way is bounded by the length of the interface. This feature is reminiscent of a similar bound in black hole set by its area due to the holographic principle and the Bekenstein entropy. Finally, we discussed the simulation of black hole evaporation by shrinking the Pfaffian region which releases quasiparticles back to the 331 liquid. We demonstrate explicitly that the corresponding entanglement entropy would follow the Page curve. Hence, the original pseudospin information is recovered and the “information paradox" in our model is resolved. Note that remnants may be left at the end of evaporation in actual astrophysical black holes <cit.>. This may provide an alternative resolution of the information paradox, which is not addressed in the present work.

It is quite surprising that the seemingly simple quantum Hall interface has a rich analogy with black hole physics. Nevertheless, we need to point out some potential differences between our model and real astrophysical black holes. For a (semi)classical black hole, the horizon is not expected to have an effect on an infalling object (the so-called “no drama scenario”), including the information carried by it. Whether this remains to be the case or not for a fully quantum-mechanical black hole is unclear. A firewall at the horizon is a possible scenario that is currently under investigation and debate <cit.>. In our model, the Abelian quasiparticles must be transmuted when they cross the interface. This is inevitable as the Abelian and non-Abelian quantum Hall liquids support different degrees of freedom. Thus, the interface in our model behaves like a firewall. In our opinion, this interface may be a very simple and accessible “black hole firewall", which deserves more attention. In future work, it will be tempting to examine possible analogy of black hole thermodynamics in quantum Hall interfaces. It is also interesting to examine whether quantum Hall interfaces can provide an easy simulation of (a topological version of) the Hayden-Preskill protocol. 

The black-hole information paradox is arguably one of the most fundamental problems in physics, which involves gravitation, quantum field theory, and in particular, quantum information science. This long-standing problem is currently being actively studied by physicists in many different areas, and from very different perspectives (but so far only theoretically). Its resolution may well pave the way for the quantum theory of gravity, the holy grail of theoretical physics. While there is a lack of complete parallel between our model and certain “believed” processes in actual black holes (especially in the description of black hole evaporation which should be spontaneous), the analogy presented here provides a simple and accessible platform to simulate (i) apparent information loss, (ii) information scrambling, and (iii) information recovery. We believe these are arguably the most important and central concepts in understanding and resolving the original information paradox. Furthermore, our work may open a new research direction of studying how local information can be transmuted and stored nonlocally in an actual black hole. Since the concept of firewall and many other aspects in the paradox are still under intense debate, it is worthwhile to have simple analogies that capture some of the relevant concepts (but not necessarily all details precisely) in the original problem. In addition, our results have established a connection between quantum information, black hole physics and quantum Hall physics, and may bring experimentalists into this exciting research area.  

Last but not least, it is worthwhile to mention that a deep connection between quantum Hall effect and gravitational physics has been revealed in previous work <cit.>, leading to the recent experimental discovery of chiral graviton<cit.>. In particular, Refs. <cit.> have suggested a possible simulation of Hawking-Unruh effect by scattering quasiparticles in quantum Hall systems. We are optimistic that more connections between black hole physics and quantum Hall physics may be discovered in the future.



§ ACKNOWLEDGMENT


We thank N. E. Bonesteel for a useful discussion in our original work. This work was supported by the National Science Foundation Grants No. DMR-1932796 and DMR-2315954, and performed at the National High Magnetic Field Laboratory which is supported by National Science Foundation Cooperative Agreement No. DMR-2128556, and the State of Florida.

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footnote-gap
The gapping of modes is realized when the tunneling amplitude in H_T is a constant instead of a random variable in Eq. (<ref>). Strictly speaking, this kind of nonrandom tunneling is unphysical as there is a momentum mismatch between different modes. However, gapped modes and localized modes discussed in Sec. <ref> have the same physical consequences in the present discussion.

footnote0
In our opinion, this is a big advantage over the Pfaffian-NASS interface <cit.>. In that case, the NASS quantum Hall liquid at ν=4/7 has quasiparticle with the smallest charge of e/7. Hence, a net charge of e/4-e/7=3e/28 is absorbed, which leads to an accumulation of charges at the interface. Furthermore, the difference between filling factors of the two quantum Hall liquids may complicate the gapping of counterpropagating charge modes at the interface. This difference also introduces additional challenges in realizing the interface experimentally.

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footnote
Although Fig. <ref> may imply the interface is a one-dimensional object, it is not. The interface actually penetrates in both quantum Hall liquids and has a finite width (but much shorter than its length), similar to the interface between Pfaffian and anti-Pfaffian liquids <cit.>. Therefore, braiding of anyons can occur at the interface.

Chamon2019
Z.-C. Yang, K. Meichanetzidis, S. Kourtis, and C. Chamon, Scrambling via Braiding of Nonabelions,
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footnote2
In the Pfaffian-NASS interface, one may consider a similar “information paradox" by dragging a charge-e/7 non-Abelian quasiparticle from the NASS liquid to the Pfaffian liquid. Depending on the species, the neutral sector is given by σ_↑ or σ_↓ in the c=6/5 CFT. Both transmute into (σ̅, σ), where σ̅ is a primary field in the ℳ(5,4) minimal model describing the interface <cit.>. Since σ̅ has a quantum dimension d=(1+√(5))/√(2)>√(2), the Hilbert space of anyons at the interface has a larger dimension.

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foot-TQC
It is reminded that one cannot perform universal TQC (or reproduce any unitary operation to arbitrary accuracy by braiding of anyons) by using anyons in the Ising CFT <cit.>. More complicated anyons such as Fibonacci anyons are required for achieving the universality <cit.>.

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 Paper Title: Dzyaloshinskii-Moriya interaction torques and domain wall dynamics in van der Waals heterostructures.




These authors contributed equally to this work

These authors contributed equally to this work

Email: sdong@seu.edu.cn
Key Laboratory of Quantum Materials and Devices of Ministry of Education, School of Physics, Southeast University, Nanjing 211189, China


§ ABSTRACT 
Since the discovery of two-dimensional ferroelectric and ferromagnetic materials, the van der Waals (vdW) heterostructures constructed by ferroelectric and ferromagnetic monolayers have soon become the ideal platforms to achieve converse magnetoelectric functions at the nanoscale, namely to use electric field to control magnetization. In this Letter, by employing density functional theory calculations and dynamic simulations of atomic spin model, we study the key role of interfacial Dzyaloshinshii-Moriya interaction (DMI) in CrI_3-In_2Se_3 vdW heterostructures. Our work demonstrates feasible DMI torques pumped by ferroelectric switching, which can drive current-free and low-energy consumption domain wall motion. Moreover, such interfacial DMI can also significantly enlarge the Walker field in magnetic field-driven domain wall technique.  


Dzyaloshinskii-Moriya interaction torques and domain wall dynamics in van der Waals heterostructures
    Shuai Dong
    August 5, 2024
====================================================================================================



Originated from spin-orbit coupling (SOC), Dzyaloshinshii-Moriya interaction (DMI) plays a vital role in modern spintronics. By coupling neighbor spins in the antisymmetric form H_D=D· (S_i×S_j)  <cit.>, DMI can lead to many noncollinear spin textures, e.g. canting antiferromagnetism, chiral domain walls, as well as skyrmions  <cit.>. It is well known that noncollinear spin textures usually produce nontrivial spin-electron scattering for transportation, which can contribute to topological (quantum) Hall effects and motion of quasi-particles <cit.>, facilitating functional manipulations at the nanoscale. Because the presence of DMI relies on the lack of space-inversion symmetry, DMI widely exists in those non-centrosymmetric magnetic materials.  In particular, helimagnets such as Mn_xFe_1-xSi, Fe_xCo_1-xSi, and FeGe, in which the DMI causes  spiral orders and/or skyrmions, have attracted lots of attentions <cit.>. Besides, the asymmetric interfaces in artificial heterostructures or superlattices, can also produce the interfacial DMI which leads to various emergent phenomena <cit.>.

Reversibility of DMI vector provides a promising route to control the chirality of magnetism. Multiferroics allows the coexistence of more than one ferroic order within a single phase material, which provide a platform to control DMI vector via electric field or mechanical methods <cit.>. For some specific multiferroic systems, e.g. BiFeO_3, due to the locking between oxygen octahedral distortions and dipole moments, the DMI vector can be reversed by ferroelectric switching, leading to the fipping of canting moment <cit.>. By reversing the DMI vector, the topological spin textures or solitons can be manipulated <cit.>. Following this mechanism, Yu et al demonstrated the chirality-dependent skyrmion-skyrmion interactions <cit.>.  

The magnetic chirality also widely exists in domain walls, and thus DMI also plays an essential role in domain wall dynamics <cit.>. For example, the reversal of DMI vector can produce a dissipative transverse motion during the chirality switching, as demonstrated in perovskite multiferroics  <cit.>. Comparing with the scenario based on current-dependent torques <cit.>, such a pure electric field control has the advantage for its lower energy consumption, which has been coined as a new class of “DMI torques” <cit.>. 





Van der Waals (vdW) heterostructures are ideal platforms to design electronic devices with controllable DMI, benefiting from their high-usage interfaces and easy control of stacking modes. As a widely studied system, α-In_2Se_3 monolayer was theoretically predicted to be ferroelectric  <cit.>, which was then fabricated and confirmed in experiments <cit.>. After that, In_2Se_3-based heterostructures have been theoretically studied and designed for nonvolatile electric field manipulations of various physical properties, including magnetism <cit.>, topological states or spin textures <cit.>, band alignments and charge transfer <cit.>. 
Although domain walls have been attracting more and more attentions in two-dimensional systems <cit.>, the DMI control of domain wall dynamics remains less touched.

In this Letter, the DMI torques and domain wall dynamics of CrI_3-In_2Se_3 vdW heterostructures are studied. Our first-principles calculations indicate the interfacial DMI in CrI_3 monolayer can be induced in these heterostructures and the full reversal of DMI vector can be achieved in the In_2Se_3/CrI_3/In_2Se_3 sandwich-like heterostructure, which can generate DMI torques to drive the motion of domain walls. Furthermore, the existing of interfacial DMI can enlarge the critical field of Walker breakdown in magnetic field-driven domain wall motion. Such magnetoelectric vdW heterostructures shed lights on the low energy-consuming domain wall nanoelectronic applications.

CrI_3 monolayer is a well-recognized two-dimensional ferromagnetic sheet with a magnetic easy axis perpendicular to its plane  <cit.>. The effective spin Hamiltonian for CrI_3's honeycomb lattice can be written as:

    H=-J∑_<i,j>S_i·S_j-∑_<i,j>D_ij·(S_i×S_j)-A∑_i(S_i^z)^2,

where S is the normalized spin vector, i/j are site indexs, and <> denotes nearest-neighbors. As depicted in Fig. <ref>(a). The first term is the ferromagnetic exchange (J>0), and the second term describes the interfacial DMI with the coefficient vector D. Due to the inversion symmetry, D=0  for isolated CrI_3 monolayer. However, the symmetry can be broken from the original D_3d to C_3v for CrI_3 monolayer on substrate, producing nonzero DMI vectors satisfy D_μ=e_z×e_μ  (μ=1, 2, 3) as depicted in Fig. <ref>(a). The last term is the magnetocrystalline anisotropy with the easy axis along the z-axis (A>0). 

Due to the presence of DMI, the magnetic domain walls should belong to the Néel-type with specific chirality (i.e. clockwise (CW) or counter-clockwise (CCW)) to reduce the energy cost from DMI, as sketched in Fig. <ref>(b).  In such a honeycomb lattice, although domain walls are possibly propagating along the zig-zag or armchair directions, i.e. the shadow stripes in Fig. <ref>(a), they can be proved being equivalent in the magnetic dynamics <cit.>. Then in the continuous limit, the model Hamiltonian for Néel-type domain walls propagating along the x-axis can be expressed as:

    H=∫[J/2(∇S)^2-De_y·(S×∇S)-4A/3(S^z)^2]dx,

where D is the amplitude of DMI vector. Therefore, the domain wall can be described as S^x=sech(x/Δ) with the characteristic width Δ=√(3J/8A)  <cit.>. It's worthy to note that the effective width of domain walls in the honeycomb lattice should be πΔ a_0 (a_0 is the Cr-Cr distance as defined in Fig. <ref>(a)).



