# Topological metals constructed by sliding quantum wire arrays

Zheng-Wei Zuo  Linxi Lv  Dawei Kang  School of Physics and Engineering, Henan
University of Science and Technology, Luoyang 471023, China

###### Abstract

A general strategy of alternated slide construction to craft topological
metals is proposed, where there is a relative slide between the odd and even
chains in the trivial spinless quantum wire array. Firstly, taking the three-
leg ladder as an example, we find that alternated slide can induce a
topological phase transition from the normal metal to topological metal
phases, which are protected by inversion symmetry. Remarkably, topological
metal without nontrivial edge states is found, and the bulk-boundary
correspondence breaks down. Secondly, the two-dimensional quantum wire arrays
with alternated slide manifests similar physical behaviors. Two types of
topological metal phases emerge, where there are gapless bulk bands with and
without nontrivial edge states. These results could be confirmed by current
experimental techniques.

##  I Introduction

Nowadays, identifying, classifying, and engineering topological quantum
matters is of extensive interest in condensed matter physics[1, 2, 3, 4, 5,
6]. The topological insulator, topological superconductors, and topological
metal (semimetals) are prototype examples of topological phases. For the
electronic materials, based on symmetry-indicator theory[7, 8] and (magnetic)
topological quantum chemistry[9, 10], a myriad of topological materials[11,
12, 13, 14, 15] have been predicted and discovered. For the topological
metals, there are many types depending on the dimensionality, dispersion
(slope and order), and degeneracy of band crossing. For example, according to
the dimensionality of nodal gap closing, topological metals can be divided
into nodal-point, nodal-line, and nodal-surface topological metals. Based on
the slope of band dispersion near the crossing, there are types I and II
topological metals. Additionally, there is another topological metal, with the
topological localized states with eigenvalues embedded in the continuum of
extended states, dubbed topological bound states in the continuum[16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. A variety of gapless symmetry-
protected topological phases[30, 31, 32, 33, 34, 35, 36] beyond the
topological metals have garnered much attention recently.

There are many theoretical and experimental tools for engineering topological
quantum matters. One of the flagships is that the magnetic field induces the
quantum Hall effect in two-dimensional (2D) electron gas[37, 38]. Combined the
electron-electron interaction, the magnetic field can also cause the
fractional quantum Hall effect[39, 40]. The modulated hoppings and potentials
provide another approach to realize different types of topological band
phases. The Su-Schrieffer-Heeger (SSH) model[41], Aubry-André-Harper (AAH)
model[42, 43], and Kekulé distortion of graphene-like structures[44] are
representative examples. On the other hand, the interplay of magnetic field
and modulated hoppings could result in the high-order topological
insulators[45, 46, 47]. More recently, by adjusting the twist angle, graphene
moiré superlattices have been widely explored as platforms for exploring
correlated physics, energy band topology, and superconductivity[48, 49].

In this paper, we propose a simple scheme to construct topological metals by
way of trivial spinless quantum wire arrays sliding against each other.
Firstly, for the simple three-leg ladder, the center leg has a relative
translation compared with the bottom and top chains. The ladder showcases
unconventional properties. By way of a unitary transformation, the Hilbert
space of the system could be separated into two independent subspaces, where a
trivial subspace and a topological nontrivial subspace coexist. Because of the
different nearest-neighbor interchain couplings (depending on interchain
distance), the system undergoes a topological phase transition from normal
metal to inversion symmetry-protected topological metal phases, where for the
topological metal, the nontrivial edge states appear or not depending on the
strength of intrachain couplings. Second, for the 2D quantum wire array with
alternated slide, similar physical phases also emerge.

Figure 1: (a) Schematic picture of 2D lattice system (trivial quantum wires
array). (b) The modified 2D lattice system with quantum wires array sliding
against each other. (c) The three-leg (labeled $A,B,$ and $C$) ladder. The
hopping in quantum wires is $J$; the coupling strengths between quantum wires
are $t_{1}$ and $t_{2}$.

##  II Model

Now one can consider a 2D lattice system of trivial spinless quantum wire
arrays (Fig.1a). Then, the even quantum wires have a gradual radial-direction
translation relative to the odd quantum wires, as shown in Fig.1b. For the
nearest-neighbor interchain coupling, there are two different strength
hoppings $t_{1}$ and $t_{2}$ (red thin and thick lines) depending on distance.
As the translation of even quantum wires pushes ahead, the coupling strength
$t_{1}$ ($t_{2}$) decreases (increases). When $t_{1}<t_{2}$, one may
intuitively think there is nothing happening. However, a topological phase
transition from normal metal to topological metal takes place, which is an
unexpected and counterintuitive theoretical discovery. We unveil two types of
topological metals, where the nontrivial edge states can appear or not. For
the strong intrachain coupling $J$, the system could enter into the
topological metal phase without nontrivial edge states. When the even quantum
wires are shifted from one lattice constant, the system returns to the 2D
normal quantum wire arrays. Furthermore, these results could be confirmed
experimentally in a photonic waveguide array, photonic crystals,
topoelectrical circuits, or coupled acoustic resonators. For example, the
coupled-waveguide arrays are manufactured by the femtosecond laser direct-
writing technique. The coupling coefficients $J$, $t_{1}$, and $t_{2}$ are
determined by the evanescent mode coupling between adjacent waveguides, which
can be tailored by waveguide spacing.

###  II.1 Three-leg ladder

For simplicity, we use the three-leg ladder (Fig.1c), which serves as a
starting point. We can write the tight-binding model (set lattice constant
$a=1$) as

$H=\sum_{j,\alpha,\beta}^{L}Jc_{\alpha,j}^{\dagger}c_{\alpha,j+1}+t_{1}c_{\beta,j}^{\dagger}c_{C,j}+t_{2}c_{\beta,j+1}^{\dagger}c_{C,j}+H.c.,$
(1)

where $c_{\alpha,j}^{\dagger}$ ($c_{\alpha,j}$) is the fermionic creation
(annihilation) operator at lattice site $j$ in the leg $\alpha$. Here $\alpha$
stands for the legs $A,B,$ or $C$, and $\beta$ denotes the legs $A,$ or $B$.
The $J$, and $t_{1}$ ($t_{2}$) are strengths of the intrachain coupling and
interchain coupling, respectively. The length of the chain is $L$.

Applying the Fourier transform, we can express the bulk momentum-space
Hamiltonian as (base ($c_{A}^{\dagger}$ , $c_{B}^{\dagger}$,
$c_{C}^{\dagger}$))

$H_{o}=\left(\begin{array}[c]{ccc}2J\cos k_{x}&0&t_{1}+t_{2}e^{ik_{x}}\\\
0&2J\cos k_{x}&t_{1}+t_{2}e^{ik_{x}}\\\
t_{1}+t_{2}e^{-ik_{x}}&t_{1}+t_{2}e^{-ik_{x}}&2J\cos k_{x}\end{array}\right)$
(2)

If we use a unitary transformation $H_{D}=Q^{-1}H_{o}Q$, the Hamiltonian can
be block-diagonalized

$\displaystyle H_{D}$ $\displaystyle=\left(\begin{array}[c]{ccc}2J\cos
k_{x}&0&0\\\ 0&2J\cos k_{x}&\sqrt{2}\left(t_{1}+t_{2}e^{ik_{x}}\right)\\\
0&\sqrt{2}\left(t_{1}+t_{2}e^{-ik_{x}}\right)&2J\cos k_{x}\end{array}\right)$
(6)

where the unitary matrix

$Q=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{ccc}-1&1&0\\\ 1&1&0\\\
0&0&\sqrt{2}\end{array}\right)$ (7)

The Hamiltonian divides into two independent blocks $h_{1}=2J\cos k_{x}$ and
$h_{2}=\left(\begin{array}[c]{cc}2J\cos
k_{x}&\sqrt{2}\left(t_{1}+t_{2}e^{ik_{x}}\right)\\\
\sqrt{2}\left(t_{1}+t_{2}e^{-ik_{x}}\right)&2J\cos k_{x}\end{array}\right)$.
Thus, the Hilbert space becomes separable. We can diagonalize the Hamiltonian
in Eq.6 and obtain the energy eigenvalues. The energy dispersions are

$\displaystyle E_{1}$ $\displaystyle=2J\cos k_{x},$ (8) $\displaystyle
E_{\pm}$ $\displaystyle=2J\cos
k_{x}\pm\sqrt{2\left(t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}\cos k_{x}\right)}$ (9)

The lower $2\times 2$ block $h_{2}$ is equivalent to the SSH model with next-
nearest-neighbor (NNN) hopping[50, 51, 52], where the chiral symmetry is
broken and the inversion symmetry is conserved. It means that the block-
diagonalization separates the Hilbert space of the system into two
independently subspaces, where a trivial subspace and a topological nontrivial
subspace coexist. The system can be viewed as a hybrid of a trivial quantum
wire and a virtual SSH model with NNN hopping. Thus, the whole system could
enter into the topological metal phase, where the gapless bulk states with
topological bound edge states appear. Here, let us firstly investigate the
topological properties of the subspace $h_{2}$. The system displays the
inversion symmetry $\mathcal{I}$ where
$\mathcal{I}h_{2}(k_{x})\mathcal{I}^{-1}=h_{2}(-k_{x})$, where the inversion
symmetry operator $\mathcal{I}$ is defined by

$\mathcal{I}=\left(\begin{array}[c]{cc}0&1\\\ 1&0\end{array}\right)$ (10)

and $h_{2}(k_{x})$ commutes with $\mathcal{I}$ at inversion-invariant momenta
$k_{x}=\left(0,\pi\right)$. Thus, eigenstates of $h_{2}(k_{x})$ have a well-
defined parity $\zeta_{i}\left(k_{\text{inv}}\right)=\pm 1$ at those points.
According to the parity value of inversion symmetry $\mathcal{I}$, we can
calculate the topological invariant $\mathcal{N}$[53](the Zak/Berry phase[54]
can also characterize this topological phase transition)

$\mathcal{N}=\left|n_{1}-n_{2}\right|$ (11)

where $n_{1}$, $n_{2}$ are the number of negative parities at $k_{x}=0$ and
$k_{x}=\pi$, respectively. So we get $\mathcal{N}=1$ (topological phase, the
Zak phase is $\pi$) for $t_{1}<t_{2}$ and the two degenerate edge states would
appear in a finite system. For $t_{1}>t_{2}$, the topological invariant
$\mathcal{N}=0$ and the subspace $h_{2}$ is trivial.

It is worthwhile noting that the topological phase transition is independent
of the intrachain hopping $J$. The intrachain hopping $J$ term in subspace
$h_{2}$ is proportional to the identity matrix and does not change the bulk
eigenstates. The eigenfunctions of $h_{2}$ can be explicitly expressed as
$u_{\pm}=(1,\pm e^{-i\phi})^{T}$ ($\phi$ is the phase angle of complex number
$t_{1}+t_{2}e^{ik_{x}}$), which is independent of intrachain hopping $J$.
Thus, the topological invariants ($\mathcal{N}$, and Zak phase) are fixed for
any values of intrachain hopping. However, for strong intrachain hopping $J$,
the energy gap (see Hamiltonian $h_{2}$ and Eq.9) could close without band
crossing owing to the indirect nature of the gap (see Appendix A for more
details). At the same time, in a finite system, the two degenerate edge states
would assimilate into the bulk bands and become delocalize. Thus, the
conventional bulk-boundary correspondence breaks down. The subsystem is still
in the topological phase (without the nontrivial edge states). For the entire
system, the topological bands ($h_{2}$) coexist with the single trivial bulk
band ($h_{1}$). Thus, there are two types of topological metals, where the
nontrivial edge states can appear or not.

Figure 2: (a) The phase diagram of the three-leg ladder with $t_{2}=1$. Region
I (II) represents that the system is in the topological metal with (without)
bound end states. Other regions are the trivial normal metal. (b) Energy
spectra as a function of hopping $t_{1}$ with system size $L=30$, $t_{2}=1$,
and $J=0.5$ under open boundary condition. The red dots indicate the
topological bound end states identified by IPR. (c) The spatial distribution
of the two topological bound end states; other parameters are $L=30$,
$t_{2}=1$, and $t_{1}=J=0.5$.

###  II.2 Phase diagram

According to the behaviors of the edge states and topological invariant
$\mathcal{N}$ in $h_{2}$ subspace, we can numerically obtain the phase diagram
of the whole system (set $t_{2}=1$ here) shown in Fig.2a. It shows that, as
the interchain coupling $t_{1}$ decreases (the slide displacement increases),
the system changes from the normal metal to topological metal. When the slide
displacement is larger (smaller) than half of one lattice constant, the system
is in topological (normal) metal states. For the interchain couplings
$t_{1}<t_{2}$, the system evolves from the topological metal with nontrivial
edge states to the topological metal without edge states (region I to region
II) when the hopping $J$ increase (the critical value is $\sqrt{2}/2$
calculated by the edge state energy curve and the energy dispersions
$E_{\pm}$). For the interchain couplings $t_{1}>t_{2}$, the system is in
trivial normal metal phase.

Next, let us investigate the topological phase transition. The energy spectra
under open boundary condition are plotted in Fig.2b with regard to the hopping
$t_{1}$ when system size $L=30$, $t_{2}=1$, and $J=0.5$. To identify the
topological edge (localized) states from bulk states, we use the inverse
participation ratio (IPR), which is defined by
$\mathrm{IPR_{n}}=\sum_{j=1}^{3L}\left|\psi_{j}(n)\right|^{4}$. The IPR of an
extended state scales as $1/3L$, thereby vanishing in the thermodynamic limit,
while remaining finite for a localized state. The red dots in Fig.2b show the
topological bound end states identified by $\mathrm{IPR_{n}}$. Thus, the
nontrivial edge states coexist with gapless bulk states at the Fermi level. At
the same time, they do not couple with each other because they belong to
different subspaces and the two subspaces are orthogonal. As the hopping
$t_{1}$ increases, the system changes from the topological metal with
nontrivial edge states to normal metal. As shown in Fig.2c, the two nontrivial
edge states with $L=30$, $t_{2}=1$, and $t_{1}=J=0.5$ are clearly localized in
the left and right boundaries. This type of topological metal is also
identified by other systems[17, 26] (known for topological bound states in the
continuum).

Figure 3: Single-particle quantum walks on the three-leg (labeled $A$, $B$,
and $C$) ladder with $J=1$, $t_{2}=1$, and $L=60$ under open boundary
condition. The quantum walker is initially positioned on the center of leg
$C$. (a) Topological metal phase case without the two bound end states with
hopping $t_{1}=0.5$. (b) Normal metal case with hopping $t_{1}=1.2$. (c)
Averaged mean density displacements for the topological metal (red line) and
normal metal (blue line) cases.

For the topological metals without nontrivial edge bound states, it is
difficult to distinguish them from the normal metal because the bulk-boundary
correspondence breaks down. Here, we use the experimentally continuous-time
quantum walks[55] to decode the topological properties of topological metal
without the edge bound states. Firstly, a single-particle state
$\left|\Psi_{0}\right\rangle$ is initially located at the center of leg $C$.
The time-dependent density distribution of the single-particle quantum walker
is given by $n(t)=\sum_{n}n(|A_{n}(t)|^{2}+|B_{n}(t)|^{2}+|C_{n}(t)|^{2})$,
where $A_{n}(t)$ ($B_{n}(t)$ and $C_{n}(t)$) is the occupation amplitudes of
the dynamical evolution state $|\Psi(t)\rangle$ in the $n$-th unit cell of the
leg A (B and C) sites, where
$|\Psi(t)\rangle=e^{-iHt}\left|\Psi_{0}\right\rangle$. The time-averaged mean
displacement in the time interval $(0,T)$ can be calculated by
$\overline{n(t)}=\int_{0}^{T}n(t)dt/T$. For large $T$, $\overline{n(t)}$
approaches a constant asymptotic value[50, 52]. Fig.3a (b) displays the
numerically time-dependent density distribution when the system is in
topological (trivial) metal phase with $t_{1}=0.5(1.2)$, $J=1$, $t_{2}=1$, and
$L=60$ under open boundary condition. The corresponding time-averaged mean
density displacements for the topological and trivial phases are shown in
Fig.3c. Numeral calculations show that the time-averaged mean density
displacement approaches the quantized value $0.5$ ($0.0$) for a large system
with long $T$ using the finite-size analysis, which is attributed to the
appearance of topological (trivial) states and indicates the system is in the
topological (normal) metal phase.

##  III 2D topological metal

Motivated and encouraged by the above-mentioned results, we straightway apply
a similar analysis for the 2D case. Firstly the real-space Hamiltonian can be
written as

$\displaystyle H$
$\displaystyle=\sum_{i,j}^{L_{y},L_{x}}J(A_{i,j}^{\dagger}A_{i,j+1}+B_{i,j}^{\dagger}B_{i,j+1})+t_{1}A_{i,j}^{\dagger}B_{i,j}$
(12)
$\displaystyle+t_{1}A_{i,j}^{\dagger}B_{i-1,j}+t_{2}A_{i,j+1}^{\dagger}(B_{i,j}+B_{i-1,j})+H.c.,$

where $A_{i,j}^{\dagger}$ ($B_{i,j}^{\dagger}$) and $A_{i,j}$ ($B_{i,j}$) are
the fermionic creation and annihilation operators of the odd (even) quantum
wire $i$ at lattice site $j$, respectively. The number of odd (even) quantum
wires is $L_{y}$ and the length of the quantum chain is $L_{x}$. The primitive
lattice vectors are defined as $a_{1}=(1,0)$ and $a_{2}=(0,1)$. For the
reciprocal lattice, the corresponding primitive lattice vectors are
$b_{1}=2\pi(1,0)$ and $b_{2}=2\pi(0,1)$. The Hamiltonian in momentum space can
be written as

$H=\left(\begin{array}[c]{cc}2J\cos
k_{x}&(1+e^{ik_{y}})(t_{1}+t_{2}e^{ik_{x}})\\\
(1+e^{-ik_{y}})(t_{1}+t_{2}e^{-ik_{x}})&2J\cos k_{x}\end{array}\right)$ (13)

Figure 4: (a) The phase diagram of 2D quantum wire array. Region I (II)
represents that the system is topological metal with (without) edge states.
(b) Energy spectra with system size $L_{x}=100$, $t_{2}=1$, and $t_{1}=J=0.5$
under open boundary condition in $x$ direction and periodic boundary condition
in $y$ direction. The red dots in the energy band gap represent the
topological nontrivial edge states. (c) The spatial distribution of a
topological nontrivial edge state under open boundary condition in both
directions. Other parameters are $L_{x}=2L_{y}=30$, $t_{2}=1$, and
$t_{1}=J=0.5$.

The Hamiltonian obeys the inversion symmetry
$\mathcal{I}H(\mathbf{k})\mathcal{I}^{-1}=H(-\mathbf{k})$ similar to the
three-leg ladder case, where inversion symmetry matrix
$\mathcal{I}=\sigma_{x}$ in terms of Pauli matrices. According to the values
of the topological invariant and behaviors of the topological edge states, the
phase diagram can be described as shown in Fig.4a. In region I (II), the
system is in topological metal phase with (without) the topological edge
states. For the interchain couplings $t_{1}>t_{2}$, the system enters into the
trivial normal metal phase. To show the topological edge states in region I,
we plot the energy spectrum under open boundary condition in $x$ direction in
Fig.4b. Clearly, the edge states (red dots) appear in the energy band gap for
some momentum. The spatial distribution of a topological nontrivial edge state
is shown Fig.4c.

Like the three-leg ladder case, the intrachain hopping $J$ modifies the energy
spectra but not the corresponding eigenfunctions of the bulk Hamiltonian in
Eq.13. Thus, the topological property of topological metal is not changed and
the energy gap closing could occur without bound touching because of the
indirect gap. However, in a finite system, the amplitude of intrachain hopping
could affect the spatial profile of the topological nontrivial edge states,
which become delocalized for the strong intrachain hopping $J$. For the
topological region II in Fig.4a, numerical calculations reveal that the time-
averaged mean displacement in the $x$ direction approaches to the value 0.5,
which indicates that the system enters the topological metal phase without
edge states.

##  IV Summary and outlook

In short, we use a three-leg ladder and 2D lattice spinless quantum wire
arrays sliding against each other as paradigmatic examples and demonstrate
that alternated slide construction provides an elegant mechanism toward
engineering topological metals. Remarkably, there is a topological metal
without nontrivial edge states. The strategy of alternated slide construction
could be experimentally confirmed in different systems such as photonic
waveguide arrays, photonic crystals, topoelectrical circuits, or coupled
acoustic resonators.

Finally, we would like to point out that the alternated slide construction can
be a specific way to realize other topological phases of matters. For
instance, the building-blocks could be extended to spin–orbit coupling (gap
and gapless) systems, and topological chains (such as SSH model, Kitaev
chain[56], and AAH chain). Different types of topological insulators and
superconductors can be constructed. On the other hand, the easily alternated
slide construction also applies to multilayer systems including graphene moiré
superlattices.

_Acknowledgements._ – We thank Weiyin Deng for helpful discussions. This work
was supported by the National Natural Science Foundation of China (Grants No.
12074101, and 11604081). Z.W. Z. is also sponsored by the Natural Science
Foundation of Henan (Grant No. 212300410040).

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##  Appendix A The energy dispersions of three-leg ladder

Figure 5: Band structure of the three-leg ladder with intrachain couplings
$t_{1}=0.5$, $t_{2}=1$ under periodic boundary condition. The blue line
indicates the energy band of Hamiltonian $h_{1}$ and the red lines stand for
the two energy band of Hamiltonian $h_{2}$. (a) Intrachain hopping $J=0.2$
case. (b) $J=0.5$. (c) $J=1.2$.

In this appendix, we investigate the energy structure for three-leg ladder in
topological metal phases. Fig.5 show the three energy dispersion curves
$E_{1}$ and $E_{\pm}$ (Eqs. 5 and 6 in main text) for three different
intrachain hoppings $J$ vs the Bloch wave number $k_{x}$ in momentum space
when the interchain couplings $t_{1}=0.5$ and $t_{2}=1$. The blue line
indicates the energy band $E_{1}$ of subspace Hamiltonian $h_{1}$ and the red
lines stand for the two energy bands $E_{\pm}$ of subspace Hamiltonian
$h_{2}$. When the intrachain hopping $J$ ($<\sqrt{2}/2$) is small, the energy
gap of subsystem $h_{2}$ is open. The subsystem changes from a direct bandgap
insulator to an indirect bandgap insulator without phase transition (see
Fig.5a-b). The entire system is still topological metal with nontrivial edge
states. As the intrachain hopping $J$ increases, the energy gap of Hamiltonian
$h_{2}$ could be closed. However, the two energy bands of $h_{2}$ do not touch
due to the indirect nature of the bandgap (see Fig.5c). As a consequence, in a
finite system, the two degenerate edge states of Hamiltonian $h_{2}$ would
assimilate into the bulk bands and become delocalize. Thus, the conventional
bulk-boundary correspondence breaks down. The whole system enters into the
topological metal phase without the nontrivial edge states.


# Hybrid rank-1 and rank-2 U(1) lattice gauge theory, the F3 model,
and its effective field theory

Jintae Kim  Department of Physics, Sungkyunkwan University, Suwon 16419, Korea
Yun-Tak Oh  Department of Physics, Korea Advanced Institute of Science and
Technology, Daejeon 34141, Republic of Korea  Division of Display and
Semiconductor Physics, Korea University, Sejong 30019, Korea  Jung Hoon Han
<EMAIL_ADDRESS>Department of Physics, Sungkyunkwan University, Suwon
16419, Korea

###### Abstract

A number of exactly solvable spin models, including the Kitaev toric code in
two and three dimensions and the X-cube model in three dimensions, can be
related to their respective parent lattice gauge theories (LGT) through the
mathematical process of ‘Higgsing’. Field theories of the low-energy
excitations of these spin models can be developed subsequently, building upon
the symmetry of the parent LGTs. Recently, two of the present authors proposed
a variant of the three-dimensional toric code which we now call the F3 model,
whose elementary excitations consist of freeon and fluxon excitations of the
three-dimensional toric code and fracton excitations of the X-cube model. In
this work, we identify the parent LGT of the F3 model as the hybrid rank-1 and
rank-2 U(1) LGT, and develop the corresponding field theory. The resulting
Lagrangian of the F3 model is that of a three-dimensional toric code with an
extra term, which ties the dynamics of fractons to that of fluxons. The matter
part of the effective action for the F3 model can be derived as well, by
carefully keeping track of the gauge invariance of the F3 model. Hydrodynamic
equations of motion of the quasiparticles are derived, which properly reflect
their constrained dynamics. Finally, we present a tight-binding model for the
quasiparticle motion in the F3 model.

##  I Introduction

Exactly solvable spin Hamiltonians such as toric codes (TC) in two and three
dimensions Kitaev (2003); von Keyserlingk et al. (2013); Freedman and Meyer
(2001) and the X-cube model in three dimensions Vijay et al. (2016); Slagle
and Kim (2017); Nandkishore and Hermele (2019) have been studied extensively.
The multi-spin interactions that define various exactly solvable spin models
seem at first novel, but in fact have well-defined roots in their
corresponding parent lattice gauge theories (LGT) Ma et al. (2018); Bulmash
and Barkeshli (2018); Tagliacozzo et al. (2013). The mutually commuting
operators in the LGT - also known as generators - become, through Higgsing,
the stabilizer operators in the spin Hamiltonian. Various rank-1 and rank-2
U(1) LGTs and their Higgsed spin models have been analyzed, and it was shown
that the parent theory of the TC is the rank-1 U(1) LGT, while that of the
X-cube model is the rank-2 U(1) LGT Bulmash and Barkeshli (2018). The
elementary excitations of the toric code are the electric and magnetic
quasiparticles exhibiting mutual anyonic statistics, while those of the X-cube
model are the fractons and lineons. Fractons are immobile but deconfined in
the sense that a single fracton cannot move without extra energy cost, but a
four-fracton cluster can be expanded arbitrarily without extra energy. Lineons
are subdimensional quasiparticles that can only move freely along a specific
axis and are also deconfined Vijay et al. (2016).