To obtain the effective coefficients from real heterostructures, the first-principles calculations based on density functional theory (DFT) are performed, and technical details of calculations can be found in Supplementary Materials (SM) <cit.>. Both CrI_3 and In_2Se_3 monolayers are hexagonal lattices, as shown in Fig. <ref>(a-b), and their lattice constants are well-matched by following the (1×1)/(√(3)×√(3)) stacking mode with less than 0.5% mismatch. The CrI_3/In_2Se_3 bilayer and In_2Se_3/CrI_3/In_2Se_3 trilayer are considered. Here, the energetically most favorable stacking modes are displayed in Fig. <ref>(c-f), while other stacking modes are discussed in SM <cit.>. The CrI_3/In_2Se_3 bilayer is polar despite the polarization direction of In_2Se_3. The In_2Se_3/CrI_3/In_2Se_3 trilayer can be polar or nonpolar, depending on the combination of polarizations of two In_2Se_3 layers, as shown in Fig. <ref>(e-f). For bilayer heterostructures CrI_3/↑ and CrI_3/↓ (here ↑ or ↓ denotes the polarization direction in In_2Se_3 layer), the energy is calculated as the function of interlayer distance L. As shown in Fig. <ref>(g), their optimal distances L_↑ and L_↓ are only slightly different. Similar conclusion is also applicable to the trilayer heterostructures, in which the optimal L's slightly depend on the polarization of In_2Se_3 layers.

As aforementioned, the DMI can be induced by the proximity effects at interfaces. Microscopically, DMI can originate from multiple factors, including the lattice distortion, orbital hybridization, electrostatic field, and charge transfer. Here, although the lattice distortions between Cr-I-Cr bonds are rather tiny after optimization, the charge density is significantly different between the upper I and lower I surfaces, due to the electrostatic field from In_2Se_3, as shown in Fig. <ref>(a). To calculate the DMI vector, two noncollinear magnetic orders with identical exchange energy but opposite chiralities as depicted in Fig. <ref>(b), are calculated with SOC. The energy difference between these two states is only contributed by the DMI according to D=(E_CCW-E_CW)/4.  


All calculated coefficients including J, D, and A are summarized in Fig. <ref>(c-d). On one hand, comparing with monolayer CrI_3, the ferromagnetic exchange J is strengthened by 15%-28% in these heterostructures, while the magnetocrystalline anisotropy A is reduced by 0-26%. Thus the ferromagnetism of CrI_3 should remain robust. On the other hand, nonzero D's appear only in those polar heterostructures. For bilayers, the amplitude of DMI's are estimated as 0.16 meV and 0.3 meV for CrI_3/↑ and CrI_3/↓ respectively. Interestingly, the sign of DMI does not change upon the ferroelectric switching, implying that the single interface itself contributes to 0.23 meV while the reversable DMI contributes to 0.07 meV. For trilayer, the DMI is completely reversed from 0.08 meV in ↑/CrI_3/↑ to -0.08 meV in ↓/CrI_3/↓, while it is zero in ↑/CrI_3/↓ or ↓/CrI_3/↑ as expected from the inversion symmetry.

To study the spin dynamics of domain walls in the CrI_3 layer, we employed the atomic simulations by solving Landau-Lifshitz-Gilbert (LLG) equation <cit.>: 

    ∂S/∂ t=γ/μ_s(S×∂ H/∂S)+α(S×∂S/∂ t),

where γ=gμ_ B/ħ is the gyromagnetic ratio. μ_s=3 μ_ B is the atomic magnetic moment for Cr^3+. The last term is for Gilbert damping with coefficient α, which is typically determined by electron spin resonance in experiments. A stripy honeycomb lattice N=600×4 is adopted, with periodic boundary conditions along the y-direction and open boundary conditions along the x-direction. The two x-ends are fixed as spin-up and spin-down. The fourth-order Runge-Kutta method is used to solve the LLG equation to obtain the time-dependent evolution of spins.  



In the trilayer heterostructure, the DMI vector can be reversed by flipping the polarization, i.e. from +0.08 meV to -0.08 meV, which is anticipated to pump the DMI torques for spin dynamics <cit.>. Here, we study an isolated domain wall at the honeycomb lattice, starting from a sharp boundary between the spin-up domain and spin-down domain. As shown in Fig. <ref>(a), due to the DMI (D=0.08 meV) in ↑/CrI_3/↑, the final stable state is a CW Néel-type domain wall. After the polarization flipping to ↓/CrI_3/↓, the DMI energy will produce an effective field acting on the domain wall, and thus pumps a DMI torque as:

    Γ_D=-2Dγ/μ_s(S·e_y)∇S,

which will reverse the chirality of domain wall and induce a transverse motion as well. As shown in Fig. <ref>(b), with a large damping coefficient α=0.2, the domain wall moves by ∼15 nm along the x-direction within 0.8 ns, accompanying its chirality reversal to CCW. This dynamic process with a smaller damping coefficient α=0.1 is also calculated for comparison. As shown in Fig. <ref>(c), the transverse motion distance d reaches ∼30 nm within ∼1.5 ns. The motion of domain wall converges to stop gradually, thus it is not precise to define the motion time and the final distance   d_s. In fact, for the honeycomb lattice, the DMI torque-driven domain wall motion can be well described by:

    d=a_0Δ/αarctan[sinh(t/Q)],

where Q=8Δ(α+α^-1)μ_s/(3πγ D) denotes the characteristic time and the maximal moving distance is d_s=a_0Δπ/α. More details about the derivatives can be found in Supporting Information, and this kind of domain wall motion is similar to the "rolling-downhill"-type motion discovered in multiferroic perovskites <cit.>. All the simulation results are in perfect agreement with the theory, in which the moving distance is proportional to α^-1 and independent on the strength of DMI. Here, we can define the effective time for the motion as ∼2π Q, thus the average velocity can be calculated by v̅=3a_0πγ D/[16(α^2+1)μ_s]. The average velocity v̅ increases with decreasing α and its maximal limitation is v_m=3a_0πγ D/16μ_s when α→0, in consistent with the simulation results as shown in Fig. <ref>(d). For the In_2Se_3/CrI_3/In_2Se_3 heterostructure, the maximal average velocity is estimated as ∼20 m/s, which is close to the typical velocity of magnetic field-driven ferromagnetic domain walls below 50 mT <cit.>. 



For completeness of domain wall motion, we also investigate the magnetic field-driven domain wall motion with different polarizations. Considering a magnetic field h along the z-direction, the Zeeman energy term can be expressed as H=-hμ_s∑_ie_z·S_i, which gives rise to a magnetic field torque as:

    Γ_h=γ he_z×S.

The snapshots of domain wall dynamics in CrI_3/↓ are displayed in Fig. <ref>(a-b), where the magnetic field torque together with the Gilbert damping produce a dissipative torque to shift the domain wall and reduce the Zeeman energy. In this process, there is a critical field, called Walker field h_W <cit.>. Below h_W, the dynamic process is mainly the steady motion, while it becomes a processional motion above h_W (Fig. <ref>(d)). Here, the time-dependent domain wall motions under various magnetic fields are plotted in Fig. <ref>(c-d), which indicates h_W∼152 mT.

It is known that this h_W orignates from the in-plane magnetic anisotropy. However, for CrI_3 monolayer itself, the fully occupied t_ 2g orbitals at Cr ion together with the D_3d point symmetry of honeycomb lattice, preserve the in-plane symmetry and thus the in-plane magnetic anisotropy is almost zero <cit.>, as confirmed in our calculations. Therefore, the magnetic in-plane anisotropy comes from the interfacial DMI. With the coarse-graining approximation, the Walker field can be derived as:

    h_W=3πα D/8Δμ_s,

which agrees with above numerical value very well as shown in Fig. <ref>(d).

Without this interfacial DMI, e.g. in CrI_3 monolayer, the domain wall motion can only be the processional motion, which leads to a velocity v_p=a_0αγΔ h/(1+α^2). In heterostructures, below h_W, the domain wall motion reaches a final steady velocity v_s after a very short speeding duration (∼0.04 ns). This v_s=a_0γΔ h/α is much higher than the v_p. The numerical results of average velocity v̅ are plotted in Fig. <ref>(d) as a function of magnetic field for different heterostructures. Compared with those D=0 cases (CrI_3 monolayer and non-polar ↑/CrI_3/↓), all polar heterostructures own larger h_W and higher v̅, as expected. In particular, below h_W, the velocity of domain wall can be up to ∼140 m/s for CrI_3/↓, which is considerably large and similar to the gate-voltage effect in those magnetic film experiments <cit.>. 

In conclusion, by employing first-principles calculations and atomic spin dynamics simulations, we have studied the mechanism of interfacial DMI and domain wall dynamics in CrI_3-In_2Se_3 heterostructures. We have demonstrated the DMI torques can be pumped into In_2Se_3/CrI_3/In_2Se_3 trilayer heterostructures by polarization flipping, which contributes an efficient domain wall motion without either magnetic field or current. Furthermore, the existing and tunable DMI within heterostructures also plays an important role in improving the critical Walker field for magnetic field-driven domain wall motion, exhibiting the tunable and nonvolatile bias effect. In addition, our results are completely portable to other vdW heterostructures, which can be described by the effective spin model as well. Our results have performed rich manipulations of magnetic domain walls, which can be achieved in magnetoelectric vdW heterostructures.



This work was supported by the National Natural Science Foundation of China (Grant Nos. 12325401 & 12274069), China Scholarship Council, Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX21_0079), and the Big Data Computing Center of Southeast University. 






 Paper Title: Numerical Study of Quantum Oscillations of the Quasiparticle Lifetime: Impurity Spectroscopy, Novel Electric Field and Strain Effects.







Technical University of Munich, TUM School of Natural Sciences, Physics Department, TQM, 85748 Garching, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany

Technical University of Munich, TUM School of Natural Sciences, Physics Department, TQM, 85748 Garching, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany
Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
 

		
§ ABSTRACT 
Quantum oscillation (QOs) measurements constitute one of the most powerful methods for determining the Fermi surface (FS) of metals, exploiting the famous Onsager relation between the FS area and the QO frequency. The recent observation of non-Onsager QOs with a frequency set by the difference of two FS orbits in a bulk three-dimensional metal can be understood as the QO of the quasiparticle lifetime (QPL) due to interorbital scattering [Huber, Leeb, et al., Nature 621 (2023)]. QPL oscillations (QPLOs) generalize magneto-intersubband oscillations (MISOs) known from coupled two-dimensional metals. They may provide a novel tool for extracting otherwise hard-to-measure intra- versus interband scattering times of quasiparticles. 
Here, we provide a numerical lattice study of QPLOs comparing transport and thermodynamic observables. We explore the effect of different imperfections like general impurities, Hall effect-induced electric fields, various forms of strain from bending, and magnetic field inhomogeneities. We confirm the basic phenomenology of QPLOs as predicted in analytical calculations and identify additional novel, non-perturbative features. Remarkably, we find that some imperfections can stabilize, or even enhance, non-Onsager QPLOs in contrast to standard QO frequencies. We discuss various avenues for identifying QPLOs in experiments and how to use their dependence on imperfections to extract material properties.

	
Numerical Study of Quantum Oscillations of the Quasiparticle Lifetime: 
 Impurity Spectroscopy, Novel Electric Field and Strain Effects
    Johannes Knolle
    August 5, 2024
=======================================================================================================================================





§ INTRODUCTION

Quantum oscillation (QO) measurements are an exceptionally sensitive tool for measuring Fermi surface (FS) geometries as well as the strength of interaction effects via extracting the effective masses m_i from the temperature dependence <cit.>. Often, the observation of a QO frequency serves as a unique and sensitive indicator for the existence of a FS. For example, in underdoped cuprates, QO studies famously confirmed the presence of a closed FS pocket in a magnetic field <cit.>. Similarly, QO experiments in the spin density wave parent phase of iron-based superconducting compounds showed the emergence of small pockets, in contrast with angle-resolved photoemission spectroscopy (ARPES) results at the time <cit.>. Agreement on the electronic structure suggested by QOs was confirmed only much later <cit.>.

The interpretation of QOs, as measured in transport or thermodynamic observables, is based on the famous Onsager relation, which ascribes each QO frequency to a semi-classical FS orbit <cit.>. Recently, unusual frequencies have been reported in the multi-fold semi-metal CoSi <cit.>, which are forbidden in the semi-classical Onsager theory and generalizations thereof <cit.>. In addition, they are at odds with known mechanisms generating non-Onsager QO frequencies like magnetic breakdown, chemical potential oscillations, magnetic interaction, and (Stark) quantum interference <cit.>. The emergence of these forbidden frequencies as combinations of two semiclassical QO frequencies, originating from underlying FSs, has been explained by a non-linear coupling of the FSs via interband impurity scattering <cit.>, dubbed quasiparticle lifetime oscillations (QPLOs) here. Ref. <cit.> argued that QPLOs are generic for bulk metals with multiple extremal orbits or multiple FS pockets. This claim is supported by the identification of many further candidate materials <cit.>, ranging from topological semi-metals to iron-based superconducting parent compounds <cit.>. A key observation of Ref. <cit.> was that QPLOs could be erroneously interpreted as FS orbits, leading to incorrect identification of the electronic structure of a given material. 