In an attempt to extend the realm of exactly solvable spin models, two of the
present authors proposed a model Kim and Han (2021) exhibiting features akin
to those of the three-dimensional toric code (3DTC) as well as the X-cube
model. The new model is obtained from the 3DTC by including an additional
twelve-spin interaction which, in the context of the X-cube model, acts as a
fracton operator. Excitations in this new model consist, at first sight, of
the two kinds of elementary excitations of the 3DTC, augmented by the fracton
excitation similar to that in the X-cube model. In a previous publication Kim
and Han (2021) these excitations were dubbed freeons, fluxons, and fractons,
respectively, and hence we will call the overall model ‘F3 model’. It was also
shown Kim and Han (2021) that the model is exactly solvable only for the
$\mathbb{Z}_{3}$ Hilbert space at the links of the lattice, and the ground
state degeneracy (GSD) of the model is $3^{3}$, same as the GSD of the
$\mathbb{Z}_{3}$ 3DTC. In other words, the F3 model remains in the same phase
as the 3DTC despite the addition of the local fracton operators. A physical
way to understand this is that fractons that are deconfined in the X-cube
model become confined in the F3 model because fluxons act as ‘gluons’ between
fractons with energy that increases linearly with separation. The confinement
of fractons in the F3 model is an indication that the low-energy field theory
of the F3 model is more likely to resemble that of 3DTC than that of the
X-cube model. We will show that this is indeed the case.

Neither the parent LGT of the F3 model nor its low-energy field theory was
properly understood in the previous work Kim and Han (2021). We show in this
paper that the F3 model is the descendant of the hybrid rank-1 and rank-2 U(1)
LGTs. The field theory of the F3 model is constructed accordingly. In Sec. II
we review the rank-1 and rank-2 U(1) LGTs, their descendant exactly solvable
spin models, and the corresponding effective field theories (EFTs). Although
the discussions in this section overlap with several works in the past, we
find it useful to collect known results and present them in a unified fashion,
thus paving the way for the development of EFT of the F3 model. Some new
insights as well as new results about the toric code and X-cube model in three
dimensions are obtained. In Sec. III we adopt the strategies developed in Sec.
II and apply them to the F3 model. The parent LGT of the F3 model is properly
identified and is shown to lead to the F3 stabilizer Hamiltonian upon
Higgsing. The gauge transformation properties of the various gauge fields are
clarified and subsequently exploited to derive the field theories of the
matter fields and the gauge fields pertinent to the F3 model. Section IV gives
the summary. Several technical themes are delegated to the appendices.

##  II Review of Rank-1 and Rank-2 U(1) LGT and their descendants

We review the rank-1 and rank-2 U(1) LGT, their descendant exactly solvable
spin models, and the corresponding EFTs.

###  II.1 Rank-1 U(1) LGT and its descendants

The rank-1 U(1) LGT has the vector gauge fields $A_{i}^{a}$ residing on the
$(i,i+\hat{a})$ links of the lattice, where $a=x,y,z$. The canonically
conjugate electric fields $E_{i}^{a}$ satisfy the commutation relation
$[A_{i}^{a},E_{j}^{b}]=i\delta_{ij}\delta_{ab}$. The gauge transformation of
$A_{i}^{a}$ and $E_{i}^{a}$ is generated as

$\displaystyle A_{i}^{a}\rightarrow U^{\dagger}_{A}A_{i}^{a}U_{A}$
$\displaystyle=A_{i}^{a}+f_{i+\hat{a}}-f_{i},$ $\displaystyle\sim
A_{i}^{a}+\partial_{a}f_{i}$ $\displaystyle E_{i}^{a}\rightarrow
U_{E}^{\dagger}E_{i}^{a}U_{E}$
$\displaystyle=E_{i}^{a}+\sum_{bc}\epsilon_{abc}(g_{i}^{c}-g_{i-\hat{b}}^{c})$
$\displaystyle\sim E_{i}^{a}+\sum_{bc}\epsilon_{abc}\partial_{b}g_{i}^{c},$
(1)

where the unitary matrices $U_{A}$ and $U_{E}$ are

$\displaystyle
U_{A}=e^{i\sum_{j}f_{j}({\bm{\nabla}}\cdot{\bm{E}})_{j}},~{}~{}~{}U_{E}=e^{i\sum_{j}{\bm{g}}_{j}\cdot{\bm{B}}_{j}}$
(2)

for arbitrary functions $f_{i}$ and $g_{i}^{a}$. The spatial indices $a$, $b$,
$c$ run over $x,y,z$. The lattice divergence of the electric field and the
magnetic field operator, called generators, are defined by

$\displaystyle({\bm{\nabla}}\cdot{\bm{E}})_{i}$
$\displaystyle\equiv\sum_{a}(E_{i}^{a}-E_{i-\hat{a}}^{a})\sim\sum_{a}\partial_{a}E_{i}^{a}$
$\displaystyle B_{i}^{a}$
$\displaystyle\equiv\sum_{bc}\epsilon_{abc}(A_{i}^{b}-A_{i+\hat{c}}^{b})\sim\sum_{bc}\epsilon_{abc}\partial_{b}A_{i}^{c}.$
(3)

Importantly, we have the commutativity of the two generators

$\displaystyle\left[\left({\bm{\nabla}}\cdot{\bm{E}}\right)_{i},B_{j}^{a}\right]=0,$
(4)

following from their definitions.

The Hamiltonian produced from Higgsing such rank-1 U(1) LGT is precisely the
3DTC. The Higgsing is given by

$\displaystyle X_{a,i}$ $\displaystyle=\exp(-2\pi iE_{i}^{a}/p),$
$\displaystyle Z_{a,i}$ $\displaystyle=\exp(iA_{i}^{a})$ (5)

for $a=1,~{}2,~{}3$ corresponding to $a=x,~{}y,~{}z$, respectively. The
generalized Pauli operators $X,Z$ obey the commutation algebra

$\displaystyle ZX=\omega XZ~{}~{}(\omega=e^{2\pi i/p})$ (6)

for integer $p\geq 2$. The $\mathbb{Z}_{p}$ 3DTC von Keyserlingk et al. (2013)
is defined as

$\displaystyle H_{\rm 3DTC}$
$\displaystyle=-\sum_{i}(\mathcal{A}_{i}+\mathcal{C}_{i}^{x}+\mathcal{C}_{i}^{y}+\mathcal{C}_{i}^{z}),$
(7)

where

$\displaystyle\mathcal{A}_{i}$
$\displaystyle=\frac{1}{p}\sum_{j=0}^{p-1}(a_{i})^{j}~{}~{}~{}~{}~{}~{}~{}\mathcal{C}_{i}^{a}=\frac{1}{p}\sum_{j=0}^{p-1}(c_{i}^{a})^{j}$
$\displaystyle a_{i}$
$\displaystyle=X_{1,i}X_{1,i-\hat{x}}^{-1}X_{2,i}X_{2,i-\hat{y}}^{-1}X_{3,i}X_{3,i-\hat{z}}^{-1}$
$\displaystyle c_{i}^{x}$
$\displaystyle=Z_{2,i}Z_{3,i+\hat{y}}Z^{-1}_{2,i+\hat{z}}Z^{-1}_{3,i}$
$\displaystyle c_{i}^{y}$
$\displaystyle=Z_{3,i}Z_{1,i+\hat{z}}Z^{-1}_{3,i+\hat{x}}Z^{-1}_{1,i}$
$\displaystyle c_{i}^{z}$
$\displaystyle=Z_{1,i}Z_{2,i+\hat{x}}Z^{-1}_{1,i+\hat{y}}Z^{-1}_{2,i}.$ (8)

Electric quasiparticle is the excitation of $A_{i}$ while magnetic
quasiparticles labeled as $m_{a}$ ($a=x,y,z$) are the excitations of
$C_{i}^{a}$.

The EFT for 3DTC is the well-known BF theory, but the treatment of the EFT for
the matter fields, i.e., electric and magnetic excitations, is not given with
equal care in the literature. Here we derive the EFTs of both matter and gauge
fields for the 3DTC and derive the corresponding continuity equations.
Furthermore, a tight-binding realization of the $e$-particle and $m$-particle
is given by invoking the gauge principle.

The effective action of the $e$ quasiparticles is the usual one,

$\displaystyle\mathcal{L}_{e}$
$\displaystyle=i\psi_{e}^{\dagger}\partial_{t}\psi_{e}-\frac{1}{2m_{e}}|{\bm{D}}_{e}\psi_{e}|^{2},$
(9)

with ${\bm{D}}_{e}\psi_{e}=({\bm{\nabla}}-i{\bm{A}})\psi_{e}$. Its form is
dictated by the requirement of invariance under the gauge transformation
$\psi_{e}\rightarrow e^{if}\psi_{e}$ and
${\bm{A}}\rightarrow{\bm{A}}+{\bm{\nabla}}f$. The mass $m_{e}$ is introduced
by hand. The EFT of the $m_{a}$ quasiparticles is more involved,

$\displaystyle\mathcal{L}_{m}$
$\displaystyle=i{\bm{\psi}}_{m}^{\dagger}\cdot\partial_{t}{\bm{\psi}}_{m}-\frac{1}{2m_{m}}\sum_{a<b}|{\bm{D}}_{m}^{ab}\psi_{m}^{a}\psi_{m}^{b}|^{2},$
(10)

with ${\bm{\psi}}_{m}=(\psi_{m}^{x},\psi_{m}^{y},\psi_{m}^{z})$ representing
the three species of magnetic quasiparticles. The covariant derivative
$D_{m}^{ab}$ must be chosen to comply with the rule of gauge invariance, as
follows (no sum on $a,b$):

$\displaystyle{\bm{D}}_{m}^{ab}\psi_{m}^{a}\psi_{m}^{b}$
$\displaystyle=\psi_{m}^{a}\partial_{a}\psi_{m}^{b}-\psi_{m}^{b}\partial_{b}\psi_{m}^{a}$
$\displaystyle-i\sum_{c}\epsilon_{abc}E^{c}\psi_{m}^{a}\psi_{m}^{b}.$ (11)

The gauge invariance of the action under $\psi_{m}^{a}\rightarrow
e^{ig^{a}}\psi_{m}^{a}$ and $E^{a}\rightarrow
E^{a}+\sum_{bc}\epsilon_{abc}\partial_{b}g^{c}$ can be readily checked. The
action is quartic in the fields and implies that the quasiparticles are
inherently interacting. It is well-known that magnetic excitations in the 3DTC
always form loops von Keyserlingk et al. (2013), which one can now understand
as a consequence of the strongly interacting nature of the magnetic
quasiparticles.

Some conservation laws of electric and magnetic quasiparticles follow from the
matter action:

$\displaystyle\partial_{t}\rho^{e}+{\bm{\nabla}}\cdot{\bm{j}}^{e}=0,$
$\displaystyle\partial_{t}\rho^{m_{a}}+\sum_{b}\partial_{b}[j^{m}]^{ab}=0.$
(12)

Explicit expressions of the charge and current densities in terms of the
matter fields can be found in the Appendix A. The electric part of the
conservation law is nothing out of the ordinary. For the magnetic part, a
vector of charges $\bm{\rho}^{m}=(\rho^{m_{x}},\rho^{m_{y}},\rho^{m_{z}})$ and
the corresponding antisymmetric tensor current $[j^{m}]^{ab}$ give rise to an
unusual form of continuity equation. In fact, one can write the current tensor
in a vector form ${\bm{j}}=([j^{m}]^{yz},[j^{m}]^{zx},[j^{m}]^{xy})$,
introduce the vector charge density
${\bm{\rho}}^{m}=(\rho^{m_{x}},\rho^{m_{y}},\rho^{m_{z}})$, and simplify the
continuity equation of the magnetic particles:

$\displaystyle\partial_{t}{\bm{\rho}}^{m}+{\bm{\nabla}}\times{\bm{j}}=0.$ (13)

A natural corollary of this equation is that
$\partial_{t}({\bm{\nabla}}\cdot{\bm{\rho}}^{m})=0$, consistent with the fact
that magnetic excitations in 3DTC only form closed loops. The construction of
EFTs for the matter parts of the 3DTC and the corresponding continuity
equations are not treated in the literature, and we presented them in some
detail here.