The basic ingredient leading to QPLOs is a coupling between two QO orbits, such as interband impurity scattering. The inverse lifetime of the quasiparticles on a given orbit then is a sum of three terms – a constant, the so-called Dingle temperature T_D1, an oscillatory contribution with frequency F_1 from intraorbit scattering, and, crucially, an oscillatory term with frequency F_2 determined by the other orbit 2 with an amplitude proportional to interorbit scattering. The constant Dingle temperature term damps the oscillations with a universal factor, known as the Dingle factor R_D = exp(-2π^2 m T_D/e B), with e the electron charge and B the magnetic field. The third term leads to cross terms of QOs in the conductivity with frequency F_1 and F_2 which appear as the sum and difference of the basis frequencies F_1 and F_2 in the conductivity

    σ_±∝cos(2πF_1 ± F_2/B) R_D1 R_D2 R_T(m_1 ± m_2),

where R_T(m) is the Lifshitz–Kosevich dependence as introduced below in (<ref>). 

Conceptually, QPLOs generalize so-called magneto-intersubband oscillations (MISOs) known in coupled 2D electron gases <cit.> as well as quasi-2D materials <cit.> and observed in several experiments <cit.>. MISOs are nowadays claimed to be understood in great detail <cit.> which is supported by several analytic works <cit.>. The new insight of QPLOs is the generalization to bulk 3D materials <cit.> and a new qualitative picture of their origin <cit.>. Hereinafter, we refer to the phenomenon of additional QO frequencies, which originate in oscillatory contributions of the self-energy due to interorbit scattering, as QPLOs. 



So far, all theories of QPLOs are based on analytical, perturbative calculations, requiring approximations like the self-consistent Born approximation for deriving (<ref>) <cit.> or even more ad-hoc assumptions. This calls for a controlled check of the analytic predictions, which numerical simulations can provide. The numerical evaluation of QOs in the presence of disorder is numerically not challenging, in the sense that the Hilbert space dimension grows only linearly with system size, which allows for an efficient numerical simulation at least for 2D systems, e.g., for transport <cit.>. Besides supporting the phenomena itself, numerical simulations offer the possibility to test predictions of analytic theories in a more controlled manner than experiments can, provding additional insights to advance our understanding of QPLOs. In addition, one can capture effects beyond the limit of perturbation theory.  

In this work, we aim to benchmark analytical theories of QPLOs by full numerical lattice simulations of the Shubnikov–de Haas (SdH) effect, i.e., the conductance as function of the magnetic field. 

In addition, we show how QPLOs respond to different types of impurities compared to standard Onsager QO frequencies, which paves the way for using QPLO measurements to understand otherwise hard to obtain impurity properties of materials. Beyond disorder, we also study the effects of i) electric fields from current-induced charge accumulation, ii) strain engineering from bending nanostructures, and iii) magnetic field inhomogeneities. The unique response of QPLOs to such imperfections may, on the one hand, serve as an additional criterion simplifying their identification and, on the other hand, allow to extract material properties. 




The work is structured as follows. In sec. <ref> we introduce the lattice model and the numerical method.
In the main part, sec. <ref>, we discuss the response of QPLOs to different types of impurities. Next, in sec. <ref>, we analyze the properties of QPLOs like the appearance of higher harmonics, the dependence of QPLO on the non-linear coupling of the FSs, the phase relation to the basis frequencies and the behavior in thermodynamic observables. In the following sections, we discuss the effect of finite temperature, sec. <ref>;  electric fields induced by a Hall effect from a varying current strength, sec. <ref>); crystal strain induced by bending, sec. <ref>; and inhomogeneities in the magnetic field, sec. <ref>. 


















§ TWO-BAND MODEL AND SET-UP


QPLOs appear in two and three-dimensions <cit.>. Here, we focus on a basic 2D two-band model that also captures the relevant physics of quasi-2D materials. The latter are characterized by weakly coupled layers and a nonzero interlayer tunneling leads to a warping of the FS such that 2 extremal orbits at k_z=0 and k_z=π appear. To include these orbits determining the QOs, it is sufficient to model only 2 layers, which would for periodic boundary conditions in z-direction (which are trivial in that case) correspond to two discrete momenta.






Hence, we consider the Hamiltonian

    H_quasi-2D =    -t∑_⟨ i j ⟩, z c^†_i,z c_j,z -t'∑_⟨⟨ i j ⟩⟩, z c^†_i,z c_j,z
       -
    t_⊥/2∑_⟨ z, z' ⟩, i c^†_i,z c_i,z'

on a square lattice, where t (t') are intralayer (next-) nearest-hoppings and t_⊥≪ t is the interlayer hopping, see Fig. <ref> (a) and (c). For simplicity we focus on  which is essentially the 2D limit, shown in Fig. <ref> (b) and (d), and introduce the  orbital (layer) basis Ψ_i=(c_i,0,c_i,1) such that 

    H_0 =    -t∑_⟨ i j ⟩Ψ̌^†_iΨ̌_j  -t' ∑_⟨⟨ i j ⟩⟩Ψ̌^†_iΨ̌_j  - t_⊥∑_jΨ̌^†_jτ^x Ψ̌_j
    
    =    -∑_k⃗Ψ̌^†_k⃗ [2t( cos k_x+cos k_y) 
    
           + 4t' cos k_x cos k_y +t_⊥τ^x ]Ψ̌_k⃗
    
    =    -∑_k⃗ E_±(k⃗) Φ̌^†_k⃗Φ̌_k⃗

where we introduce the bands E_±(k⃗) = -2t cos k_x -2t cos k_y -4t' cos k_x cos k_y ± t_⊥ and the fields in the band basis Ψ̌_k⃗ = U Φ̌_k⃗ which are just bonding and antibonding states 

    U = 1/√(2)[  1  1;  1 -1 ].






 §.§ Peierls substitution

We apply an orbital magnetic field B in the out of plane z-direction, by Peierls substitution, which transforms the hopping t_r⃗,r⃗' from site r⃗ to site r⃗'

    t_r⃗,r⃗'→ t_r⃗,r⃗'exp(- e ∫_r⃗^r⃗'A⃗(l⃗) ̣⃗l).

Here, A⃗ = (- y B,0,0) is the vector potential. For this gauge choice, only the hoppings t_x in x-direction and the diagonal next-nearest neighbor hoppings t_xy acquire a phase. 



Additionally, we consider potential impurities

    H_imp = ∑_j Ψ̌^†_jΛ_j^ΨΨ̌_j

where Λ_i is, in general, a random hermitian matrix. In sec. <ref> we specify different types of the scattering matrix Λ_j^Ψ and study their effect on QOs and QPLOs.





 §.§ Numerical method


Numerically, it is more convenient to work with dimensionless quantities. We set the lattice constant and ħ to 1. The dimensionless magnetic field B/Φ_0 can be interpreted as the amount of flux through a single plaquette measured in units of the flux quantum Φ_0 = 2π/e.


Our goal is a full numerical lattice calculation of the SdH effect. This requires computing the conductance G(B) of the model given by H = H_0 + H_imp as function of the magnetic field B and averaged over impurity samples. 

We evaluate the conductance through the built-in Landauer–Büttiker algorithm <cit.> of the python software package kwant <cit.>. For this, we construct an open lattice, dubbed scattering region, of L× L sites with 2 orbitals per site and implement H including the Peierls phases. Additionally, we attach two leads of width L on the right and left side of the scattering region, see Fig. <ref> (a). The leads realize H_0 without magnetic field and are characterized by translational invariance in x-direction. The Landauer–Büttiker algorithm evaluates the transmission probability T_nm(E) from an eigenstate n with energy E of the left lead to an eigenstate m with energy E of the right lead via an S-matrix approach. The conductance is the sum over all channels at the Fermi energy μ <cit.>

    G = e^2/2π∑_n,m T_nm(μ).


We evaluated the conductance G numerically for 2500 equidistant values of Φ_0/B from 10 to 250 and observed SdH oscillations as a function of Φ_0/B, see Fig. <ref> (b). We then analyzed the Fourier transform in Φ_0/B with standard QO techniques, which include subtraction of a 5th-order polynomial background to extract the oscillatory signal, shown in Fig. <ref> (c), zero padding to increase the point density of the spectrum and windowing with a Blackman–Harris window to decrease spectral leakage. The representative steps are shown in Fig. <ref>. Fourier spectra, see Fig. <ref> (d), are shown with frequencies in units of the BZ area. They show sharp peaks at the contributing QO frequencies.

We also implemented a numerical simulation of the de Haas–van Alphen (dHvA) effect. For this, we use the scattering region without additional leads. We then compute the density of states (DOS) ρ(E) via the kernel polynomial method <cit.>, which is based on a Chebyshev expansion of the spectral density. For sampling, we used 30 randomly chosen vectors such that only the bulk of the system is sampled. Cutting off the edges suppresses finite-size boundary effects. We defined the bulk by the set of all lattice points at least 40 sites away from the edges and used 7000 Chebyshev moments.

The DOS can be used to determine thermodynamic observables. First, the zero temperature thermodynamic potential is computed 

    Ω(μ) = ∫_-∞^μẸρ(E) ( E- μ).

The magnetization M can then be computed via the derivative M = -∂Ω/∂ B. Here, we experienced that a numerical evaluation of the derivative of the data obtained with the kernel polynomial method is unstable. A different thermodynamic observable, the particle number N, is also defined via a derivative N = -∂Ω(μ)/∂μ, however it can be carried out analytically to obtain 

    N(μ) = ∫_-∞^μẸρ(E).

The particle number can hence be evaluated directly from ρ without increasing the noise on the data by an additional numerical derivative.

The oscillating part of particle number and magnetization are related by N ∝ B m/F M for a single frequency F with effective mass m. Hence, the main difference is that the QO amplitude of N reduces for decreasing magnetic field B. This does not pose a problem in numerical simulations since the magnetic fields are relatively large. 



§ IMPURITY SPECTROSCOPY WITH QPLOS


Within our basic model, different types of impurities can be classified according to their orbital/layer structure. We first discuss various types of impurity vertices and then show how they lead to qualitatively different effects. 



 §.§ Impurity models




  §.§.§ Identical impurities

In the simplest impurity model, the vertex at each point is taken to be proportional to the identity Λ_i^Ψ = λ_i τ^0 and the prefactor is drawn from λ_i ∈ [-Λ^0/2,Λ^0/2] randomly and uniformly. The physical picture is that each layer is an identical copy of the other layer, which may arise from systematic defects when growing the material or atoms sitting between the layers, hence having the same effect on both. A transformation to the band basis Λ_i^Φ = U^†Λ^Ψ_i U shows that the scattering vertex remains diagonal in the band basis, therefore leading only to pure intraband scattering.



  §.§.§ Opposite identical impurities

The impurities at each lattice point are again perfectly correlated but with opposite signs in each layer, Λ_i^Ψ = λ_i τ^z. Their strength is drawn from λ_i ∈ [-Λ^z/2,Λ^z/2] randomly and uniformly. The corresponding physical picture could be dipoles located between the layers. A transformation to the band basis Λ^Φ = λ_i τ^x shows that it gives rise to a pure interband coupling term.



  §.§.§ Independent impurities

Both impurity models above are somewhat artificial. Therefore, we also consider impurities randomly distributed in each layer, i.e.,

    Λ_i^Ψ   = [ λ_i   0;   0 η_i ]

where λ_i,η_i ∈ [-Λ/2,Λ/2] are distributed randomly, uniformly, and independently. Crucially, the vertex is a superposition Λ_i = (λ_i+η_i)/2 τ^0 + (λ_i-η_i)/2 τ^z of the identical and the opposite identical impurities. The probability distribution of (λ_i ±η_i)/2 is the convolution of the 2 uniform (rectangular) probability distributions and hence a triangular probability distribution. Transforming this to the band basis shows that independent impurities in the layers lead to equally strong intra- and interband scattering, as expected from the form of Λ_i^Ψ.