The Lagrangian reflecting the commutation relation of the gauge fields
$(A_{i}^{a},E_{j}^{b})$ and the conservation laws in Eq. (12) is the three-
dimensional BF theory. The action can be cast in the four-vector continuum
notation ($\epsilon_{0123}=+1$) Hansson et al. (2004):

$\displaystyle\mathcal{L}_{\text{BF}}=\frac{p}{2\pi}\epsilon_{\alpha\beta\mu\nu}\frac{1}{2}E^{\alpha\beta}\partial_{\mu}A^{\nu}-A\cdot
j^{e}-\frac{1}{2}E^{\alpha\beta}{[j^{m}]}^{\alpha\beta},$ (14)

where $A$ and $j^{e}$ are four-vectors, $A=(A^{0},{\bm{A}})$,
$j^{e}=(\rho^{e},{\bm{j}}^{e})$, and $E^{\alpha\beta}$ and
${[j^{m}]}^{\alpha\beta}$ are antisymmetric tensors whose definitions are
written out in Appendix A. The four-current densities of electric and magnetic
quasiparticles are denoted $j^{e}$ and $[j^{m}]^{\mu\nu}$, respectively. The
commutation relation and the gauge transformations of the $A^{a}$ and $E^{ab}$
are

$\displaystyle[A^{a}({\bm{r}}),$ $\displaystyle
E^{bc}({\bm{r}^{\prime}})]=\frac{2\pi
i}{p}\epsilon_{abc}\delta^{3}({\bm{r}}-{\bm{r}^{\prime}})$
$\displaystyle{\bm{A}}({\bm{r}})$ $\displaystyle\rightarrow
U_{A}^{\dagger}{\bm{A}}({\bm{r}})U_{A}={\bm{A}}({\bm{r}})+{\bm{\nabla}}f({\bm{r}})$
$\displaystyle E^{ab}({\bm{r}})$ $\displaystyle\rightarrow
U_{E}^{\dagger}E^{ab}({\bm{r}})U_{E}=E^{ab}({\bm{r}})-\partial_{a}g^{b}({\bm{r}})+\partial_{b}g^{a}({\bm{r}}),$
(15)

and the gauge transformations of the temporal components $A^{0}$ and $E^{a0}$
are

$\displaystyle A^{0}(\bm{r})$ $\displaystyle\rightarrow
A^{0}(\bm{r})+\partial_{t}f(\bm{r})$ $\displaystyle E^{a0}(\bm{r})$
$\displaystyle\rightarrow
E^{a0}(\bm{r})-\partial_{a}g^{0}(\bm{r})+\partial_{t}g^{a}(\bm{r}).$ (16)

Accordingly, $A\cdot j^{e}$ and
$\frac{1}{2}E^{\alpha\beta}{[j^{m}]}^{\alpha\beta}$ transform under the gauge
transformations as

$\displaystyle A\cdot j^{e}$ $\displaystyle\rightarrow A\cdot
j^{e}+\sum_{\mu}\partial_{\mu}f[j^{e}]^{\mu}$
$\displaystyle\frac{1}{2}E^{\alpha\beta}{[j^{m}]}^{\alpha\beta}$
$\displaystyle\rightarrow\frac{1}{2}E^{\alpha\beta}{[j^{m}]}^{\alpha\beta}+\frac{1}{2}(-\partial_{\alpha}g^{\beta}+\partial_{\beta}g^{\alpha})[j^{m}]^{\alpha\beta}.$
(17)

Applying integration by parts to the two equations in Eq. (17) leads to the
conservation laws of $e$ and $m$ quasiparticles as well as the identity
$\bm{\nabla}\cdot\bm{\rho}^{m}=0$. The BF action for the 3DTC is well-known,
but the approach that starts from the explicit construction of the matter
Lagrangian followed by the identification of various conservation laws is new,
and we outline the derivation in Appendix A.

We can construct a lattice version of the matter field action of the 3DTC by
following the gauge principle. For the electric quasiparticle, the usual
Peierls substitution

$\displaystyle[\psi_{e}]_{i+\hat{a}}^{\dagger}[\psi_{e}]_{i}e^{iA^{a}_{i}}$
(18)

yields gauge-invariant hopping of the electric operator $\psi_{e}$. On the
other hand, the gauge transformations of the ${\bm{E}}$ field and magnetic
quasiparticle operators
${\bm{\psi}}_{m}=(\psi_{m}^{x},\psi_{m}^{y},\psi_{m}^{z})$ on the lattice are
given by

$\displaystyle E_{i}^{ab}$ $\displaystyle\rightarrow
E_{i}^{ab}-g^{b}_{i+\hat{a}}+g^{b}_{i}+g^{a}_{i+\hat{b}}-g^{a}_{i},$
$\displaystyle[{\bm{\psi}}_{m}]_{i}$
$\displaystyle\rightarrow(e^{ig^{x}_{i}}[\psi_{m}^{x}]_{i},e^{ig^{y}_{i}}[\psi_{m}^{y}]_{i},e^{ig^{z}_{i}}[\psi_{m}^{z}]_{i}),$
(19)

and the corresponding gauge-invariant hopping terms must take the form

$\displaystyle[\psi_{m}^{x}]_{i+\hat{y}}^{\dagger}[\psi_{m}^{y}]_{i}^{\dagger}[\psi_{m}^{x}]_{i}[\psi_{m}^{y}]_{i+\hat{x}}e^{iE^{xy}_{i}},$
$\displaystyle[\psi_{m}^{y}]_{i+\hat{z}}^{\dagger}[\psi_{m}^{z}]_{i}^{\dagger}[\psi_{m}^{y}]_{i}[\psi_{m}^{z}]_{i+\hat{y}}e^{iE^{yz}_{i}},$
$\displaystyle[\psi_{m}^{z}]_{i+\hat{x}}^{\dagger}[\psi_{m}^{x}]_{i}^{\dagger}[\psi_{m}^{z}]_{i}[\psi_{m}^{x}]_{i+\hat{z}}e^{iE^{zx}_{i}}$
(20)

and their Hermitian conjugates. No single-particle hopping is allowed for
magnetic particles; only their correlated hopping involving two species of $m$
quasiparticles at a time is allowed under the gauge principle.

###  II.2 Rank-2 U(1) LGT and its descendants

The particular kind of rank-2 U(1) LGT we are interested in utilizes symmetric
rank-2 gauge fields $A_{i}^{ab}=A_{i}^{ba}$ $(a,b=x,y,z)$ obeying the
hollowness condition $A_{i}^{aa}=0$ (no sum on $a$) for the diagonal
components. The conjugate fields $E_{i}^{ab}$ satisfying the commutation
$[A_{i}^{ab},E_{j}^{ab}]=i\delta_{ij}$ share the same symmetry and hollowness
properties: $E_{i}^{ba}=E_{i}^{ab}$ and $E_{i}^{aa}=0$ Bulmash and Barkeshli
(2018). The transformation rule for the gauge fields are

$\displaystyle A_{i}^{ab}$ $\displaystyle\rightarrow
U^{\dagger}_{A}A_{i}^{ab}U_{A}$
$\displaystyle=A^{ab}_{i}-f_{i-\hat{a}-\hat{b}}+f_{i-\hat{a}}+f_{i-\hat{b}}-f_{i}$
$\displaystyle\sim A^{ab}_{i}-\partial_{a}\partial_{b}f_{i},$
$\displaystyle{E}_{i}^{ab}$ $\displaystyle\rightarrow
U_{E}^{\dagger}{E}_{i}^{ab}U_{E}$
$\displaystyle={E}_{i}^{ab}+\sum_{c}\epsilon_{abc}(g^{b}_{i+\hat{c}}-g_{i}^{b}-g^{a}_{i+\hat{c}}+g_{i}^{a})$
$\displaystyle\sim{E}_{i}^{ab}+\sum_{c}\epsilon_{abc}(\partial_{c}g^{b}_{i}-\partial_{c}g^{a}_{i}),$
(21)

where the unitary operators are defined by

$\displaystyle U_{A}=e^{i\sum_{j}f_{j}({\bf
DE})_{j}},~{}~{}U_{E}=e^{i\sum_{j}{\bm{g}}_{j}\cdot{\bm{B}}_{j}}$ (22)

for arbitrary functions $f_{i}$ and $\bm{g}_{i}$. Generators of the gauge
transformations are

$\displaystyle({\bf DE})_{i}$ $\displaystyle\equiv
E_{i+\hat{x}+\hat{y}}^{xy}-E_{i+\hat{x}}^{xy}-E_{i+\hat{y}}^{xy}+E_{i}^{xy}$
$\displaystyle+E_{i+\hat{y}+\hat{z}}^{yz}-E_{i+\hat{y}}^{yz}-E_{i+\hat{z}}^{yz}+E_{i}^{yz}$
$\displaystyle+E_{i+\hat{x}+\hat{z}}^{xz}-E_{i+\hat{x}}^{xz}-E_{i+\hat{z}}^{xz}+E_{i}^{xz}$
$\displaystyle\sim\partial_{x}\partial_{y}E_{i}^{xy}+\partial_{y}\partial_{z}E^{yz}_{i}+\partial_{z}\partial_{x}E_{i}^{xz},$
$\displaystyle B_{i}^{a}$
$\displaystyle\equiv\sum_{bc}\epsilon_{abc}(A_{i}^{ba}-A_{i-\hat{c}}^{ba})$
$\displaystyle\sim\sum_{bc}\epsilon_{abc}\partial_{c}A_{i}^{ba}.$ (23)

One can easily prove the commutativity $[({\bf DE})_{i},B_{j}^{a}]=0$.

The Hamiltonian produced from Higgsing the rank-2 U(1) LGT is the X-cube
model. Before introducing the Higgsing formula, we need to redefine the
notation of tensor fields. There are, in fact, only three independent
components in the tensors $A^{ab}$ and $E^{ab}$ and they can be mapped to
vector fields

$\displaystyle(A_{i}^{yz},A_{i}^{xz},A_{i}^{xy})$
$\displaystyle\rightarrow(A_{i}^{x},A_{i}^{y},A_{i}^{z})$
$\displaystyle(E_{i}^{yz},E_{i}^{xz},E_{i}^{xy})$
$\displaystyle\rightarrow(E_{i}^{x},E_{i}^{y},E_{i}^{z}).$ (24)

Then, the Higgsing of the rank-2 U(1) LGT is performed by ($a=x$, $y$, $z$ or
$a=1$, $2$, $3$)

$\displaystyle X_{a,i}$ $\displaystyle=\exp(iA_{i}^{a}),$ $\displaystyle
Z_{a,i}$ $\displaystyle=\exp(2\pi iE_{i}^{a}/p).$ (25)

Compared to the Higgsing formula used in Eq. (5) for the toric codes, the
roles of $X$ and $Z$ operators have been switched in Eq. (25). This is by
design and will be crucial in constructing the hybrid (F3) model. The
$\mathbb{Z}_{p}$ X-cube model Vijay et al. (2016) is defined as

$\displaystyle
H_{\text{XC}}=-\sum_{i}(\mathbb{A}_{i}^{x}+\mathbb{A}_{i}^{y}+\mathbb{A}_{i}^{z}+\mathcal{B}_{i}).$
(26)

The mutually commuting Hermitian projectors $\mathbb{A}_{i}^{a}$ and
$\mathcal{B}_{i}$ are represented as

$\displaystyle\mathbb{A}_{i}^{a}$
$\displaystyle=\frac{1}{p}\sum_{j=0}^{p-1}(\mathsf{a}_{i}^{a})^{j}$
$\displaystyle\mathcal{B}_{i}$
$\displaystyle=\frac{1}{p}\sum_{j=0}^{p-1}(b_{i})^{j},$ (27)

using a set of mutually commuting unitary operators
$(\mathsf{a}_{i}^{x},\mathsf{a}_{i}^{y},\mathsf{a}_{i}^{z},b_{i})$ in the
$p$-dimensional Hilbert space of spins constructed as

$\displaystyle\mathsf{a}_{i}^{x}=$ $\displaystyle
X_{3,i}X_{3,i-\hat{y}}^{-1}X_{2,i}^{-1}X_{2,i-\hat{z}},$
$\displaystyle\mathsf{a}_{i}^{y}=$ $\displaystyle
X_{1,i}X_{1,i-\hat{z}}^{-1}X_{3,i}^{-1}X_{3,i-\hat{x}},$
$\displaystyle\mathsf{a}_{i}^{z}=$ $\displaystyle
X_{2,i}X_{2,i-\hat{x}}^{-1}X_{1,i}^{-1}X_{1,i-\hat{y}},$ $\displaystyle
b_{i}=$ $\displaystyle
Z_{1,i+\hat{y}+\hat{z}}Z_{1,i+\hat{y}}^{-1}Z_{1,i+\hat{z}}^{-1}Z_{1,i}$
$\displaystyle\times$ $\displaystyle
Z_{2,i+\hat{x}+\hat{z}}Z_{2,i+\hat{x}}^{-1}Z_{2,i+\hat{z}}^{-1}Z_{2,i}$
$\displaystyle\times$ $\displaystyle
Z_{3,i+\hat{x}+\hat{y}}Z_{3,i+\hat{x}}^{-1}Z_{3,i+\hat{y}}^{-1}Z_{3,i}.$ (28)

In particular, $\mathsf{a}_{i}^{x}\mathsf{a}_{i}^{y}\mathsf{a}_{i}^{z}=1$
following from $B_{i}^{x}+B_{i}^{y}+B_{i}^{z}=0$ of the parent LGT.

The two types of elementary excitations in the X-cube model are called
$l_{a}$-quasiparticles ($a=x,y,z$) and fractons. The $l_{a}$-quasiparticles
are the excitations of the $\mathbb{A}_{i}^{a}$ operators while fractons are
those of the $\mathcal{B}_{i}$ operator defined in Eq. (27). The integer-
valued charges of $l_{a}$-quasiparticles and fractons are denoted by
$\rho^{l_{a}}$ and $\rho^{f}$, and are respectively determined as the
eigenvalues of the operators

$\displaystyle\mathbb{A}_{i}^{a}(n)$
$\displaystyle=\frac{1}{p}\sum_{j=0}^{p-1}(\omega^{-n}\mathsf{a}_{i}^{a})^{j},$
$\displaystyle\mathcal{B}_{i}(n)$
$\displaystyle=\frac{1}{p}\sum_{j=0}^{p-1}(\omega^{-n}b_{i})^{j}.$ (29)

The eigenstate $\ket{\psi_{l_{a}}}$ with a $l_{a}$ quasiparticle is defined as
$\mathbb{A}_{i}^{a}(\rho^{l_{a}})|\psi_{l_{a}}\rangle=|\psi_{l_{a}}\rangle$
and $\mathcal{B}_{i}(\rho^{f})|\psi_{f}\rangle=|\psi_{f}\rangle$ defines the
eigenstate with a fracton excitation. In other words,
$\ket{\psi_{l_{a}}}$($\ket{\psi_{f}}$) is an eigenstate of
$\mathsf{a}_{i}^{a}$($b_{i}$) with eigenvalue
$\omega^{\rho^{l_{a}}}$($\omega^{\rho^{f}}$).