  §.§.§ Random interlayer coupling

Finally, there are interlayer impurities Λ_i^Ψ = λ_i τ^x or Λ_i^Ψ = x_i τ^y
where λ_i ∈ [-Λ/2,Λ/2] randomly and uniformly. Unlike the other disorder models, random interlayer coupling constitutes a form of bond disorder and can hence be complex. A varying distance between the layers will only influence the real part of the hoppings. A complex disordered part of the hopping may, for example, arise from locally trapped fluxes or random spin-orbit coupling (i.e., heavy impurities). 

Real interlayer impurities, i.e. Λ_i^Ψ = λ_i τ^x, transform into the band basis as Λ^Φ∝τ^z, hence, only leading to intraband contributions. In contrast, complex interlayer impurities, i.e., Λ_i^Ψ = λ_i τ^y, contribute only to interband scattering because the scattering vertex transforms to Λ^Φ∝τ^y. 










 §.§ SdH oscillations

Next, we show numerical results of the QO spectra for different impurity types.
QPLOs only emerge for an effective coupling of the semiclassical orbits <cit.>. In sec. <ref>, we argue that in our model, only opposite identical and independent impurities lead to an effective interpocket coupling, whereas identical or interlayer impurities only contribute to intrapocket scattering processes. Therefore, we expect a strong dependence of the QO spectra on the types of impurities. 

We computed the SdH effect numerically for the different impurity models, following sec. <ref>. Fig. <ref> demonstrates that the main frequencies F_1, F_2 consistently appear for any model whereas the combination frequencies F_2 ± F_1 only appear for sufficiently strong interpocket coupling, i.e., for opposite identical, independent and complex interlayer impurities. Indeed, the presence of a difference frequency peak from QPLOs may serve as tool discriminating between different forms of impurities. 

In the remainder of this work, we will focus on opposite identical and identical impurities, where the strength of interband scattering is Λ^z and the strength of intraband scattering Λ^0.



§ ANALYSIS OF QPLOS




 §.§ Higher harmonics

QOs are not perfectly harmonic, i.e., their shape is not a perfect cosine. In the spectrum, this becomes visible by additional peaks at integer multiples k F, k ∈ℕ of the basis frequency F. The kth harmonic always appears together with k^th-order of the Dingle damping factor R_D^k and is, therefore, typically decreasing in magnitude.

Research on QPLOs has so far focused on 2nd order frequencies, i.e., F_2 ± F_1 which are of second order in the Dingle factor, namely governed by R_D1 R_D2. Higher harmonics also exist for QPLOs as pointed out in Ref. <cit.>. For a model with two orbits, like our model here, higher harmonics of order |k_1|+|k_2| can be found at k_1 F_1 + k_2 F_2, k_i ∈ℤ.

We analyzed the spectra for various different Fermi energies μ. Fig. <ref> (a) shows that a complicated QO spectrum with several peaks arises. We extract the peak positions of the QO spectrum and compare the results to analytic predictions, which we obtain from calculating the area of the semiclassical orbits and their integer and sum combinations. In panel (b), we demonstrate that nearly all peaks may be explained by higher harmonics of QPLOs up to 5th order following the analytic predictions <cit.>.

We note that for isolated parameters, additional peaks appear in the QO spectrum, which higher harmonics cannot explain. However, these do not seem to exist consistently for different Fermi energies. These unexplained peaks could, for example, be artifacts of the Fourier transformation, e.g., side lobes. We do not have an explanation for the peaks at low frequencies and high Fermi energies. Finally, we observe a weak splitting of some combination frequencies towards low Fermi energies, which we discuss in sec. <ref>.





 §.§ dHvA oscillations

The appearance of QOs in thermodynamic quantities, historically typically the magnetization/susceptibility, are known as the dHvA effect. dHvA oscillations are in many aspects similar to the ones appearing in the SdH effect and for standard Onsager QOs most aspects carry over. 

However, it turns out that QPLO are substantially different in transport compared to thermodynamic quantities. In analytic calculations for parabolic bands, all difference frequency combinations cancel exactly in the dHvA effect, whereas sum combinations do not <cit.>. This result also holds qualitatively true for relativistic (linear) dispersions, i.e., the amplitudes of difference frequency QOs are strongly suppressed. From the analytic theory, it is, therefore, unclear whether the absence of difference frequency QPLOs in the dHvA effect is generic. 

In Fig. <ref>, we compare quantities from the SdH effect (conductance) and from the dHvA effect (particle number and DOS) for increasing interband scattering rate. Fig. <ref> confirms that difference frequency combinations of QPLOs are strongly suppressed in thermodynamic quantities for generic band structures, if not entirely absent. 

Our high-quality spectrum of the DOS, Fig.<ref> (c), demonstrates that QPLOs cannot be understood on the basis of the DOS alone. Additional knowledge about the lifetime, hence the full spectral function, is required to explain them.









 §.§ Amplitude relation


Next, we study the dependence of the QO amplitudes on the interband scattering Λ^z. In order to rule out the effect of a varying Dingle temperature, i.e., effects induced by the changing broadening of the LLs, we change the intra- Λ^0 and interband scattering Λ^z such that their square sum (Λ^0)^2 + (Λ^z)^2, which is proportional to the Dingle temperature <cit.>, remains constant. 

Even though the Dingle temperature is kept constant, we find an increase of the main frequencies by roughly a factor of 8 when comparing pure intraband coupling to pure interband coupling. Hence, we analyze the relative amplitudes of the higher harmonics with respect to the algebraic mean of the peak height of the two basis frequencies, see Fig. <ref>. The second harmonics 2F_1, 2F_2 remain roughly constant as expected. The combination frequencies originating from QPLO vanish for decreasing interband coupling Λ^z. We fitted the amplitudes of the frequencies F_2± F_1 with (Λ^z)^α from 0 to 0.6 and found α≈ 1. This result is at odds with analytic predictions. QPLOs arise in second order in the Born approximation and should, therefore, show a scaling behavior of α = 2. The stronger scaling points towards the fact that QPLOs are either a lower order phenomenon, which does not hold in analytic calculations, or that higher order scattering contributes more strongly than expected. The stronger scaling indicates that the scattering expansion is non-perturbative, rendering the second-order approximation only qualitatively correct.

A second important observation from Fig. <ref> is the non-monotonic behavior of difference frequency combinations. The amplitudes of F_2-F_1 (2F_1-F_2) go to zero at around Λ^z ≈ 1.5 Λ^0 (Λ^z ≈ 2 Λ^0). We have checked that the zeros do not depend on the absolute strength of Λ_z^2 + Λ_0^2. It is unclear what the origin of these zeros is. 





 §.§ Peak splitting


A detailed analysis of the major peaks of the spectrum shows that, in some cases, they are actually split into two subpeaks, shown in detail in Fig. <ref> (a). The peak splitting is also visible in Fig. <ref> (b) F_2-F_1 at low μ, in Fig. <ref> (a) F_2-F_1 for Λ/t=0.2, in Fig. <ref> (a) F_1, F_2 at e R_H I = 0.06 t and F_1+F_2 at e R_H I = 0.03 t. 

We show in the appendix sec. <ref> that peak splitting is independent of the impurity type. The physical origin is that due to the oscillating quasiparticle lifetime, the QO amplitudes are polynomial functions of the magnetic field <cit.>. At the zeros of the amplitudes, the phase of the QOs jumps. Therefore, the phase itself becomes a function of the magnetic field. In the spectrum, a phase jump shows up as a weak peak splitting. However, peak splitting is of minor experimental relevance, because the zeros are located at values of T_D/B which are mostly not tractable in experiment, see appendix sec. <ref>.









 §.§ Phase relation


The phase of QOs is often of interest because of its relation to the Berry phase <cit.>. However, an absolute determination of the phase is usually difficult in experiment because large fields close to the quantum limit, i.e., B ∼ F, of a QO frequency F are required. In contrast, the relative phase between different frequencies can already be estimated by comparing the raw oscillatory data. 

Ref. <cit.> established, in contradiction with Ref. <cit.>, that there is a fixed phase relation between the phase ϕ_± of the QPLOs and the phases ϕ_i of the basis frequencies 

    ϕ_± = ϕ_2 ±ϕ_1   or  ϕ_± = ϕ_2 ±ϕ_1 + π.

The additional term π accounts for the fact that the amplitudes may change sign, as also discussed in the appendix sec. <ref>.

We extracted the phases ϕ_i, ϕ_± as function of the magnetic field by filtering, see Fig. <ref> (d,e). We also compared the relative phase of the, beating pattern, which is typical for close frequencies, with the filtered sum and difference frequencies, see Fig. <ref> (a-c). For a description of the filtering method, see appendix sec. <ref>. We observe constant phases over large magnetic field ranges. ϕ_- experiences an abrupt phase change of π at ϕ_0/B=100. At the same magnetic field, ϕ_+ shows a ≈ 0.16 π phase change.

Our analysis shows that the phase of QPLOs is indeed the sum/difference of the basis frequencies as described by (<ref>). Our numerical results indicate that the even stronger relation 

    ϕ_- = ϕ_2 - ϕ_1   and  ϕ_+ = ϕ_2 + ϕ_1 + π  or
       ϕ_+ = ϕ_2 + ϕ_1   and  ϕ_- = ϕ_2 - ϕ_1 + π

might hold, i.e., that the amplitudes of difference and sum frequency have always reversed signs. To our knowledge, this has not been discussed in the literature before. We conclude that a relative phase analysis, as done in Fig. <ref>, constitutes a strong indicator to confirm that a frequency in the spectrum can be assigned to QPLOs and not as a standard Onsager frequency.




§ FINITE TEMPERATURE


The finite temperature conductance can be obtained within the Landauer-Büttiker formalism from  

    G_T = 2 π/e^2∫_-∞^∞Ẹ  (-n_F'(E-μ)) ∑_n,m T_nm(E),

a convolution with the derivative of the Fermi distribution function <cit.>

    -n_F'(ϵ) = ^ϵ/T/T(1+^ϵ/T)^2.

For a numerical evaluation of this integral, we use that n_F'(E-μ) is strongly peaked around μ and limit the integration boundaries to ≈μ± 6T to capture more than 99.5% of the spectral weight.


We evaluated the zero temperature conductance G(E) for several energies E sampled logarithmically around the Fermi energy μ=-t. Applying (<ref>) we evaluated G_T at μ=-t for several temperatures and then extracted the temperature dependence of the main peaks of the spectrum of G_T, see Fig. <ref>.



Fig. <ref> shows a perfect Lifshitz–Kosevich dependence with

    R_T(m) = π m T Φ_0/B/sinh(π m T Φ_0/B)

of all QO frequencies. The effective mass  can be computed from the band structure by taking the derivative of the FS area with respect to the Fermi energy, obtaining m_1 = 0.47/Φ_0 and m_2 = 0.57/Φ_0. The theory of QPLO predicts for combination frequencies k_1 F_1 + k_2 F_2 a Lifshitz–Kosevich temperature dependence with an effective mass k_1 m_1 + k_2 m_2. The numerical data in Fig. <ref> confirm this result with high precision. Most importantly, it confirms the presence of the nearly temperature-independent difference frequency for m_1 ≈ m_2. 

We note that small deviations between the numerical results and the expected theory curve appear for increasing temperature. The deviations increase with decreasing mean of I_B, i.e, with increasing magnetic field. We suggest that this is due to our method of extracting the temperature dependence from the Fourier transform, which is to some extent inaccurate because of the dependence of R_T(m) on the magnetic field B. We verified that higher order corrections to the Lifshitz–Kosevich factor are much smaller and are therefore not responsible for discrepancies between numerics and analytics. [Lifshitz–Kosevich dependence assumes that the frequency can be linearly expanded around the chemical potential F(μ+ϵ) = F(μ) + ϵ F'(μ) where F'(μ) determines the effective mass and that higher order corrections are sufficiently small.]




§ ELECTRIC FIELD EFFECTS


Next, we explore the effect of different types of additional perturbations on the QO spectrum. 

In a typical magneto-transport set-up to measure the SdH effect, a current I is sent through the sample to measure the resistivity. Naturally, the set-up is susceptible to a classical Hall effect, where charge accumulates on the edges parallel to the current directions. The strength of the Hall effect can be characterized by a sample-dependent Hall coefficient R_H. The generated perpendicular Hall voltage U_H = R_H I B/Φ_0 leads to an electric field affecting the QO spectrum.

We model the effect of a Hall voltage-induced electric field in our system by a spatially varying Fermi energy μ→μ(y) = μ + e U_H y/L = e R_H I B y/L Φ_0. The leads are kept at constant Fermi energy.