The term lineon is used to refer to the excitation in the X-cube model that
can move freely only in a particular direction. They can be defined in terms
of the $l_{a}$-quasiparticles just defined, as the eigenstates satisfying

$\displaystyle(x):$
$\displaystyle\mathbb{A}_{i}^{x}(0)|\psi_{x}\rangle=\mathbb{A}_{i}^{y}(-\rho^{x})|\psi_{x}\rangle=\mathbb{A}_{i}^{z}(\rho^{x})|\psi_{x}\rangle=\ket{\psi_{x}},$
$\displaystyle(y):$
$\displaystyle\mathbb{A}_{i}^{x}(\rho^{y})|\psi_{y}\rangle=\mathbb{A}_{i}^{y}(0)|\psi_{y}\rangle=\mathbb{A}_{i}^{z}(-\rho^{y})|\psi_{y}\rangle=\ket{\psi_{y}},$
$\displaystyle(z):$
$\displaystyle\mathbb{A}_{i}^{x}(-\rho^{z})|\psi_{z}\rangle=\mathbb{A}_{i}^{y}(\rho^{z})|\psi_{z}\rangle=\mathbb{A}_{i}^{z}(0)|\psi_{z}\rangle=\ket{\psi_{z}}.$
(30)

The $x$, $y$, $z$ lineons denoted by $(x),(y),(z)$ with charges
$\rho^{x},\rho^{y},\rho^{z}$ are respectively defined as having the
eigenvalues $(0,-\rho^{x},\rho^{x})$, $(\rho^{y},0,-\rho^{y})$, and
$(-\rho^{z},\rho^{z},0)$ of the $l_{a}$-quasiparticles. In other words,
lineons are composite excitations consisting of a pair of
$l_{a}$-quasiparticles. Hence the $l_{a}$-quasiparticles are the more
fundamental excitations and we proceed to construct field theories for them,
with the lineon dynamics emerging as a natural by-product.

The matter field Lagrangian of fractons and $l_{a}$-quasiparticles can be
derived as

$\displaystyle\mathcal{L}_{f}$
$\displaystyle=i\psi_{f}^{\dagger}\partial_{t}\psi_{f}-\frac{1}{2m_{f}}\sum_{a<b}|{\bm{D}}_{f}^{ab}\psi_{f}|^{2},$
$\displaystyle\mathcal{L}_{l}$
$\displaystyle=i{\bm{\psi}}_{l}^{\dagger}\cdot\partial_{t}{\bm{\psi}}_{l}-\frac{1}{2m_{m}}\sum_{a}|{\bm{D}}_{l}^{a}{\bm{\psi}}_{l}|^{2},$
(31)

where $\psi^{f}$ and
${\bm{\psi}}_{l}=(\psi_{l}^{x},\psi_{l}^{y},\psi_{l}^{z})$ represent the
fracton and the three species of $l_{a}$-quasiparticle fields, respectively.
The covariant derivatives are given as

$\displaystyle{\bm{D}}_{f}^{ab}\psi_{f}$
$\displaystyle=\psi_{f}\partial_{a}\partial_{b}\psi_{f}-\partial_{a}\psi_{f}\partial_{b}\psi_{f}+i\sum_{c}|\epsilon_{abc}|A^{c}\psi_{f}$
$\displaystyle{\bm{D}}_{l}^{x}{\bm{\psi}}_{l}$
$\displaystyle=\psi_{l}^{z}\partial_{x}\psi_{l}^{y}-\psi_{l}^{y}\partial_{x}\psi_{l}^{z}+iE^{x}\psi_{l}^{y}\psi_{l}^{z}$
$\displaystyle{\bm{D}}_{l}^{y}{\bm{\psi}}_{l}$
$\displaystyle=\psi_{l}^{z}\partial_{y}\psi_{l}^{x}-\psi_{l}^{x}\partial_{y}\psi_{l}^{z}+iE^{y}\psi_{l}^{x}\psi_{l}^{z}$
$\displaystyle{\bm{D}}_{l}^{z}{\bm{\psi}}_{l}$
$\displaystyle=\psi_{l}^{x}\partial_{z}\psi_{l}^{y}-\psi_{l}^{y}\partial_{z}\psi_{l}^{x}+iE^{z}\psi_{l}^{x}\psi_{l}^{y}.$
(32)

One can readily check their gauge covariance under $\psi_{f}\rightarrow
e^{if}\psi_{f}$, $\psi_{l}^{a}\rightarrow e^{ig^{a}}\psi_{l}^{a}$, as well as
$A^{ab}\rightarrow A^{ab}-\partial_{a}\partial_{b}f$ and $E^{ab}\rightarrow
E^{ab}+\sum_{c}\epsilon_{abc}(\partial_{c}g^{b}-\partial_{c}g^{a})$. Various
covariant derivatives given here, while straightforward to show, were not
derived previously.

The continuity equations of $l_{a}$-quasiparticles and fractons follow from
the matter Lagrangians (31) as

$\displaystyle\partial_{t}\rho^{f}-\sum_{a<b}\partial_{a}\partial_{b}[j^{f}]^{ab}=0$
$\displaystyle\partial_{t}\rho^{l_{a}}+\sum_{bc}\epsilon_{abc}\partial_{c}[j^{l}]^{c}=0.$
(33)

Here $\rho^{f}$ and $\rho^{l_{a}}$ are respectively the densities of fractons
and $l_{a}$-quasiparticles, and $[j^{f}]^{ab}$ and $\bm{j}^{l}$, respectively
a symmetric tensor and a vector, represent the current density of the fracton
and $l_{a}$-quasiparticles. These conservation laws, first mentioned in Ref.
Slagle and Kim, 2017, are rigorously derived from the matter field Lagrangian
in Eq. (31). Charge and current density expressions in terms of matter fields
are written in Appendix B.

The EFT of the XC model can be constructed by incorporating the conservation
laws of quasiparticles by means of Lagrange multipliers:

$\displaystyle\mathcal{L}_{\text{XC}}$
$\displaystyle=\frac{p}{2\pi}\sum_{abc}|\epsilon_{abc}|\frac{1}{2}A^{ab}\partial_{t}E^{c}-\frac{1}{2}\sum_{ab}A^{ab}[j^{f}]^{ab}-{\bm{E}}\cdot\bm{j}^{l}$
$\displaystyle+\frac{p}{2\pi}E^{a0}\Bigl{(}\sum_{bc}\epsilon_{abc}\partial_{c}A^{ab}-\frac{2\pi}{p}{\rho}^{l_{a}}\Bigl{)}$
$\displaystyle+A^{0}\Bigl{(}\frac{p}{2\pi}\sum_{abc}|\epsilon_{abc}|\frac{1}{2}\partial_{a}\partial_{b}E^{c}-\rho^{f}\Bigl{)}.$
(34)

The sum is over spatial indices. The commutation relation and the gauge
transformation of the $A^{ab}$ and $E^{a}$ in Eq. (34) are

$\displaystyle[E^{a}(\bm{r}),A^{bc}(\bm{r}^{\prime})]=|\epsilon_{abc}|\frac{2\pi
i}{p}\delta^{3}(\bm{r}-\bm{r}^{\prime})$ $\displaystyle
A^{ab}(\bm{r})\rightarrow A^{ab}(\bm{r})+\partial_{a}\partial_{b}f$
$\displaystyle E^{a}(\bm{r})\rightarrow
E^{a}(\bm{r})-\epsilon_{abc}\partial_{a}g^{c},$ (35)

We have given the derivation of the Lagrangian (34) starting from the matter
Lagrangian and the subsequent conservation laws in Appendix B. The Lagrangian
appears identical to the one given in Ref. Slagle and Kim, 2017, but there is
a subtle difference in the way the constraints are enforced. In order to
properly embed the conservation laws of $l_{a}$-quasiparticles and fractons as
well as the identity $\sum_{a}\rho^{l_{a}}=0$ in the Lagrangian, the gauge
transformations of $A^{0}$ and $E^{a0}$ must be

$\displaystyle A^{0}(\bm{r})$ $\displaystyle\rightarrow
A^{0}(\bm{r})+\partial_{t}f(\bm{r})$ $\displaystyle E^{a0}(\bm{r})$
$\displaystyle\rightarrow
E^{a0}(\bm{r})+\partial_{t}g^{a}(\bm{r})+g^{0}(\bm{r}).$ (36)

Conservation laws of $l_{a}$-quasiparticles and fractons follow from requiring
gauge invariance of the Lagrangian. In particular, the gauge transformation
involving $g^{0}$ gives the term $g^{0}\sum_{a}\rho^{l_{a}}$, which leads to
the constraint $\sum_{a}\rho^{l_{a}}=0$. 111A different method was used in
Ref. Slagle and Kim (2017) to enforce this constraint.

The lattice model for fracton fields $\psi^{f}_{i}$ can be constructed from
the gauge principle. Given that fracton fields couple to ${\bm{A}}$ which
transforms as $A_{i}^{ab}\rightarrow
A^{ab}_{i}-f_{i-\hat{a}-\hat{b}}+f_{i-\hat{a}}+f_{i-\hat{b}}-f_{i}$ [Eq.
(21)], we write down the gauge-invariant hopping of the fractons as

$\displaystyle(\psi^{f})^{\dagger}_{i-\hat{b}}(\psi^{f})^{\dagger}_{i-\hat{a}}\psi^{f}_{i-\hat{a}-\hat{b}}\psi^{f}_{i}e^{iA^{ab}_{i}}$
(37)

and its Hermitian conjugate. The fracton motion clearly preserves the center-
of-mass, or the dipole moment. Note that this is a three-dimensional extension
of the quadrupole model studied in the context of quadrupole topological
insulators Dubinkin et al. (2020); You et al. (2021).

The $l_{a}$-quasiparticles coupled to the $E^{ab}$ fields which transform as
in Eq. (21) have the gauge-invariant hopping terms

$\displaystyle(\psi^{l_{x}})_{i}^{\dagger}(\psi^{l_{y}})_{i+\hat{z}}^{\dagger}\psi^{l_{x}}_{i+\hat{z}}\psi^{l_{y}}_{i}e^{iE^{xy}_{i}},$
$\displaystyle(\psi^{l_{y}})_{i}^{\dagger}(\psi^{l_{z}})_{i+\hat{x}}^{\dagger}\psi^{l_{y}}_{i+\hat{x}}\psi^{l_{z}}_{i}e^{iE^{yz}_{i}},$
$\displaystyle(\psi^{l_{z}})_{i}^{\dagger}(\psi^{l_{x}})_{i+\hat{y}}^{\dagger}\psi^{l_{z}}_{i+\hat{y}}\psi^{l_{x}}_{i}e^{iE^{zx}_{i}}$
(38)

and their Hermitian conjugates. The first line, for instance, clearly shows
that a $z$-lineon motion is made by the coordinated motion of the $l_{x}$ and
$l_{y}$ quasiparticles in the opposite directions along the $z$-axis.

The lattice formulation of the fracton and lineon dynamics coupled to the
rank-2 gauge fields was not discussed previously, while their derivation is
quite straightforward from the gauge principle as shown above.

##  III Hybrid rank-1 and rank-2 LGT and F3 model as its descendant

We finally come to the discussion of the F3 model that we proposed in an
earlier publication. We first lay out the underlying LGT of the F3 model and
its Higgsing, and then construct the relevant effective field theory.

###  III.1 F3 model from Higgsing the hybrid rank-1 and rank-2 LGT

There are two types of terms in the X-cube model, and we ask if any of them
can be added onto the 3DTC without violating the exact solvability. One sees
readily that the addition of the magnetic field terms $\mathsf{a}_{i}^{a}$
from the X-cube model [Eq. (28)] to the 3DTC poses a problem because they in
general fail to commute with the magnetic flux operators $c_{i}^{a}$ in the
3DTC: $[\mathsf{a}_{i}^{a},c_{j}^{a^{\prime}}]\neq 0$.

The Gauss’s law operator ${\cal B}_{i}$ of the X-cube model obviously commutes
with the magnetic fields ${\cal C}_{i}^{a}$ of the 3DTC as they are built out
of the same $Z$ operators. On the other hand, the ${\cal B}_{i}$’s commute
with the Gauss’s law operator ${\cal A}_{i}$ of the 3DTC only if the local
Hilbert space dimension is $p=3$ since commuting the two operators results in
the phase $\omega^{3}$, which equals one only for $p=3$ or its multiple. This
is the compatibility condition between the Gauss’s laws of the rank-1 and
rank-2 U(1) LGTs. When the compatibility condition is fulfilled, we obtain a
new, exactly solvable spin Hamiltonian

$\displaystyle H_{\text{F3}}$
$\displaystyle=-\sum_{i}\mathcal{A}_{i}-\sum_{i}\mathcal{B}_{i}-\sum_{i,a}\mathcal{C}_{i}^{a},$
(39)

which is the F3 model Kim and Han (2021). This model is the sum of the 3DTC
($\mathcal{A}_{i}$ and $\mathcal{C}_{i}^{a}$) and the Gauss’s law
$\mathcal{B}_{i}$ of the X-cube model. The quasiparticle excitations
associated with $\mathcal{A}_{i},\mathcal{B}_{i}$ and $\mathcal{C}^{a}_{i}$
were respectively dubbed the freeon, fracton, and the fluxon, hence the F3
model. One can think of F3 as the 3DTC modified by the presence of the cube
term $-\sum_{i}\mathcal{B}_{i}$. Despite the additional term, the logical
operators of the F3 model are identical to those of the 3DTC and consequently
they share the same GSD. This observation will permeate the field-theoretic
discussion that follows as well.

We can derive the F3 model by Higgsing the gauge theory of two vector gauge
fields $\bm{A}_{i}$ and $\bm{E}_{i}$ which satisfy the commutation relation
$[A_{i}^{a},E_{j}^{b}]=i\delta_{ij}\delta_{ab}$. One can identify the three
“mutually commuting” generators, $(\bm{\nabla}\cdot\bm{A})_{i}$,
$(\bm{\nabla}\times\bm{E})_{i}$, and $({\bf DE})_{i}$, given by

$\displaystyle({\bm{\nabla}}\cdot{\bm{A}})_{i}$
$\displaystyle\equiv\sum_{a}(A_{i}^{a}-A_{i-\hat{a}}^{a})$
$\displaystyle(\bm{\nabla}\times\bm{E})_{i}$
$\displaystyle\equiv\sum_{b,c}\epsilon_{abc}(E_{i}^{b}-E_{i+\hat{c}}^{b})$
$\displaystyle({\bf DE})_{i}$ $\displaystyle\equiv
E_{i+\hat{x}+\hat{y}}^{z}-E_{i+\hat{x}}^{z}-E_{i+\hat{y}}^{z}+E_{i}^{z}$
$\displaystyle~{}~{}~{}~{}+E_{i+\hat{y}+\hat{z}}^{x}-E_{i+\hat{y}}^{x}-E_{i+\hat{z}}^{x}+E_{i}^{x}$
$\displaystyle~{}~{}~{}~{}+E_{i+\hat{x}+\hat{z}}^{y}-E_{i+\hat{x}}^{y}-E_{i+\hat{z}}^{y}+E_{i}^{y}.$
(40)

One can recognize the first two generators as those of the rank-1 LGT in Eq.
(3), except for the reversal of the roles ${\bm{A}}\leftrightarrow{\bm{E}}$.
The rank-2 U(1) generator $({\bf D}{\bf E})_{i}$ is taken from Eq. (23). In
fact, we do not have strict commutativity of all three operators since the
explicit calculation shows

$\displaystyle[({\bm{\nabla}}\cdot{\bm{A}})_{i},({\bf DE})_{j}]$
$\displaystyle=3i(\delta_{ij}+\delta_{i,j+\hat{x}+\hat{y}}+\delta_{i,j+\hat{x}+\hat{z}}$
$\displaystyle~{}~{}~{}+\delta_{i,j+\hat{y}+\hat{z}}-\delta_{i,j+\hat{x}}-\delta_{i,j+\hat{y}}$
$\displaystyle~{}~{}~{}-\delta_{i,j+\hat{z}}-\delta_{i,j+\hat{x}+\hat{y}+\hat{z}}).$
(41)

Nevertheless, we can show that the commutativity is recovered after the
Higgsing, thus justifying the use of “mutually commuting” operators in
parenthesis.

The Higgsing formula is given by

$\displaystyle X_{a,i}$ $\displaystyle=\exp(iA_{i}^{a}),$ $\displaystyle
Z_{a,i}$ $\displaystyle=\exp(2\pi iE_{i}^{a}/p).$ (42)

One can easily identify

$\displaystyle\exp((\bm{\nabla}\cdot\bm{A})_{i})$ $\displaystyle\equiv a_{i}$
$\displaystyle\exp(2\pi i({\bf DE})_{i}/p)$ $\displaystyle\equiv b_{i}$
$\displaystyle\exp(2\pi i(\bm{\nabla}\times\bm{E})^{a}_{i}/p)$
$\displaystyle\equiv c_{i}^{a}$ (43)

as the $a_{i},~{}b_{i},~{}c_{i}^{a}$ operators of the F3 model. By virtue of
Eq. (41), we see that the commutator $[({\bm{\nabla}}\cdot{\bm{A}})_{i},({\bf
DE})_{j}]$ equals $+3i,-3i$ or 0, and therefore $[a_{i},b_{j}]=0$ provided
$p=3$. That is, although the generators themselves do not commute, their
Higgsed versions do commute in the special case of $p=3$.

The gauge transformation rules of $\bm{A}_{i}$ and $\bm{E}_{i}$ are

$\displaystyle A^{a}_{i}$ $\displaystyle\rightarrow
U_{A}^{\dagger}A_{i}^{a}U_{A}$ $\displaystyle A^{x}_{i}$
$\displaystyle\rightarrow
A^{x}_{i}-f^{0}_{i-\hat{y}-\hat{z}}+f^{0}_{i-\hat{y}}+f^{0}_{i-\hat{z}}-f^{0}_{i}$
$\displaystyle~{}~{}~{}-f_{i}^{z}+f_{i-\hat{y}}^{z}+f_{i}^{y}-f_{i-\hat{z}}^{y}$
$\displaystyle\sim
A_{i}^{x}-\partial_{y}\partial_{z}f_{i}^{0}-\partial_{y}f_{i}^{z}+\partial_{z}f_{i}^{y}$
$\displaystyle A^{y}_{i}$ $\displaystyle\rightarrow
A^{y}_{i}-f^{0}_{i-\hat{x}-\hat{z}}+f^{0}_{i-\hat{x}}+f^{0}_{i-\hat{z}}-f^{0}_{i}$
$\displaystyle~{}~{}~{}-f_{i}^{x}+f_{i-\hat{z}}^{x}+f_{i}^{z}-f_{i-\hat{x}}^{z}$
$\displaystyle\sim
A_{i}^{y}-\partial_{z}\partial_{x}f_{i}^{0}-\partial_{z}f_{i}^{x}+\partial_{x}f_{i}^{z}$
$\displaystyle A^{z}_{i}$ $\displaystyle\rightarrow
A^{z}_{i}-f^{0}_{i-\hat{x}-\hat{y}}+f^{0}_{i-\hat{x}}+f^{0}_{i-\hat{y}}-f^{0}_{i}$
$\displaystyle~{}~{}~{}-f_{i}^{y}+f_{i-\hat{x}}^{y}+f_{i}^{x}-f_{i-\hat{y}}^{x}$
$\displaystyle\sim
A_{i}^{z}-\partial_{x}\partial_{y}f_{i}^{0}-\partial_{x}f_{i}^{y}+\partial_{y}f_{i}^{x}$
$\displaystyle E_{i}^{a}$ $\displaystyle\rightarrow
U_{E}^{\dagger}E_{i}^{a}U_{E}=E_{i}^{a}-g_{i+\hat{a}}+g_{i}$
$\displaystyle\sim E_{i}^{a}-\partial_{a}g_{i}.$ (44)

The two unitary operators are

$\displaystyle U_{A}$ $\displaystyle=\exp\left(i\sum_{j}(f^{0}_{j}({\bf
DE})_{j}+{\bm{f}}_{j}\cdot(\bm{\nabla}\times\bm{E})_{j})\right)$
$\displaystyle U_{E}$
$\displaystyle=\exp\left(i\sum_{j}g_{j}({\bm{\nabla}}\cdot{\bm{A}})_{j}\right),$
(45)

for $\bm{f}=(f^{x},f^{y},f^{z})$. (When $f^{0}_{j}=0$, these unitary operators
are those of the 3DTC with the roles of $\bm{A}$ and $\bm{E}$ interchanged.)
Since the generators do not commute, $[({\bm{\nabla}}\cdot{\bm{A}})_{i},({\bf
DE})_{j}]\neq 0$, neither do the unitary operators $U_{A}U_{E}\neq
U_{E}U_{A}$. An explicit calculation shows

$\displaystyle[\sum_{j}f^{0}_{j}({\bf
DE})_{j},\sum_{i}g_{i}({\bm{\nabla}}\cdot{\bm{A}})_{i}]$
$\displaystyle=-3i\sum_{i}g_{i}[f^{0}_{i}+f^{0}_{i-\hat{x}-\hat{y}}+f^{0}_{i-\hat{y}-\hat{z}}+f^{0}_{i-\hat{z}-\hat{x}}-f^{0}_{i-\hat{x}}-f^{0}_{i-\hat{y}}-f^{0}_{i-\hat{z}}-f^{0}_{i-\hat{x}-\hat{y}-\hat{z}}]$
$\displaystyle=-3i\sum_{i}f^{0}_{i}[g_{i}+g_{i+\hat{x}+\hat{y}}+g_{i+\hat{y}+\hat{z}}+g_{i+\hat{z}+\hat{x}}-g_{i+\hat{x}}-g_{i+\hat{y}}-g_{i+\hat{z}}-g_{i+\hat{x}+\hat{y}+\hat{z}}].$
(46)

In the continuum limit one can show that the commutator reduces to
$-3i\sum_{i}g_{i}\partial_{x}\partial_{y}\partial_{z}f^{0}_{i}=+3i\sum_{i}f^{0}_{i}\partial_{x}\partial_{y}\partial_{z}g_{i}$.
By choosing $f^{0}$ or $g$ to be functions of a special type, i.e.

$\displaystyle
f^{0}_{i}=f^{0}_{1}(i_{x},i_{y})+f^{0}_{2}(i_{y},i_{z})+f^{0}_{3}(i_{z},i_{x})$
(47)

(or similarly for $g_{i}$), one can prove the commutator vanishes
$[\sum_{j}f^{0}_{j}({\bf
DE})_{j},\sum_{i}g_{i}({\bm{\nabla}}\cdot{\bm{A}})_{i}]=0$. We thus obtain
valid definitions of unitary operators $U_{A}$ and $U_{E}$ with
$[U_{A},U_{E}]=0$ under the (mild) restriction on the gauge functions
$f^{0}_{i}$ or $g_{i}$. Such a restriction is lifted when we turn off
$f^{0}_{i}=0$ and the model becomes identical to 3DTC. These transformation
rules (and the constraint on the gauge functions) will play a vital role in
identifying the field theory description of the F3 model.

###  III.2 Field theory of F3 model

The F3 model was derived from the hybrid of the rank-1 and rank-2 U(1) LGTs.
We now apply the gauge principle to derive the appropriate EFT of the F3
model. There are three types of quasiparticle excitations in the F3 model
called the freeons, fractons, and fluxons, which are the excitations of the
$\mathcal{A}_{i}$, $\mathcal{B}_{i}$, and $\mathcal{C}^{a}_{i}$ operators. We
will use $e$, $f$, $m$ to express the quantities related to freeons, fractons,
fluxons, respectively.

First of all, the gauge transformation of $\bm{A}$ and $\bm{E}$ in the
continuum field theory follows directly from the lattice transformation rules
given in Eq. (44)

$\displaystyle A^{a}({\bm{r}})$ $\displaystyle\rightarrow
U_{A}^{\dagger}A^{a}({\bm{r}})U_{A}$
$\displaystyle=A^{a}({\bm{r}})-\frac{1}{2}\sum_{bc}|\epsilon_{abc}|\partial_{b}\partial_{c}f^{0}({\bm{r}})-({\bm{\nabla}}\times\bm{f}({\bm{r}}))^{a}$
$\displaystyle\bm{E}({\bm{r}})$ $\displaystyle\rightarrow
U_{E}^{\dagger}\bm{E}({\bm{r}})U_{E}=\bm{E}({\bm{r}})-{\bm{\nabla}}g({\bm{r}}).$
(48)

The two unitary operators

$\displaystyle U_{A}$ $\displaystyle=\exp\left[i\int
d^{3}{\bm{r}}\Bigl{(}f^{0}({\bm{r}}){\bf
DE}(\bm{r})+{\bm{f}}(\bm{r})\cdot\bm{\nabla}\times\bm{E}(\bm{r})\Bigl{)}\right]$
$\displaystyle U_{E}$ $\displaystyle=\exp\left[i\int
d^{3}{\bm{r}}g({\bm{r}}){\bm{\nabla}}\cdot{\bm{A}}({\bm{r}})\right],$ (49)

$\bm{f}=(f^{x},f^{y},f^{z})$, are the continuum descendants of the lattice
operators given in Eq. (45). In particular, we have ${\bf
DE}=\partial_{x}\partial_{y}E^{z}+\partial_{y}\partial_{z}E^{x}+\partial_{x}\partial_{y}E^{z}$.
As in the lattice consideration, the two unitary operators commute provided we
choose either $\partial_{x}\partial_{y}\partial_{z}f^{0}({\bm{r}})=0$ or
$\partial_{x}\partial_{y}\partial_{z}g({\bm{r}})=0$. Such condition is
fulfilled by choosing $f^{0}$ or $g$ to be the sum of functions that depend on
at most two of the coordinates but not on all three.