A qualitative picture of the effect of a spatially varying Fermi energy can be obtained by a phase smearing argument <cit.>. Note that formally, the application of a phase smearing argument requires that variations appear only on a length scale longer than the size of a cyclotron orbit. Here, μ(y) varies continuously within the cyclotron orbit. According to phase smearing, the damping of the QO frequencies can be evaluated by averaging over all present Fermi energies

    1/L∫_-L/2^L/2ỵcos(2πF(μ+e U_H y/L)/B)
    
    ≈   ∫_-1/2^1/2ụcos(2πF +m U_H u/B)
    
    
    =   cos(2πF/B) (πm R_H I/Φ_0)


where x = sin x/x.


The numerical results are shown in Fig. <ref>.
We observe a stronger decay of the main frequencies with respect to e R_H I compared to the analytical prediction in Eq. (<ref>). We attribute this to the fact that the cyclotron orbits are not sufficiently large compared to the variation of the Fermi energy. Hence, the assumptions for the derivation of (<ref>) are not met. Nevertheless, the derivation provides an intuitive picture of the qualitative effect. With increasing electric field (or Hall coefficient/current), the QO amplitude should monotonically decrease to zero with a scale which is determined by the effective mass of the QO frequency. If this semiclassical phase smearing picture can be applied similarly to finite temperature, it implies that the amplitude of the difference frequency decreases weakly with m_2-m_1.

The main finding of this section is the anomalous behavior of the difference frequency combinations F_2-F_1 and 2F_1-F_2 in Fig. <ref>. Both amplitudes show a significant non-monotonic increase of more than a factor 6 before dropping again. This behavior is at odds with the present understanding of QPLOs or QOs in general.  






§ STRAIN EFFECTS FROM BENDING


Intense material research in recent years has led to new developments, which allow strain tuning of materials in a controlled manner <cit.>. Specifically, there exist new experimental set-ups for continuously bending materials <cit.>. 

Bending may have different effects depending on the microscopic details of the set-up and material. However, two effects appear quite generically and are straightforward to incorporate in our numerical implementation: (i) Bending leads effectively to a spatially dependent magnetic field by deforming the area of every single plaquette of the crystal differently, which changes the flux piercing through a plaquette. This effect becomes dominant when bending around an axis perpendicular to the magnetic field, e.g., the y-axis, see Fig <ref> (a) and (c). (ii) The distance between the atoms in the crystal and, hence, the overlap of the atomic orbitals changes. This leads to spatially dependent hoppings, see Fig. <ref> (b) and (d). This effect becomes dominant when bending in the plane perpendicular to the magnetic field.

Note that bending inside a material cannot only arise from external application of forces. Crystal imperfections may lead to similar phenomena, which become apparent in domains of slightly different material parameters and grain boundaries.

Here, we consider the effects of bending types (i), (ii) separately in order to understand their effects on QOs in a controlled manner. For simplicity, we focus on lattices with uniform curvature. We quantify the bending strength by the curvature κ_j around the axis-j in the middle of the lattice. The leads are kept at zero curvature.  





 §.§ Bending-induced effective magnetic field

A non-zero value of κ_y leads to a spatially dependent magnetic field B(x) = B(0) cos(κ_y x/L_x) through each plaquette, where B(0) = B is the flux per plaquette of the relaxed model, see Fig. <ref> (a) and (c). Note that the spatial dependence of the magnetic field is weak on the scale of the lattice, hence the vector potential A⃗ inherits the spatial dependence of the magnetic field. 

The naive expectation is that the electrons only feel the average magnetic field acting on the system B = ∫_0^1 B(x) (̣x/L) = B κ_y as it has already been discussed in the standard literature <cit.>. Better estimates can be made by employing a phase smearing argument as (<ref>). However, even for κ_y ≪ 1 only small analytic progress can be made favoring a qualitative picture. The varying magnetic field damps the oscillations. Higher frequencies are damped stronger than lower frequencies.










Our numerical observations contradict this naive expectations, see Fig. <ref>. At first, the amplitudes of semiclassical frequencies and sum combinations of QPLO remain relatively stable with increasing curvature κ_y before eventually decreasing. Most remarkably is the behavior of the amplitudes of the difference frequency combinations F_2-F_1 and 2F_1-F_2. The amplitudes first decrease until κ_y=0.1 but then increase by roughly a factor of 3.









 §.§ Spatially varying hoppings

Next, we study spatially dependent hoppings, see Fig. <ref> (b) and (d). Similar to above, we expect that bending leads to a monotonic decrease of all QO amplitudes. 

We observe a strong, almost monotonic decrease of the QO amplitude of F_1, whereas the amplitude of F_2 remains stable with the tendency to even increase (factor ≈ 1.3), see Fig. <ref>. Similar to above, the difference frequency increases in amplitude by a maximum factor of roughly 2, peaking around κ_z ≈ 0.35. 

We have verified that the stability of F_2 is also present without interband coupling. The stability probably arises due to the quantum Hall limit where transport is mediated by edge states and is not generic for QOs.






§ MAGNETIC FIELD INHOMOGENEITIES


Finally, we study the effect of inhomogeneous magnetic fields. Historically, unwanted field inhomogeneity posed a technical challenge in the early days of QOs, as already noted in Landau's seminal work with the original prediction of QOs  <cit.>. We explore the possibility of magnetic fields which are not constant over the sample. A possible reason for this is a spatially dependent magnetic susceptibility χ(x,y), which changes the magnetic field B → B(1+χ(x,y)). For simplicity we model χ(x,y) = χ_0 (sin(2π x/L_x)+sin(2 π y/L_y)).

An increasing magnetic field inhomogeneity should decrease the QO amplitude, as already discussed in the literature <cit.>. The limit at which the damping becomes relevant is χ_0  ≈ B/π F. From this condition, it is apparent that higher frequencies are affected more strongly or earlier than lower frequencies.

Fig. <ref> shows the QO spectrum and the amplitudes for various χ_0. As expected, the QO amplitudes decrease monotonically. However, yet again, we find an unusual non-monotonic behavior for the difference frequency F_2-F_1. After an initial decay, it peaks at roughly the same value as for zero inhomogeneity before decaying below the detection limit. 








§ SUMMARY AND DISCUSSION

In this work, we studied SdH and dHvA QOs in the presence of different types of impurities, electric fields, bending-induced strain, and magnetic field inhomogeneities. The motivation for this quantitative study comes from the analytical  prediction <cit.> and observation <cit.> of a new mechanism of non-Onsager difference frequency QO dubbed QPLOs, which we benchmarked here in a full numerical lattice implementation. 

We focused on a minimal but generic model featuring two FSs leading to two QO frequencies F_1 and F_2. Different types of impurities are introduced and by transforming the impurity vertex into the band basis, we evaluated the strength of intra- and interband coupling between the two semiclassical QO orbits F_1, F_2. We confirmed numerically that new QPLO frequencies at F_2 ± F_1 appear only in the presence of nonzero interband coupling. We also observed higher harmonics k_1 F_1 + k_2 F_2 of the QPLO up to 5th order. We confirmed that the temperature dependence of k_1 F_1 + k_2 F_2 is an exact Lifshitz–Kosevich with an effective k_1 m_1 + k_2 m_2. In particular, we observed the temperature stability of F_2-F_1, making it the only observable frequency for T>0.025t, a tell-tale prediction for their experimental identification. 

In accordance with the theory for parabolic and linear bands <cit.>, all difference frequency contributions F_2-F_1, 2 F_2 - F_1, ... vanish in the dHvA effect. For parabolic bands, the cancellation of difference frequency contributions in analytics appears somewhat fine-tuned. Therefore, the exact absence in numerics points towards a more general reason that transport and thermodynamics quantities behave qualitatively differently for QPLOs.

In sec. <ref>, we showed that the amplitudes of  scale linearly with interband coupling and not as expected quadratically. This shows that the expansion of the self-energy in the self-consistent Born approximation is non-perturbative, rendering the approximation only qualitatively correct. Additionally, we observe that difference frequencies may cancel even for finite interband couplings for particular parameter values.

In the remainder of our work, we showed that the relative amplitude of QPLOs increases, i.e., it has a qualitatively different behavior from the main Onsager frequencies under several different types of imperfections. The imperfections include electric fields induced by currents through the sample, i.e., a finite Hall effect, bending of the sample such that hoppings or the magnetic field acquire a spatial dependence and fluctuations of the magnetic field. In the case of the Hall effect and spatially fluctuating magnetic field, we find that the absolute amplitude of F_2-F_1 increases substantially. The amplitude dependencies are at odds with existing analytical theories, i.e., they cannot be explained by simple phase smearing arguments, as for finite temperature, or by a combination of the amplitudes of the basis frequencies. We verified that none of the above imperfections induces combination frequencies on its own. The unique response of the frequency F_2-F_1 to these imperfections demonstrates the non-perturbative nature of this QO frequency arising from intriguing interference effects not captured by semi-classical arguments. 

In the absence of interband scattering, we never observe combination frequencies in our exact numerics, even for non-trivial imperfections. Thus, interband scattering is key for inducing these new non-Onsager frequencies. In turn, an observation thereof can be used to perform 'impurity spectroscopy,' e.g., extracting otherwise hard-to-measure properties of intra- versus interband/orbital impurity contributions <cit.>.

Our work has established rigorously the key aspects of difference frequency QPLOs and unearthed a whole range of new phenomena beyond the perturbative theory of Ref. <cit.> pointing to effects beyond the lowest order Born approximation of interband scattering, which will be very worthwhile to explore in the future. In addition, interband scattering may also arise naturally from interactions and collective bosonic excitations like phonons or (para-)magnons. While similar QPLOs are expected in these cases neither analytical nor numerical works exist and are an important, yet challenging, avenue for future research. Again, the hope is that QPLOs may serve as a novel tool for extracting otherwise hard-to-obtain material properties. 





Data and code availability.–
Code and data related to this paper are available on Zenodo <cit.> from the authors
upon reasonable request.



We acknowledge helpful discussions and related collaborations with  N. Huber, P. Bieniek, M. A. Wilde, and C. Pfleiderer. We would like to thank P Moll for pointing us to the idea of studying QO in strained materials. 
 V. L. acknowledges support from the Studienstiftung des deutschen Volkes. 
We acknowledge support from the Imperial-TUM flagship partnership. JK acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy–EXC–2111–390814868, DFG grants No. KN1254/1-2, KN1254/2- 1, and TRR 360 - 492547816; as well as the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus.
 


 
 




§ FILTERING




For the filtering process, we first compute a complex-valued spectrum s(F) with a rectangular window, as described in Fig. <ref>. To filter a frequency f, we scale the spectrum s(F) with a narrow Gaussian window exp(-(F-f)^2/Δ f^2) centered around f, see Fig. <ref> (a), and compute the inverse Fourier transform of exp(-(F-f)^2/Δ f^2) × s(F). The real part of the inverse Fourier transform is the filtered oscillation, see Fig. <ref> (b). The absolute (complex phase) of the inverse Fourier transform of exp(-(F-f)^2/Δ f^2) × s(F-f) are the filtered envelope (phase) of the filtered oscillation, see Fig. <ref> (b) (Fig. <ref> (c)).



§ PEAK SPLITTING


We filter the frequencies as described in the appendix sec. <ref>. In Fig. <ref> we show that in the case of t_⊥=0, i.e., the case of a single degenerate QO frequency, there is a remarkable agreement of the conductance for interband and intraband scattering. Especially, peak splitting is not a result of interband scattering but of scattering in general, as expected from analytics <cit.>. For inter- and intraband scattering, a phase change around Φ_0/B=150 is the origin of the peak splitting. The phase change is continuous and roughly 0.6π. It is unclear why the phase change differs from π.

The magnetic field at which the phase jump occurs increases as scattering, i.e., T_D increases. This is expected from the zeros of the polynomial amplitudes. E.g. for the first harmonic, the zeros are at T_D ≈ e B/2π m. Inserting the zero in R_D ≈ 0.04 shows that the Dingle factor strongly damps the amplitude of QOs around the zero.






 Paper Title: Singlet-Triplet Kondo Effect in Blatter Radical Molecular Junctions: Zero-bias Anomalies and Magnetoresistance.