The freeon field $\psi_{e}$ transforms as
$U_{E}^{\dagger}\psi_{e}U_{E}=e^{ig}\psi_{e}$ under the constraint

$\displaystyle{\bm{\nabla}}\cdot{\bm{A}}=\rho_{e}=\psi^{\dagger}_{e}\psi_{e}.$
(50)

It behaves in the same way as the electric quasiparticle in the 3DTC with the
matter field Lagrangin in Eq. (9) and the continuity equation
$\partial_{t}\rho^{e}+\nabla\cdot\bm{j}^{e}=0$. The other quasiparticles, i.e.
fracton and three species of fluxons, can be grouped as $\psi^{\mu}$ ($\mu=0$
for fracton, $\mu=x,y,z$ for fluxons) and transform as

$\displaystyle\psi^{\mu}\rightarrow
U_{A}^{\dagger}\psi^{\mu}U_{A}=e^{if^{\mu}}\psi^{\mu}$ (51)

under the constraints

$\displaystyle{\bf
DE}(\bm{r})=\rho^{f},~{}~{}{\bm{\nabla}}\times{\bm{E}}={\bm{\rho}}^{m}$ (52)

where ${\bm{\rho}}^{m}=(\rho^{m_{x}},\rho^{m_{y}},\rho^{m_{z}})$. A gauge-
invariant Lagrangian can be constructed as

$\displaystyle\mathcal{L}_{f}=i\sum_{\mu}(\psi^{\mu})^{\dagger}\partial_{t}\psi^{\mu}-\frac{1}{2m}\sum_{a<b}|\bm{D}^{ab}\psi|^{2}$
(53)

with the covariant derivatives given by

$\displaystyle\bm{D}^{xy}\psi$
$\displaystyle=\psi^{x}\psi^{y}\psi^{0}\partial_{x}\partial_{y}\psi^{0}-\psi^{x}\psi^{y}\partial_{x}\psi^{0}\partial_{y}\psi^{0}$
$\displaystyle+{(\psi^{0})}^{2}[\psi^{x}\partial_{x}\psi^{y}-\psi^{y}\partial_{y}\psi^{x}+iA^{z}\psi^{x}\psi^{y}]$
$\displaystyle\bm{D}^{xz}\psi$
$\displaystyle=\psi^{x}\psi^{z}\psi^{0}\partial_{x}\partial_{z}\psi^{0}-\psi^{x}\psi^{z}\partial_{x}\psi^{0}\partial_{z}\psi^{0}$
$\displaystyle+{(\psi^{0})}^{2}[\psi^{z}\partial_{z}\psi^{x}-\psi_{f}^{x}\partial_{x}\psi^{z}+iA^{y}\psi^{y}\psi^{z}]$
$\displaystyle\bm{D}^{yz}\psi$
$\displaystyle=\psi^{y}\psi^{z}\psi^{0}\partial_{y}\partial_{z}\psi^{0}-\psi^{y}\psi^{z}\partial_{y}\psi^{0}\partial_{z}\psi^{0}$
$\displaystyle+{(\psi^{0})}^{2}[\psi^{y}\partial_{y}\psi^{z}-\psi^{z}\partial_{z}\psi^{y}+iA^{x}\psi^{y}\psi^{z}].$
(54)

They are quartic in the matter fields and involve the interaction between
fractons and fluxons. (The Lagrangian itself is octic in the matter fields.)
Once the fracton field is turned off by taking $\psi^{0}=1$, we recover the
covariant derivative of the 3DTC given in Eq. (11), which again reveals the
3DTC root of the F3 model.

The conservation laws following from the Lagrangian in Eq. (53) are

$\displaystyle\partial_{t}\rho^{f}$
$\displaystyle=\partial_{x}\partial_{y}j^{z}+\partial_{x}\partial_{z}j^{y}+\partial_{y}\partial_{z}j^{x},$
$\displaystyle\partial_{t}\bm{\rho}^{m}$ $\displaystyle=\nabla\times\bm{j}.$
(55)

The quasiparticle density and the current operators are given by

$\displaystyle\rho^{f}$ $\displaystyle={(\psi^{0})}^{\dagger}\psi^{0}$
$\displaystyle{\bm{\rho}}^{m}$
$\displaystyle=((\psi^{x})^{\dagger}\psi^{x},(\psi^{y})^{\dagger}\psi^{y},(\psi^{z})^{\dagger}\psi^{z})$
$\displaystyle j^{x}$
$\displaystyle=\frac{1}{2mi}\left[{(\psi^{y})}^{\dagger}{(\psi^{z})}^{\dagger}{(\psi^{0})}^{\dagger}{(\psi^{0})}^{\dagger}{\bm{D}}^{yz}\psi-h.c.\right]$
$\displaystyle j^{y}$
$\displaystyle=\frac{1}{2mi}\left[{(\psi^{z})}^{\dagger}{(\psi^{x})}^{\dagger}{(\psi^{0})}^{\dagger}{(\psi^{0})}^{\dagger}{\bm{D}}^{xz}\psi-h.c.\right]$
$\displaystyle j^{z}$
$\displaystyle=\frac{1}{2mi}\left[{(\psi^{x})}^{\dagger}{(\psi^{y})}^{\dagger}{(\psi^{0})}^{\dagger}{(\psi^{0})}^{\dagger}{\bm{D}}^{xy}\psi-h.c.\right].$
(56)

First of all, note that
$\partial_{t}{\bm{\rho}}^{m}={\bm{\nabla}}\times{\bm{j}}$ is identical to the
conservation of magnetic charges in Eq. (13). Furthermore, the constraint
${\bm{\rho}}^{m}={\bm{\nabla}}\times{\bm{E}}$ given in Eq. (52) implies
$\bm{\nabla}\cdot\bm{\rho}^{m}=0$, i.e. the magnetic excitations must form
closed loops. Time derivatives of both fracton ($\rho^{f}$) and the fluxon
($\bm{\rho}^{m}$) densities are related to the same current density vector
$\bm{j}$ in Eq. (55), which goes to show that fractons and fluxons are not
independent entities in the F3 model, but are intertwined objects through the
definitions of the current operators ${\bm{j}}$ given above and the continuity
equation given in Eq. (55).

Having established the EFT of the freeon, fracton, and fluxon matter fields,
we proceed to construct the gauge field Lagrangian. The starting point, as
usual, is the constraints relating the matter and the gauge degrees of
freedom:

$\displaystyle(\bm{\nabla}\cdot\bm{A})_{i}$
$\displaystyle=\frac{2\pi}{p}\rho^{e}_{i},$
$\displaystyle(\bm{\nabla}\times\bm{E})_{i}$
$\displaystyle=\bm{\rho}^{m}_{i},$ $\displaystyle({\bf DE})_{i}$
$\displaystyle=\rho^{f}_{i}.$ (57)

Taking the time derivatives on both sides of the equations and invoking the
continuity equation of freeons in Eq. (12), and of fluxons and fractons in Eq.
(55), we obtain three relations

$\displaystyle(\bm{\nabla}\cdot\partial_{t}{\bm{A}})_{i}$
$\displaystyle=-(2\pi/p)(\nabla\cdot\bm{j}^{e})_{i}$
$\displaystyle(\bm{\nabla}\times\partial_{t}{\bm{E}})_{i}$
$\displaystyle=(\nabla\times\bm{j})_{i}$ $\displaystyle\partial_{t}({\bf
DE})_{i}$
$\displaystyle=(\partial_{x}\partial_{y}j^{z}+\partial_{y}\partial_{z}j^{x}+\partial_{x}\partial_{z}j^{y})_{i}$
(58)

that are solved by

$\displaystyle\partial_{t}{{\bm{A}}}_{i}$
$\displaystyle=-\frac{2\pi}{p}{\bm{j}}^{e}_{i}$
$\displaystyle\partial_{t}{\bm{E}}_{i}$ $\displaystyle={\bm{j}}_{i}.$ (59)

Luckily the second and the third equations in Eq. (58) share the same
solution, providing consistency to the whole approach we take. The Lagrangian
that embodies both Eqs. (57) and (59) is readily constructed:

$\displaystyle\mathcal{L}_{\rm F3}$
$\displaystyle={\bm{E}}_{i}\cdot\partial_{t}{\bm{A}}_{i}+\bm{A}_{i}\cdot\bm{j}_{i}+\frac{2\pi}{p}{\bm{E}}_{i}\cdot\bm{j}^{e}_{i}$
$\displaystyle+\frac{p}{2\pi}E^{0}_{i}\Bigl{(}\bm{\nabla}\cdot\bm{A}-\frac{2\pi}{p}{\rho}^{e}\Bigl{)}_{i}+\bm{A}^{0}_{i}\cdot\Bigl{(}\bm{\nabla}\times\bm{E}-{\bm{\rho}}^{m}\Bigl{)}_{i}$
$\displaystyle+A^{0}_{i}\Bigl{(}{\bf DE}-\rho^{f}\Bigl{)}_{i}.$ (60)

Several Lagrange multipliers, $(E^{0},A^{0},{\bm{A}}^{0})$, are introduced to
implement the Gauss constraints relating the gauge fields to the matter
fields. By redefining $\bm{A}\rightarrow-\bm{A}$ and
$-(2\pi/p)\bm{E}\rightarrow\bm{E}$, one can rewrite the F3 Lagrangian as

$\displaystyle\mathcal{L}_{\text{F3}}$
$\displaystyle=\frac{p}{2\pi}{\bm{E}}\cdot\partial_{t}{\bm{A}}-\bm{A}\cdot\bm{j}-{\bm{E}}\cdot\bm{j}^{e}$
$\displaystyle-\frac{p}{2\pi}E^{0}\Bigl{(}\bm{\nabla}\cdot\bm{A}+\frac{2\pi}{p}{\rho}^{e}\Bigl{)}-\bm{A}^{0}\cdot\Bigl{(}\frac{p}{2\pi}\bm{\nabla}\times\bm{E}+{\bm{\rho}}^{m}\Bigl{)}$
$\displaystyle-A^{0}\Bigl{(}\frac{p}{2\pi}{\bf DE}+\rho^{f}\Bigl{)}$
$\displaystyle=\mathcal{L}_{\rm BF}-A^{0}\Bigl{(}\frac{p}{2\pi}{\bf
DE}+\rho^{f}\Bigl{)},$ (61)

clearly showing $\mathcal{L}_{\rm F3}$ as the action of the 3D toric code
$\mathcal{L}_{\rm BF}$ given in Eq. (14), supplemented with an additional
constraint through the Lagrange multiplier $A^{0}$. The constraint in turn
ties the fracton density $\rho^{f}$ to the local configuration of the ${\bf
E}$ through the relation ${\bf DE}+\rho^{f}=0$. Note that there is no other
term in the effective action that depends on the fracton degrees of freedom.
The current ${\bf j}$ is also defined as a fracton-fluxon composite object as
seen in Eq. (56). All these features are indications that fracton dynamics is
‘parasitic’ to that of fluxons, and the overall dynamics resembles that of 3D
toric code.

The commutation relation and the gauge transformation of the $\bm{A}$ and
$\bm{E}$ in Eq. (61) are

$\displaystyle[A^{a}(\bm{r}),E^{b}(\bm{r}^{\prime})]=\frac{2\pi
i}{p}\delta_{ab}\delta^{3}(\bm{r}-\bm{r}^{\prime})$ $\displaystyle
A^{a}({\bm{r}})\rightarrow U_{A}^{\dagger}A^{a}({\bm{r}})U_{A}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=A^{a}({\bm{r}})+\frac{1}{2}\sum_{bc}|\epsilon_{abc}|\partial_{b}\partial_{c}f^{0}({\bm{r}})+({\bm{\nabla}}\times\bm{f}({\bm{r}}))^{a}$
$\displaystyle\bm{E}(\bm{r})\rightarrow
U_{A}^{\dagger}A^{a}({\bm{r}})U_{A}=\bm{E}({\bm{r}})+{\bm{\nabla}}g({\bm{r}})$
(62)

where $U_{A}$ and $U_{E}$ are

$\displaystyle U_{A}$ $\displaystyle=\exp\left[-\frac{ip}{2\pi}\int
d^{3}{\bm{r}}\Bigl{(}f^{0}({\bm{r}}){\bf
DE}(\bm{r})+{\bm{f}}(\bm{r})\cdot\bm{\nabla}\times\bm{E}(\bm{r})\Bigl{)}\right]$
$\displaystyle U_{E}$ $\displaystyle=\exp\left[-\frac{ip}{2\pi}\int
d^{3}{\bm{r}}g({\bm{r}}){\bm{\nabla}}\cdot{\bm{A}}({\bm{r}})\right].$ (63)

The gauge transformations of $\bm{A}^{0}$, $A^{0}$, and $E^{0}$ are

$\displaystyle\bm{A}^{0}(\bm{r})$
$\displaystyle\rightarrow\bm{A}^{0}(\bm{r})+\partial_{t}\bm{f}(\bm{r})+\bm{\nabla}f^{\prime}(\bm{r})$
$\displaystyle A^{0}(\bm{r})$ $\displaystyle\rightarrow
A^{0}(\bm{r})+\partial_{t}f^{0}(\bm{r})$ $\displaystyle E^{0}(\bm{r})$
$\displaystyle\rightarrow E^{0}(\bm{r})+\partial_{t}g^{a}(\bm{r}).$ (64)

Conservation laws of quasiparticles in Eq. (55) as well as the identity
$\bm{\nabla}\cdot\bm{\rho}^{m}=0$ follow from requiring the gauge invariance
of the Lagrangian in Eq. (61) under the gauge transformations in Eqs. (62) and
(64). This completes the derivation of the EFT for the F3 model.

We only need to discuss the lattice model for fracton-fluxon matter fields
$\psi^{\mu}_{i}$ because the lattice model for the freeon field is simply the
expression in Eq. (18). Considering the transformation of ${\bm{A}}$ field
given in Eq. (44), the appropriate gauge-invariant hoppings are

$\displaystyle(\psi^{0})^{\dagger}_{i-\hat{y}}(\psi^{0})^{\dagger}_{i-\hat{z}}\psi^{0}_{i-\hat{y}-\hat{z}}\psi^{0}_{i}(\psi^{z})^{\dagger}_{i-\hat{y}}(\psi^{y})^{\dagger}_{i}\psi^{z}_{i}\psi^{y}_{i-\hat{z}}e^{iA^{x}_{i}}$
$\displaystyle(\psi^{0})^{\dagger}_{i-\hat{z}}(\psi^{0})^{\dagger}_{i-\hat{x}}\psi^{0}_{i-\hat{z}-\hat{x}}\psi^{0}_{i}(\psi^{x})^{\dagger}_{i-\hat{z}}(\psi^{z})^{\dagger}_{i}\psi^{x}_{i}\psi^{z}_{i-\hat{x}}e^{iA^{y}_{i}}$
$\displaystyle(\psi^{0})^{\dagger}_{i-\hat{x}}(\psi^{0})^{\dagger}_{i-\hat{y}}\psi^{0}_{i-\hat{x}-\hat{y}}\psi^{0}_{i}(\psi^{y})^{\dagger}_{i-\hat{x}}(\psi^{x})^{\dagger}_{i}\psi^{y}_{i}\psi^{x}_{i-\hat{y}}e^{iA^{z}_{i}}$
(65)

and their Hermitian conjugate. By comparing to Eq. (20) and Eq. (37), one can
see that the fracton part $\psi^{0}$ hops in the manner of a fracton in the
X-cube model, while $\psi^{a}$ ($a=x,y,z$) hop as the three magnetic charges
in the 3D toric code.

As seen in Eq. (40), the same ${\bf E}$ field is used to define both the
fluxon and the fracton charges, but these two charges are tightly bound and
move as composite objects, as the above tight-binding model clearly shows.
This, as well as the effective field theory of the F3 model given as the sum
of the BF model plus a constraint as in Eq. (61) indicates that both F3 and
the toric code share the same universal physics.

##  IV Summary

We have developed an easy-to-implement scheme for deriving field theories of
toric code and X-cube model in three dimensions, and applied the method to
derive the field theory for the F3 model that two of the present authors wrote
down earlier Kim and Han (2021). We show that the F3 model follows from
Higgsing the hybrid of rank-1 and rank-2 U(1) lattice gauge theories in three
dimensions. By consistently applying the gauge principle, we are able to
derive the matter and the gauge field Lagrangian for the F3 model in the
continuum and also in the lattice form. It turns out the effective field
theory of the F3 model is that of the 3D toric code modified by a term, and
describe the same universal physics. As a by-product, we construct the matter
field Lagrangians both in the continuum and in the lattice form for the well-
known 3D toric code and the X-cube model as well.

###### Acknowledgements.

H. J. H. was supported by the Quantum Computing Development Program (No.
2019M3E4A1080227). He also acknowledges financial support from EPIQS Moore
theory centers at Harvard and MIT. J. K. was supported by the education and
training program of the Quantum Information Research Support Center, funded
through the National research foundation of Korea(NRF) by the Ministry of
science and ICT(MSIT) of the Korean government(No.2021M3H3A103657313).
Additional support came from National Research Foundation of Korea under Grant
NRF-2021M3E4A1038308 and NRF-2021M3H3A1038085. Y.-T. O. was supported by
National Research Foundation of Korea under Grant NRF-2021R1A2C4001847,
NRF-2020R1A4A3079707, and NRF-2022R1I1A1A01065149.

##  Appendix A Derivation of the field theory of three-dimensional toric code

From the matter action of $e$ quasiparticles in Eq. (9), the conservation law
for the $e$ quasiparticle is, as expected,
$\partial_{t}\rho^{e}+{\bm{\nabla}}\cdot{\bm{j}}^{e}=0$ with
$\rho^{e}=(\psi_{e})^{\dagger}\psi_{e}$ and
${\bm{j}}^{e}=[\psi_{e}^{\dagger}{\bm{D}}_{e}\psi_{e}-h.c.]/(2m_{e}i)$. For
the magnetic quasiparticle part, we obtain

$\displaystyle\partial_{t}\rho^{m_{a}}+\sum_{b}\partial_{b}[j^{m}]^{ab}$
$\displaystyle=$ $\displaystyle 0,$ (66)

where

$\displaystyle\rho^{m_{a}}$
$\displaystyle=(\psi_{m}^{a})^{\dagger}\psi_{m}^{a},$
$\displaystyle{[j^{m}]}^{ab}$
$\displaystyle=\frac{1}{2m_{m}i}[(\psi_{m}^{a})^{\dagger}(\psi_{m}^{b})^{\dagger}{\bm{D}}_{m}^{ab}\psi_{m}^{a}\psi_{m}^{b}-h.c.].$
(67)

The gauge part of the EFT is constructed by first examining the various
constraints on the gauge fields and the matter fields. We derive the necessary
constraints directly from the Higgsing formulas in Eq. (8). One can read off

$\displaystyle({\bm{\nabla}}\times{\bm{A}})_{i_{d}}$
$\displaystyle=\frac{2\pi}{p}({\bm{\rho}}^{m})_{i_{d}},$
$\displaystyle({\bm{\nabla}}\cdot{\bm{E}})_{i}$ $\displaystyle=-\rho^{e}_{i},$
(68)

relating the lattice curl of $\bm{A}$ to the vector charge density
$({\bm{\rho}}^{m})_{i_{d}}=(\rho^{m_{x}},\rho^{m_{y}},\rho^{m_{z}})_{i_{d}}$
and the lattice divergence ${\bm{\nabla}}\cdot{\bm{E}}$ to the scalar charge
density $\rho^{e}$. Taking the time derivatives on both sides of the equations
and invoking the quasiparticle conservation laws in Eq. (66), we get
$({\bm{\nabla}}\times\partial_{t}{\bm{A}})_{i_{d}}^{a}=-(2\pi/p)(\sum_{b}\partial_{b}[j^{m}]^{ab})_{i_{d}}$
and
$({\bm{\nabla}}\cdot\partial_{t}{\bm{E}})_{i}=({\bm{\nabla}}\cdot{\bm{j}}^{e})_{i}$
whose solutions are

$\displaystyle\partial_{t}{A}^{a}_{i}$
$\displaystyle=-\frac{\pi}{p}\sum_{bc}\epsilon_{abc}[j^{m}]_{i_{d}}^{bc},$
$\displaystyle\partial_{t}{\bm{E}}_{i}$ $\displaystyle={\bm{j}}^{e}_{i}.$ (69)

The equations (68) and (69) formalize the constraints that are inherent in the
3DTC and can be encoded in the Lagrangian

$\displaystyle\mathcal{L}$
$\displaystyle={\bm{E}}_{i}\cdot\partial_{t}{\bm{A}}_{i}+{\bm{A}}_{i}\cdot{\bm{j}}^{e}_{i}+\frac{\pi}{p}\sum_{abc}\epsilon_{abc}E^{a}_{i}[j^{m}]_{i_{d}}^{bc}$
$\displaystyle+\frac{p}{2\pi}{\bm{E}}^{0}_{i}\cdot\Bigl{(}{\bm{\nabla}}\times{\bm{A}}-\frac{2\pi}{p}{\bm{\rho}}^{m}\Bigl{)}_{i_{d}}$
$\displaystyle-A^{0}_{i}\Bigl{(}{\bm{\nabla}}\cdot{\bm{E}}+\rho^{e}\Bigl{)}_{i}.$
(70)

Here, ${\bm{E}}^{0}_{i}=(E^{x0}_{i},E^{y0}_{i},E^{z0}_{i})$ and $A^{0}_{i}$
are the Lagrange multipliers. Carrying out the re-definition

$\displaystyle-(2\pi/p)\sum_{a}\epsilon_{abc}E^{a}_{i}$
$\displaystyle\rightarrow E^{bc}_{i}$
$\displaystyle\bm{A}_{i}\rightarrow-\bm{A}_{i},$ (71)

and taking the continuum limit, we can get the BF action in Eq. (14). $A$ and
$j^{e}$ are

$\displaystyle A$ $\displaystyle=(A^{0},{\bm{A}})$ $\displaystyle j^{e}$
$\displaystyle=(\rho^{e},{\bm{j}}^{e}),$ (72)

and $E^{\mu\nu}$ and ${[j^{m}]}^{\mu\nu}$ are

$\displaystyle E^{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&-E^{x0}&-E^{y0}&-E^{z0}\\\ E^{x0}&&&\\\
E^{y0}&&E^{ab}&\\\ E^{z0}&&&\end{pmatrix}$ $\displaystyle{[j^{m}]}^{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&-\rho^{m_{x}}&-\rho^{m_{y}}&-\rho^{m_{z}}\\\
\rho^{m_{x}}&&&\\\ \rho^{m_{y}}&&[j^{m}]^{ab}&\\\
\rho^{m_{z}}&&&\end{pmatrix}.$ (73)

##  Appendix B Derivation of the field theory of X-cube model

The conservation laws of fracton and $l_{a}$-quasiparticles follow from the
effective Lagrangians (31) as

$\displaystyle\partial_{t}\rho^{f}-\sum_{a<b}\partial_{a}\partial_{b}[j^{f}]^{ab}=0,$
$\displaystyle\partial_{t}\rho^{l_{a}}+\sum_{bc}\epsilon_{abc}\partial_{c}[j^{l}]^{c}=0.$
(74)

The quasiparticle density and the current operators are given by

$\displaystyle\rho^{f}$ $\displaystyle=\psi_{f}^{\dagger}\psi_{f},$
$\displaystyle[j^{f}]^{ab}$
$\displaystyle=\frac{1}{2m_{f}i}\left[\psi_{f}^{\dagger}\psi_{f}^{\dagger}{\bm{D}}_{f}^{ab}\psi_{f}-h.c.\right],$
$\displaystyle\rho^{l_{a}}$
$\displaystyle=(\psi_{l}^{a})^{\dagger}\psi_{l}^{a},$
$\displaystyle[j^{l}]^{a}$
$\displaystyle=-\frac{1}{4m_{l}i}\left[\sum_{bc}|\epsilon_{abc}|(\psi_{l}^{b})^{\dagger}(\psi_{l}^{c})^{\dagger}{\bm{D}}_{l}^{a}\bm{\psi}_{l}-h.c.\right].$
(75)

The fracton current density tensor is symmetric, $[j^{f}]^{ab}=[j^{f}]^{ba}$.

Similar to the derivation of EFT of 3DTC, we identify the constraints directly
from the Higgsing formula in Eq. (28). The first of the constraints is

$\displaystyle\frac{1}{2}\sum_{abc}|\epsilon_{abc}|(\partial_{a}\partial_{b}E^{c})_{i}=\rho^{f}_{i}.$
(76)

Taking the time derivative on both sides and invoking the fracton conservation
given by Eq. (33), we get
$(1/2)\sum_{abc}|\epsilon_{abc}|(\partial_{a}\partial_{b}\dot{E}^{c})_{i}=\sum_{a<b}(\partial_{a}\partial_{b}{[j^{f}]}^{ab})_{i}$
and its solution

$\displaystyle\dot{E}^{a}_{i}=\frac{1}{2}\sum_{bc}|\epsilon_{abc}|{[j^{f}]}^{bc}_{i}.$
(77)

The second constraint is

$\displaystyle\sum_{bc}\epsilon_{abc}(\partial_{c}A^{c})_{i}=\frac{2\pi}{p}\rho^{l_{a}}_{i}.$
(78)

Invoking the conservation law of the $l_{a}$-quasiparticles given in Eq. (33)
we get
$\sum_{bc}\epsilon_{abc}(\partial_{c}\dot{A}^{c})_{i}=-(2\pi/p)\sum_{bc}\epsilon_{abc}(\partial_{c}[j^{l}]^{c})_{i}$
and the solution is

$\displaystyle\dot{\bm{A}}_{i}=-\frac{2\pi}{p}{\bm{j}}^{l}_{i}.$ (79)

Equations (76)-(79) formalize the constraints in the X-cube model, which are
embodied in the Lagrangian

$\displaystyle\mathcal{L}$
$\displaystyle={\bm{E}}_{i}\cdot\partial_{t}{\bm{A}}_{i}+\frac{1}{2}\sum_{abc}|\epsilon_{abc}|A_{i}^{a}[j^{f}]_{i}^{bc}+\frac{2\pi}{p}{\bm{E}}_{i}\cdot\bm{j}^{l}_{i}$
$\displaystyle+\frac{p}{2\pi}E^{a0}_{i}\Bigl{(}\sum_{bc}\epsilon_{abc}\partial_{c}A^{c}-\frac{2\pi}{p}{\rho}^{l_{a}}\Bigl{)}_{i}$
$\displaystyle-A^{0}_{i}\Bigl{(}\sum_{abc}|\epsilon_{abc}|\frac{1}{2}\partial_{a}\partial_{b}E^{c}-\rho^{f}\Bigl{)}_{i},$
(80)

where $E^{a0}_{i}$ and $A_{i}^{0}$ are the Langrange multipliers. After re-
defining

$\displaystyle\sum_{a}|\epsilon_{abc}|A^{a}_{i}$ $\displaystyle\rightarrow
A^{bc}_{i}$ $\displaystyle-(2\pi/p)E^{a}_{i}$ $\displaystyle\rightarrow
E^{a}_{i}$ $\displaystyle(\rho^{f}_{i},{\bm{j}}^{f}_{i})$
$\displaystyle\rightarrow(-\rho^{f}_{i},-{\bm{j}}^{f}_{i}),$ (81)

and taking continuum limit, we can get Eq. (34).

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