Singlet-Triplet Kondo Effect in Blatter Radical Molecular Junctions: Zero-bias Anomalies and Magnetoresistance 









 
Gautam Mitra^1,2, Jueting Zheng^1,3, Karen Schaefer^4, Michael Deffner^4,5, Jonathan Z. Low^6, Luis M. Campos^6, Carmen Herrmann^4,5, Theo A. Costi^7 and Elke Scheer^1





^1 Department of Physics, University of Konstanz, 78464 Konstanz, Germany

^2 Institute of Condensed Matter and Nanosciences, Université Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium

^3 State Key Laboratory of Physical Chemistry of Solid Surfaces, College of Chemistry
and Chemical Engineering, Xiamen University, 361005 Xiamen, China

^4 Department of Chemistry, University of Hamburg, 22761 Hamburg, Germany

^5The Hamburg Centre for Ultrafast Imaging, Hamburg 22761, Germany


^6 Department of Chemistry, Columbia University,  New York, NY 10027, USA

^7 Peter Grünberg Institut, Forschungszentrum Jülich, 52425 Jülich, Germany








§ ABSTRACT 
The Blatter radical has been suggested as a building block in future molecular spintronic devices due to its radical character and expected long-spin lifetime. However, whether and how the radical character manifests itself
in the charge transport and magnetotransport properties seems to depend on the environment or has not yet been studied. 
 Here, we investigate single-molecule junctions of the Blatter radical molecule in a mechanically controlled break junction device at low temperature. Differential conductance spectroscopy on individual junctions shows two types of zero-bias anomalies attributed to Kondo resonances revealing the ability to retain the open-shell nature of the radical molecule in a two-terminal device. Additionally, a high negative magnetoresistance is also observed in junctions without showing a zero-bias peak. We posit that the high negative magnetoresistance is due to the effect of a singlet-triplet Kondo effect under magnetic field originating from a double-quantum-dot system consisting of a Blatter radical molecule with a strong correlation to a second side-coupled molecule. Our findings not only provide the possibility of using the Blatter radical in a two-terminal system under cryogenic conditions but also reveal the magnetotransport properties emerging from different configurations of the molecule inside a junction. 








§ INTRODUCTION


Stable organic radical molecules are emerging to become key components for spin-based molecular electronic materials and devices <cit.>. They are open-shell systems which do not readily decompose or chemically react 
under standard conditions and whose stability can be engineered by virtue of chemical design <cit.>. The 1,2,4-benzotriazin-4-yl radical commonly known as the Blatter radical is one of such species, first reported in 1968 <cit.>, gaining interest owing to its many exciting physical and chemical properties <cit.>.  The stability of the Blatter radical under various circumstances enables to incorporate it into manifold electronic devices <cit.>, controlled polymerization setups<cit.>, photodetectors <cit.> and thermoelectric devices<cit.>. In addition, the presence of an unpaired electron in this  molecule can lead to interesting magnetic properties,  

when interfaced with metallic substrates and electrodes <cit.>.





Blatter radicals typically have the unpaired electron delocalized on the molecule (Figure <ref>a). This spatial delocalization of the unpaired electron gives rise to the exceptionally high stability of the radical. Recently, these systems have been subject to studies under different environments of metal-molecule interfaces and have shown to retain the open-shell nature in thin films <cit.>, and upon adsorption on the Au(111) surface under high vacuum <cit.>. Typically, Kondo spectroscopy is used as an evidence to interrogate the  molecular spin of the radical <cit.>. Previously, using scanning tunneling microscopy (STM) techniques under ultrahigh vacuum conditions, a Kondo resonance arising from the interaction between the unpaired electron of the Blatter radical with the conduction electrons was revealed by Repp and coworkers <cit.>. The singly occupied molecular orbital (SOMO) of the Blatter radical lies close to the Fermi level of Au and was suggested to be very sensitive to the environment around the molecule <cit.>. Little emphasis has been given to the study of Blatter junctions in a two-terminal metal-molecule-metal system mainly due to the suspected loss of the open-shell character upon oxidation, as reported in solution-based STM-break junction experiments <cit.>. Later, this issue was tackled by functionalizing the radical by adding an electron-withdrawing group, which lowers the energy levels inside the molecule giving more stability and preventing oxidation <cit.>. Recently, Jiang et al. explored this problem theoretically using first-principles quantum transport calculations <cit.>. Their calculations suggest that the Blatter radical can successfully retain its open-shell character in a junction between Au electrodes irrespective of the environmental conditions since the dative bonding at the molecule electrode interface does not alter the spin states of the radical. 




We posit that stable organic radical junctions can also manifest unconventional magnetotransport behavior in combination with spectroscopic evidence for the Kondo effect and that both can be tuned by the coupling strength to the electrodes. To the best of our knowledge there are no experimental magnetoresistance (MR) studies to date that focus on radicals with a delocalized unpaired electron in the current pathway using in-situ tunable, yet sufficiently stable electrodes. To better understand the intrinsic properties of open shell systems, here we carry out the first systematic transport study of the Blatter radical in a mechanically controlled break junction (MCBJ) operated at low temperature. We were able to produce Blatter molecule junctions with increased stability in a two-terminal device and the signature of the open-shell nature of the radical is confirmed using dI/dV spectroscopy.  The Blatter radical was synthesized with two methylthioether units (–SCH_3) to enable coupling with the Au electrodes <cit.>. We have deposited the molecule by using a dropcasting technique and carried out the electronic transport measurements of the individual junctions in cryogenic vacuum at a temperature of 4.2 K and under magnetic fields. Kondo resonances and magnetoresistance (MR) originating from stable organic radical molecule junctions have been studied with the MCBJ technique before <cit.>. But in all previous cases, the electronic lone-pair was located in a specific site of the molecule and was sterically protected.  We have found two types of Kondo resonances, presumably originating from  different configurations of the individual junctions of the Blatter radical molecule. One type of the observed Kondo resonances features a Kondo temperature (T_ K) of 30 - 35 K with a broad and asymmetric line shape and the other type has a T_ K of 11 - 17 K with a more symmetric and narrow line shape. We have confirmed the Kondo nature with the evolution of the zero bias peak as a function of the magnetic field. These exhibit only a very weak negative finite-bias MR, as predicted by the usual Kondo effect at finite-bias (see Figure S1 in SI). However, we have also observed a significant negative finite-bias MR in certain junctions without having a zero-bias peak at the measurement temperature of 4.2 K. We find a maximum change of 21% in resistance with an applied magnetic field of ± 8 T and a subsequent saturation of MR for higher fields. These results highlight the stability of the unpaired electron in the Blatter radical molecule junctions in low temperature measurements with a MCBJ device. Finally, the negative MR at finite bias is explained in the framework of a singlet-triplet Kondo model. We have illustrated that the presence of an extra side-coupled molecule in the Blatter junction with an antiferromagnetic exchange interaction between the unpaired spins on the two radicals results in a possible scenario of negative MR at finite bias voltage and at a temperature similar to the experiment. A summary of our results from different types of junction formation is schematically shown in Figure <ref>b. 




§ RESULTS AND DISCUSSIONS




 §.§ Statistical investigations


The Blatter radical molecule with two methylthioether units was synthesized and characterized as reported earlier <cit.>. The freshly prepared solution of the molecule is prepared in tetrahydrofuran (THF) and dropcast onto a gold (Au) MCBJ sample at room temperature in a nitrogen atmosphere and properly dried for 45 min. The sample is then characterized using a dipstick equipped with a MCBJ mechanism pumped to high vacuum and cooled down to 4.2 K inside a liquid-He bath cryostat. More details of sample preparation and measurement techniques are described in the Methods section.





First, we measure and analyze the opening traces of the Blatter radical molecule with an applied bias of 100 mV at 4.2 K. Figure <ref>a shows typical examples of these opening traces as a function of electrode displacement. Au single-atom contacts with 1G_0 are observed in all the opening curves, where G_0 = 2e^2/h is the conductance quantum. Below 1G_0, upon further stretching of the electrodes, conductance plateaus are formed corresponding to the single-molecule junctions of the Blatter radical molecule bridged using methylthioether anchoring units. These plateaus shows a maximum displacement up to 0.6 nm before breaking down to the tunneling regime, with conductance G below 10^-6G_0. In Figure <ref>b, we have constructed a conductance histogram with logarithmic binning of 595 opening traces without any data selection. We observed major conductance peaks between 10^-2 and 10^-3G_0. We attribute multiple peaks observed in this conductance range to different configurations of molecule junctions formed between the Au electrodes. Flat plateaus observed in the opening traces correspond to fully stretched junctions of individual molecules whilst partially connected junctions show fluctuations in conductance as the electrode displacement is increased. This behavior is also clearly observed in the two-dimensional conductance-distance histogram as shown in Figure <ref>c. We note that there is also the possibility of dimers with two parallel or side-coupled molecules which are supported by our quantum chemical calculations using density functional theory (DFT, see the Supporting Information (SI)), the main results of which can be summarized as follows: 
First, the  Blatter radicals investigated here bind substantially to each other, as reported earlier for similar species <cit.>. Second, the simulated zero-bias conductance values are in the range of 10^-2 to 10^-3 G_0, with the monomers (denoted as M) at the lower end of this range (in line with <cit.>, see SI). These values agree well with the experimentally observed range. Knowing that DFT has the tendency to overestimate the conductance by up to around an order of magnitude <cit.>, it appears likely that the structures giving rise to higher conductance values in the simulations, hence dimers (denoted as D), correspond to the ones realized in the experiments. This interpretation is in line with the relatively strong binding between the radicals. We will come back to this aspect further below.



The observed conductance peak in the histogram is also in close agreement with the previous studies of the same molecule on a solution-based scanning electron microscopy break junction (STM-BJ) experiment <cit.> and STM studies on thin films of the Blatter radical on a Au surface at room temperature <cit.>. The small conductance obtained from the Blatter radical molecule junctions were very recently tackled theoretically by Jiang et al. using first-principles quantum transport calculations <cit.>. They attribute their unexpectedly low conductance to singly occupied frontier orbitals (SOMO and SUMO, respectively) being weakly electronically coupled to the Au electrodes. Hence, the Blatter radical may retain its unpaired spin upon contacting with Au electrodes in more instances than previously thought. In order to verify this hypothesis, we have carried out several magnetotransport measurements on individual junctions as explained below. 





 §.§ Transport spectroscopy in individual junctions: Kondo resonances


Differential conductance spectroscopy is performed in order to understand the properties of individual Blatter radical molecule junctions in detail.  We discuss the dI/dV characteristics observed in different junctions which are measured using a lock-in amplifier technique 
with 0.1 mV AC amplitude.
These measurements reveal a clear zero-bias anomaly indicating an electronic resonance formed near the Fermi level for about 20% of single-molecule junctions measured in our study. We have observed that the line width and the line shape of this resonance varies among different junctions under the same measurement conditions. Figures <ref>a and <ref>b show examples of two junctions exhibiting different dI/dV spectra. Junction M1 (<ref>a) shows a broad and asymmetric line shape and junction M2 (<ref>b) shows a narrower and symmetric resonance. We attribute the origin of these zero-bias anomalies to Kondo resonances due to the presence of an unpaired electron in the current pathway, hence indicating that the open-shell radical nature of the molecule within the two-terminal device is maintained at low temperature. Furthermore we argue that these junctions are monomer junctions, as we will discuss below.





For the junction M1, in Fig. <ref>a the zero-bias peak resembles a Fano resonance line shape as observed in many previous mesoscopic and nanoelectronics devices at low temperature <cit.>. This type of spectrum was also previously reported with STM-based studies of the Blatter radical molecule <cit.>. A Fano line shape typically occurs when there is an interference between resonant transport channels and non-resonant direct tunneling pathways through the device under study <cit.>. Here, these are the transport assisted through the SOMO due to an unpaired spin, interfering with the continuum tunneling channels. The first pathway is responsible for the Kondo resonance while the latter corresponds to the broad conductance background. 
This can be described by the commonly used equation <ref>, 


    dI/dV = α(q+ϵ)^2/1+ϵ^2 + β; ϵ = eV-E_ K/Γ_ K


where the Fano parameter, q, defines the ratio of transmission amplitudes through the resonant and non-resonant tunneling channels and hence, the peak asymmetry. E_ K is the energy of the resonant state and Γ_ K gives the halfwidth of the resonance, which is a function of temperature <cit.>. Since the experiment is at k_ BT≪Γ_ K (verified a posteriori by our fits), Γ_ K is the zero-temperature halfwidth of the resonance. The constants α and β correspond to the peak and background amplitudes, V is the bias voltage, and e is the elementary charge. The above, however, assumes a Lorentzian lineshape for the Kondo resonance. In reality, the Kondo resonance is known from accurate numerical renormalization group (NRG) calculations to have a non-Lorentzian lineshape, and can only be approximated by a Lorentzian at very low energies |eV|≪ k_ BT_ K <cit.>. In particular, the Kondo resonance has logarithmic tails decaying as 1/ln^2(|eV|/k_ BT_ K) at |eV|≫ k_ BT_ K<cit.>. It is possible to use the essentially exact numerical NRG spectra, together with the Fano effect, to accurately describe Kondo resonance lineshapes in dI/dV <cit.>. Here, we use an approximate analytic approach, the modified
Frota function <cit.>, which simulates well the non-Lorentzian lineshape of the Kondo resonance in the NRG spectra, while including also the Fano effect,

    dI/dV     = b -c  Im[i e^iϕ√(iΓ_ F/E-E_ K+iΓ_ F)],
 
where Γ_ F=Γ_ K/2.542 and ϕ=ϕ(q) is related to the Fano parameter q (see SI and Fig. S10). The constants b and c describe the background and Kondo peak amplitude, respectively. Note that both the Fano and modified Frota functions can describe any type of measured resonance, ranging from a symmetric, asymmetric, antisymmetric or even an antiresonance, depending on the Fano parameter q (or ϕ(q)), see SI. A comparison between the Lorentzian based procedure in Eq. (<ref>) and the more accurate procedure in Eq. (<ref>) for Kondo resonances is given in the SI, together with a table of all parameters extracted from fitting junctions M1 and M2 to Eq. (<ref>).


For the junction shown in Fig. <ref>a we find the energy width Γ_ K = 4.5 mV. We can extract the Kondo temperature, using the result k_ BT_ K=2 Γ_ K/π <cit.>, and found it to be approximately 34 K. This is in good agreement with the values obtained in previous STM-based studies <cit.>. For junction M2, with Γ_ K= 2.2 mV, we find T_ K  to be approximately 17 K.


 The Kondo conductance and the lineshape depends on the strength of the coupling between the molecule and the electrodes and hence on the bonding sites of the molecule with the electrodes <cit.>. Hence, we argue that the resonances in cases M1 and M2 originate from different configurations of the molecular junction. Since we have observed both flat and slanted opening traces in our study, the formation of single-molecule junctions with different configurations giving rise to different T_ K is a likely scenario. For Blatter radicals, it is known that the spin density is mostly centered around the fused benzene ring, benzotriazinyl core and is partially delocalized to one of the phenyl rings attached with the thiomethyl groups <cit.>. The third aromatic ring is spin-isolated. Hence, the difference in electronic coupling with both ends of the electrodes can lead to varying coupling strengths and transport pathways. Consequently, the junctions with metal electrodes having a direct interaction with one of the higher spin density sites can display strong zero-bias resonances. We analyzed many individual junctions with conductances between 10^-2 and 10^-3 G_0 but found no clear correlation with T_ K and G while 70% of them show asymmetric resonance with a T_ K of more than 30 K (type M1). An in-depth theoretical calculation is necessary to distinguish between different Kondo states and junction structures.
 


To further validate the presence of the Kondo resonance and the extracted 
Γ_ K (and T_ K=2Γ_ K/π k_ B) in these junctions, we have recorded the evolution of dI/dV spectra at various magnetic fields from 0 to 10 T at 4.2 K.  A magnetic field typically splits the Kondo resonance due to the Zeeman effect. Figures <ref>c and <ref>d show the corresponding spectra for junction M1 and junction M2, respectively. We have observed no splitting or suppression of the zero-bias peak in junction M1 while a clear splitting of the peak is visible for junction M2. Theoretical predictions suggest that this splitting follows the relation, gμ_BB_c ≈Γ_ K/2 <cit.>. For the junction M2, the zero-bias peak shows a clear splitting beyond the splitting field B_c of ≈ 7 T compared to the theoretical prediction 9.9 T using the extracted Γ_ K=2.29 mV. We do not observe any changes with increasing magnetic field until 10 T for junction M1, in agreement with the estimation from theory (using the extracted Γ_ K=4.57 mV) that the splitting in this case should occur only after the ≈ 20 T which is above our experimental limit. Further examples for the evolution of dI/dV spectra under magnetic field for both types of junctions is given in the SI. We have found that the experimental field dependence of the dI/dV for all the measured junctions shows a remarkable agreement with the theoretical predictions <cit.>, showing a splitting close to the predicted B_c when this lies in the experimentally accessible field range and no splitting when B_c is predicted to lie outside this range. 






 §.§ Negative magnetoresistance and singlet-triplet Kondo effect

  It was reported earlier that for single-radical molecular junctions without showing any Kondo features, a very pronounced MR can be observed at larger bias <cit.>. MR is defined here as the relative change in resistance of a molecule junction at an applied magnetic field with respect to zero field. In order to study the magnetotransport on molecular junctions which do not show any zero-bias peak in our measurements (denoted as type D junctions), we have studied the evolution of the resistance under magnetic field applied perpendicular to the sample plane from -10 to 10 T (red) and 10 to -10 T (black) at a rate of 50 mT/min and measured with a constant bias voltage of 30 mV at 4.2 K.  Figure <ref> shows MR curves obtained from three such type D junctions.
  

We have observed a strong negative MR (i.e. decrease of resistance with increasing field strength) with a maximum change of ≈ 21%. Besides a small hysteresis we observed no clear change in MR amplitude with the sweep direction. MR curves tend to saturate after approximately ± 8 T. We note that in previous studies of radical molecule junctions, this kind of saturation or maxima/minima of the resistance were also observed, but at different field value. We also found that all measured Blatter radical single-molecule junctions show exclusively negative MR in our study. 
In addition, MR measured on type M1 or M2 junctions show only a very small change in resistance of less than 2%, as given in the SI, hence revealing an anti-correlation between Kondo behavior and strong MR. This behavior was first explored in junctions of the perchlortrityl radical molecule <cit.> and is now confirmed for Blatter junctions. 

  
  

  
  The origin and mechanism of large MR in molecule junctions are still under debate. Various mechanisms have been reported based on the studies of different organic radical molecules with regard to the exact structure of the molecule and anchoring groups. Most of these mechanisms rely heavily on the influence of molecule-electrode interface scattering and changes in the electronic coupling strength <cit.>. In our study, we used methylthioether anchoring units to couple the Blatter radical to the Au electrodes. The previous study on the perchlortrityl radical molecule using the same anchoring groups yielded both positive and negative MR that was assigned to spin-dependent scattering at the molecule-electrode interfaces and resulting in a quantum interference effect <cit.>. In the case of Blatter molecule junctions, we have observed exclusively negative MR and the theoretically reported weak electronic coupling of SOMO and SUMO orbitals with gold electrodes suggests the above MR mechanisms may not be prominent. 

  




  
  Hence, we propose a possible explanation of the significant negative MR of type D junctions assuming the junctions actually consist of a double quantum dot system: a Blatter molecule in the junction side-coupled to a second 
  quantum dot, also having a S = 1/2, resulting in an antiferromagnetic exchange interaction of strength J between the two (see Eq. (<ref>) in Theoretical Methods and the SI). The second quantum dot could be, e.g., a gold atom or a cluster with unpaired spin, a molecular fragment, or a second Blatter radical. As mentioned above, the existence of  structures in which a second Blatter radical couples with the first is supported by our quantum chemical calculations (see SI). Furthermore, we found most such pairs to preferentially couple antiferromagnetically, with a coupling constant J' (for a Hamiltonian of the form H = -2 J' S_a S_b) in the range of around -90 to -140 cm^-1, which translates in our Kondo dimer model notation to antiferromagnetic couplings J=-2J' of order 20-40 meV, i.e., 200-400 K.  Examples of the optimized structures of the possible dimer (D) junctions, the resulting transmission functions and the information about energies and spin population are given in the SI. 
  Weak ferromagnetic coupling could also occur for some junctions based on our DFT calculations, but could be pushed to more antiferromagnetic by binding to the electrodes. The latter possibility is also interesting as the D system would form a S=1 which would result in an underscreened Kondo effect <cit.>.


For antiferromagnetic couplings J, the double-dot system would exhibit the so called singlet-triplet Kondo effect, which would provide a mechanism for understanding the observed negative MR in the absence of a zero-bias peak in dI/dV. 

Organic radical dimers, exhibiting a field-induced singlet-triplet Kondo effect have been investigated previously, however, the MR was not discussed <cit.>. Within this picture, see Fig. <ref>a, the ground state of the isolated dimer at B = 0 is a singlet S = 0 state with a triplet S = 1 state lying at an energy J above it. Upon coupling the dimer to the leads, the Kondo effect at low energies, temperatures and voltages (E, k_ BT, eV ≪ J) is therefore suppressed and a zero-bias peak in dI/dV is absent for eV < J. Instead, the differential conductance exhibits a gap of order 2J, which at high temperatures k_ BT>J could be smeared out (see Fig. S7a in SI). Despite the non-magnetic ground state, the system is still strongly affected by a magnetic field via the effect of the latter on the polarizable triplet state: the magnetic field splits the triplet state and decreases the splitting δ(B) = E_S=1,|S_z|=1 - E_S=0,S_z=0 between the lowest component of the triplet state (S_z = +1 for B > 0 or S_z = -1 for B < 0) and the singlet state as shown in Figure <ref>a. This splitting eventually vanishes at a sufficiently large magnetic field B = B^* where B^* is of the order of J/gμ_B (B^* = J/gμ_B exactly when the molecule plus side-coupled quantum dot is detached from the leads, but is slightly reduced when it is attached to the leads <cit.>). Fluctuations between the degenerate states at B = B^* lead to a fully developed singlet-triplet Kondo effect via the hybridization of these states to the leads <cit.>. In this scenario, increasing B from zero towards B^* induces the singlet-triplet Kondo effect and thereby enhances the conductance G(B)=dI/dV, implying that G(B = 0) - G(B) is negative (i.e., positive magnetoconductance), resulting in a negative MR for the field range -J < gμ_BB < +J (see Fig. <ref>b and Fig. S8). This is opposite to the usual S = 1/2 Kondo effect where a magnetic field reduces the conductance (resulting in a positive MR, see Fig. S9 in the SI).  

  

  We now illustrate the generic MR behavior due to the singlet-triplet Kondo effect via numerical renormalization group calculations (see Theoretical Methods and SI). Two energy scales determine the quantitative aspects of the 
  singlet-triplet Kondo effect, the antiferromagnetic dimer coupling J and the Kondo scale T_ K
  in the absence of the side-coupled quantum dot in (<ref>). The former (J) is determined by the coupling strength between the two Blatter molecules, while the latter (T_ K) depends on the tunnel couplings of the Blatter molecule in the junction to the electrodes. 
  However, even with reasonable estimates for J, the Kondo scale T_ K remains largely unknown for the experimental system: while theoretically, T_ K is easily obtained by switching off J in our model, this is clearly not possible for the experimental system (one cannot remove the side-coupled Blatter molecule in order to extract the T_ K for the monomer).  Thus, we consider a wide range of J/k_ BT_ K = 100, 30, and 10 to illustrate the overall qualitative results for the MR. This uncertainty in T_ K
  for the dimer systems, also means that T/T_ K is unknown. We shall use T/T_ K=0.4 with T_ K=10 K for an experimental temperature T=4 K as a typical value, but it should be noted that the MR depends sensitively on the precise value of T/T_ K (see SI). The bias-voltage dependence of the MR will also be calculated. 

  
  Figure <ref>b shows the calculated MR at several dimensionless voltages eV/k_BT_ K. For all voltages, the MR exhibits a similar dependence on the magnetic field as in the experiment. It is largest at V = 0. Finite V introduces decoherence, leading to a suppression of the MR. Nevertheless, even at eV/k_ BT_ K = 30, the MR reaches 10% at gμ_BB/k_ BT_ K = 10. The magnitude of the MR is sensitive to the precise J/k_ BT_ K and T/T_ K (see Fig. S8 for J/k_ BT_ K=30 and T/T_ K=10). The effect of a spin-exchange anisotropy (J_∥/J_⊥ = 2 and J_∥/J_⊥ = 1/2) is shown for the largest bias voltage (eV/k_ KT_ K = 30) and found to be small. Its effect is even smaller for smaller bias voltages. In general, for eV < J its effect is to enhance the magnitude of the MR. The effect of the Dzyaloshinskii-Moriya (DM) interaction, shown in Fig. <ref>b for two bias voltages (dotted lines) and a large D/J=0.5, is also to enhance the magnitude of the MR, while not affecting its overall B-dependence (for smaller D/J=0.2, the enhancement is much smaller, see Fig. S8a). The enhancement in MR ≈ 1-G(V, 0)/G(V, B) is due to the reduction of G(V, B) in the presence of a finite Dzyaloshinskii vector (D), relative to its value for D = 0 (since, a finite D enhances the pseudo-magnetic field in the singlet-triplet Kondo effect, thereby reducing G as discussed in <cit.>).  For comparison, we briefly also show the results for the temperature and bias-voltage dependence of a single Kondo-correlated quantum dot described by the Anderson impurity model in the SI, which is also in clear agreement with the previously reported studies <cit.>. 
  
  
  

 §.§ Conclusions

 To summarize, we have developed single-molecule junctions of the Blatter radical molecule using a mechanically controlled break junction setup at low temperature in a cryogenic vacuum environment. 
 
 Our detailed magnetotransport measurements suggest that the open-shell nature of the Blatter junction remains intact in a two-terminal junction.
 Differential-conductance spectroscopy studies on individual contacts reveal two different types of zero-bias anomalies originating from the different configurations of the metal-molecule-metal junction. They are attributed to a Kondo resonance arising from the delocalized unpaired electron orbital situated in the current pathway. In addition, Blatter radical molecular junctions without a zero-bias peak showed strong negative magnetoresistance (MR) with a maximum change of 21% in resistance measured at high bias voltage. We modeled the origin of these high negative MR towards junctions involving a Blatter molecule with strong correlation with a side-coupled quantum dot configuration which results in a singlet-triplet Kondo effect. The detailed quantum chemical calculations show qualitative agreement between the experiment and theoretical predictions. Our findings reconcile earlier, seemingly contradictory findings on Blatter radical molecular junctions studied in different environments and will open up several new experimental and theoretical avenues to revisit the transport and magnetotransport properties of other radical molecular junctions. They also pave the way for studying various fundamental aspects of the Blatter radical and its derivatives to tailor their properties for spintronic applications.
  
  
  

 §.§ Methods





 §.§ Experimental Methods


Device Fabrication. 
The break junction devices were fabricated on a 500 μm thick, polished insulating Cirlex (Kapton laminate) substrate.  2 μm polyimide sacrificial layer was spin-coated on the substrate, baked at 130°C for 5 min, then hard-baked at 430°C for 90 min under vacuum. Next, a double-layer resist consisting of methylmetacrylat-methacrylat acida/polymethylmethacrylat (MMA-MAA/PMMA) is spincoated on top and baked in an oven at 170^∘C for 30 min. Electron beam lithography was performed on these substrates of size 3 × 18 mm^2 in a Zeiss Cross Beam machine at 10 kV acceleration voltage and are developed using 1:3 methyl isobutylketon: isopropylalcohol (MIBK:IPA) solution for 30 s and rinsed with pure IPA. 80 nm of Au were later deposited using the electron beam evaporation. The samples were then etched using anisotropic reactive ion etching in a mixture of oxygen and sulfur hexafluoride at 1 mbar with a power of 50 W for 30 min. This process removes about 600 nm of the polyimide layer below the electrodes.  We prepared a 0.2 mM solution of Blatter molecules in tetrahydrofuran (THF) and then dropcast it onto a freshly prepared MCBJ substrate in nitrogen atmosphere at room temperature and dried for ≈ 1 hour. 


Transport Measurements. All experimental measurements were carried out using a custom made cryogenic vacuum dipstick equipped with a MCBJ at 4.2 K in a He bath cryostat. The sample is shielded using a copper cap to prevent stray electromagnetic fields. The electrode displacement is calibrated by analyzing the opening curves of a bare Au break junction using the tunneling conductance expression, G(δ x) ∝ exp (-2 r δ x √(2mϕ)/ħ) where ϕ is the work function of Au approximated as 5.0 eV, m is the electron mass, δ x is the pushing rod movement of the break junction setup, and the reduction ratio r has the relation with electrode displacement, δ d = r δ x <cit.>. We found a reduction ratio r = 3.6 × 10^-5 which we use to calculate the electrode displacement. At base temperature (4.2 K), sample is repeatedly opened and closed to form atomically sharp Au electrodes and this method allows to form molecular junctions at low temperature <cit.>.  Electronic transport measurements were carried out using a Yokogawa 7651 as a DC voltage source. The wiring is composed of homemade coaxial cables and SMA connectors. Current and voltage across the sample are recorded using Femto DLCPA and Femto DLPVA low noise amplifiers and measured with the help of Agilent 34410A multimeters. dI/dV measurements were carried out using HF2LI Zurich Instruments lock-in amplifiers by applying an AC modulation voltage of 0.1 mV  at 317 Hz. Magnetotransport measurements were performed with an applied magnetic field up to ± 10 T perpendicular to the sample plane produced from a superconducting magnet inside the liquid-He dewar.  All measurements and devices were remotely controlled using a custom Python program. Control experiments utilizing helium exchange gas for thermalization confirmed that eddy-current heating effects were negligible in our experiments. 
  



 §.§ Theoretical Methods
  
Singlet-triplet model.
  We model the  singlet-triplet Kondo effect by the Anderson impurity model side-coupled to a second S=1/2 quantum dot via an antiferromagnetic exchange interaction of strength J. The Hamiltonian is given by

    H    =H_ dot+H_ leads+ H_ tunneling+H_ J+H_ B.

The first term, H_ dot=∑_σε_0d_σ^†d_σ +Un_d↑n_↓, describes the dot Hamiltonian, representing the
S=1/2 degrees of freedom of the Blatter molecule,
where ε_0 is the level energy, measured relative to  the Fermi level E_ F, n_dσ is the occupation number for spin σ=↑,↓ electrons on the dot, and U is the local Coulomb repulsion 
on the dot. The second term describes the Hamiltonian of the leads and is given by H_ leads=∑_kασϵ_kc_kασ^†c_kασ, where α=L,R labels the two leads, and ϵ_kσ is the kinetic energy of the lead electrons.  The third term, H_ tunneling=∑_kασt_α(c_kασ^†d_σ +H.c.), describes the tunneling of electrons from the leads onto and off the dot with tunneling amplitudes t_α and tunneling rates Γ_α=πρ_ Ft_α^2, with ρ_ F the lead electron density of states at the Fermi level. The term H_ J=J  s_d· S describes an antiferromagnetic (J>0) coupling between the spin s_d of the Blatter molecule and the spin S of the side-coupled quantum dot. Finally, H_ B=-gμ_ BBS_z^ tot describes a magnetic field acting on the combined spin of the quantum dot and side-coupled quantum dot via the total z-component S_z^ tot=s_z,d+S_z. The above is a minimal model for the singlet-triplet Kondo effect <cit.>. It can be generalized to include the effects of anisotropic spin exchange J  s_d· S→J_⊥/2(s_d^+S^-+s_d^-S^+)+J_∥s_d,zS_z, and spin-orbit coupling via a Dzyaloshinskii-Moriya (DM) term H_DM= D· s_d× S. The former (anisotropic spin exchange) does not significantly alter the MR predictions of the isotropic model. The main effect of the latter (DM) is to lift the degeneracy of singlet and triplet states at B=B^*, resulting in an avoided crossing at this field value <cit.>, see Fig. <ref>(a), and therefore a reduced conductance for magnetic fields close to B=B^* <cit.>. Hence, the  largest effect of the DM term will be in the magnitude of the MR in the vicinity of B=B^*≈ J.  For D/J≪ 1, the overall qualitative dependence of the MR on field is close to that of the pure singlet-triplet Kondo effect (isotropic J and D=0), see Fig. <ref>(b).


The total dot-lead tunneling rate is denoted by Γ=Γ_ L+Γ_ R. For a strongly correlated quantum dot we have that U/Γ≫ 1. We shall use U/Γ=24 for the calculations, a typical value expected for molecular junctions. For example, estimates of Γ from the experiment yielded for the Blatter molecular junction 50-60 meV. For U/Γ=24 this implies that U=1.2-1.4 eV, which is a reasonable value for a single-molecule device. The relevant energy scales of the above model are J and k_ BT_ K, where T_ K is the Kondo temperature for the Anderson impurity model (J=D=0). More precisely, our T_ K is defined as k_ BT_ K= (gμ_ B)^2/4χ(0), where χ(0) is the zero-temperature spin susceptibility of the Anderson model. This T_ K is close to that of the well known Haldane expression T_ K^ H=√(Γ U/2)exp(πε_0(ε_0+U)/2Γ U) <cit.>. See SI for other definitions of Kondo scales used in the literature and their interrelationships.
Calculations will be presented for several values of the dimensionless ratio J/k_ BT_ K ranging from 1 to 100.


Numerical renormalization group method.
The model (<ref>) is diagonalized to obtain its many-body eigenstates and eigenvalues via the numerical renormalization group method <cit.>. Briefly stated, this approach consists of three steps: (i) the conduction electron kinetic energy of the leads ∑_kασϵ_k c_kασ^†c_kασ is logarithmically discretized about the Fermi level E_ F, taken as zero of energy, ϵ_k →ϵ_k_n=± DΛ^-n, n= 0,1,…, with D the half-bandwidth of the leads and Λ>1 the logarithmic discretization parameter, (ii), the discretized leads are then transformed to a (tridiagonal) linear chain form using a Lanczos procedure, and, (iii), the resulting linear chain form of (<ref>) is iteratively diagonalized on successively lower energy scales to obtain the eigenvalues E_p=1,2,…^n and eigenstates |p⟩_n on all energy scales ξ_n =DΛ^-n/2,n=0,1,…. Knowledge of the eigenstates and eigenvalues on all energy scales then allows physical properties such as thermodynamics and Green functions to be calculated. Within this approach, with refinements <cit.> described in the SI, we calculate the magnetic field, bias voltage and temperature dependence of the differential conductance G(V,T,B) = dI/dV via

    G(V,T,B)     = ∫_-∞^+∞-∂ f(E-eV,T)/∂ E  A(E,T,B) dE,

where  f is the Fermi function and A(E,T,B) is the spectral function of the impurity level. The general dependence of the spectral function and differential conductance on field, temperature and energy E=eV is shown in Figs. S5-S7. From G(V,T,B) we extract the magnetoresistance (MR) value at bias voltage V and temperature T via
   
    MR[%]    = 100 × (R(B)-R(0))/R(0),
        = 100 × (G(V,T,B=0)-G(V,T,B))/G(V,T,B).


First Principles Calculations.

All molecular structures were optimized using Kohn-Sham density functional theory (KS-DFT) as implemented in the program package Turbomole 6.6 <cit.>. The B3LYP <cit.> exchange–correlation functional was used, along with Ahlrichs' def2-TZVP <cit.> single-particle atom-centered basis set, and Grimme's empirical dispersion corrections (DFT-D3) <cit.> with Becke–Johnson damping <cit.>. Convergence criteria of 10^-7 a.u. for the energy in the self-consistent-field (SCF) algorithm and 10^-4 for the gradient in the molecular structure optimizations were applied, in combination with m4 integration grids. 
The coupling constants J are obtained using the Yamaguchi formula <cit.> based on the energy difference between an open-shell singlet and open-shell triplet (broken-symmetry) determinant <cit.>.


The conductance of the junctions was evaluated assuming coherent tunneling as the dominant transport mechanism (Landauer regime). The transmissions are modeled via a Green's function approach combined with DFT calculations for zero-bias electronic structures, employing the wide-band limit for the self-energies of the electrodes, as described, e.g., in Refs.<cit.>. A local density of states (LDOS) of 0.036 eV^-1 was used. Transmission functions were evaluated with Artaios <cit.>, using the Fock and overlap matrices from the KS-DFT electronic structure calculations on the molecular junctions. The value of the Fermi energy is not straightforward to predict in first-principles calculations on molecular junctions, as it will be affected by factors such as the irregular atomistic shapes of the electrodes and the adsorption of non-bridging molecules on the electrodes. We, therefore, evaluate zero-bias conductances for a range of reasonable Fermi energies (-5 eV, -4.5 eV, -4 eV) and check whether our conclusions are robust with respect to this choice. 
For further information please refer to the SI.
  
  

 §.§ Acknowledgements

 We thank the staff of the nano.lab at the University of Konstanz for technical support. G. M. and E.S. acknowledge funding by the Deutsche Forschungsgemeinschaft  (DFG) through  the Collaborative Research Center SFB 767 “Controlled Nanosystems” (funding ID 32152442).  K. S. and C. H. acknowledge funding via the Graduate School 2536 “Nanohybrid” of the Deutsche Forschungsgemeinschaft (DFG), funding ID 408076438. M. D. and C. H. acknowledge support by the Cluster of Excellence “CUI: Advanced Imaging of Matter” of the Deutsche Forschungsgemeinschaft (DFG) (EXC 2056, funding ID 390715994). T. A. C. gratefully acknowledges computing time on the supercomputer JURECA <cit.> at Forschungszentrum Jülich under grant no. JIFF23.
  
  

 

