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# Topological Phase Transitions of Interacting Fermions in the Presence of a
Commensurate Magnetic Flux

Axel Fünfhaus<EMAIL_ADDRESS>Institute of Theoretical
Physics, Goethe University Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt
am Main, Germany  Marius Möller  Institute of Theoretical Physics, Goethe
University Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany
Thilo Kopp  Center for Electronic Correlations and Magnetism, Experimental
Physics VI, Institute of Physics, University of Augsburg, 86135 Augsburg,
Germany  Roser Valentí  Institute of Theoretical Physics, Goethe University
Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany

(August 28, 2024)

###### Abstract

Motivated by recently reported magnetic-field induced topological phases in
ultracold atoms and correlated Moiré materials, we investigate topological
phase transitions in a minimal model consisting of interacting spinless
fermions described by the Hofstadter model on a square lattice. For
interacting lattice Hamiltonians in the presence of a commensurate magnetic
flux it has been demonstrated that the quantized Hall conductivity is
constrained by a Lieb-Schultz-Mattis (LSM)-type theorem due to magnetic
translation symmetry. In this work, we revisit the validity of the theorem for
such models and establish that a topological phase transition from a
topological to a trivial insulating phase can be realized but must be
accompanied by spontaneous magnetic translation symmetry breaking caused by
charge ordering of the spinless fermions. To support our findings, the
topological phase diagram for varying interaction strength is mapped out
numerically with exact diagonalization for different flux quantum ratios and
band fillings using symmetry indicators. We discuss our results in the context
of the LSM-type theorem.

##  I Introduction

Chern and topological insulators in condensed matter physics [1, 2] are
closely linked to the discovery of the integer quantum Hall effect (IQHE) [3,
4]. The problem of free electrons in a constant external magnetic field can be
solved analytically and it leads to an understanding of the quantization of
the Hall conductivity in terms of single-particle Landau levels [5, 6]. The
characteristic high degeneracy and Chern numbers of the Landau levels are a
result of the magnetic translation symmetry of electrons subject to an
external magnetic field [7, 8, 9]. Similar symmetry constraints arising from
magnetic translation symmetry also extend to non-interacting lattice
Hamiltonians in the presence of a commensurate magnetic flux, such as the
Hofstadter model on a square lattice [10]. Actually, it can be shown that for
a gapped Hall insulator the Chern number $C$ of the occupied bands is
constrained by a Diophantine equation of the form [7, 11]

$\operatorname{e}^{2\pi\mathrm{i}(\frac{p}{q}C-\rho)}=1,$ (1)

where $p/q$ is the flux quantum ratio per unit cell and $\rho$ is the number
of particles per unit cell. In particular, the Chern number cannot be equal to
zero if $\rho$ is not an integer, excluding a trivial band insulating phase.

Recent theoretical predictions [12, 13, 14, 15] of additional topological
invariants protected by translation symmetry in an external magnetic field
have ignited a renewed interest in the Hofstadter model. Furthermore, there
has been important experimental progress in cold atom gases [16, 17] and
twisted bilayer materials [18, 19] where the range of the flux quantum ratio
can now go up to one elementary flux quantum $\Phi_{0}$ per unit cell. In
particular, a bosonic fractional quantum Hall phase has been recently detected
in the former [20], and the formation of odd-denominator fractional quantum
Hall states in graphene at high magnetic fields have been optically detected
using excited Rydberg excitons in an adjacent transition metal dichalcogenide
monolayer [21].

For interacting systems, the classification of topological phases is more
challenging than in the non-interacting case since the ground state is no
longer given by a product state of single-particle Bloch wave functions.
However, even without resorting to Bloch states in the Brillouin zone (BZ), it
is possible to define a many-body Chern number using twisted boundary
conditions (TBC) which corresponds to adiabatic flux insertion and is thus
related to the Hall conductivity [22]. This is now a well-established
technique in mapping out topological phase diagrams of interacting Chern
insulators [23, 24] and especially relevant for fractional Chern insulators,
which are beyond a conventional mean-field description [25, 26] and hence
require a genuine many-body treatment. Due to the similarity of the BZ and the
set of possible TBC, both forming a torus, it is a natural question to ask
whether magnetic translation operators impose the same constraint, Eq. (1), on
interacting systems.

An earlier and more well-known example of a filling-enforced constraint on
quantum many-body systems is the (generalized) Lieb-Schultz-Mattis (LSM)
theorem. Originally formulated for a 1D half-integer spin chain [27], it has
been generalized to higher dimensions [28] and many-body systems with
conserved particle number on a periodic lattice [29]. Resting on the
commutation relation between the “large gauge transformation” and the
translation operator [29, 30], the theorem determines whether a translation
invariant Hamiltonian at a certain filling is allowed to have a non-degenerate
ground state or not. Specifically, a unique ground state, separated by a gap
from excited states, is prohibited for non-integer fillings.

In the presence of an external magnetic field the algebra of translation
operators changes to the magnetic translation algebra and integer fillings now
correspond to $q\rho\in\mathbb{Z}$ due to an enlarged magnetic unit cell [31].
Even more importantly is the possibility to generalize Eq. (1) to genuine
many-body systems [32, 31, 33, 34] and to exclude trivial band insulators at
certain fillings. The proof is based on using within Laughlin’s gauge argument
the commutation relation of the large gauge transformation with the product of
the magnetic translation operator and a flux insertion operator [31] in the
thermodynamic limit. It may therefore be viewed as a corollary or variation of
the generalized LSM theorem. This generalization can also be proven exactly
for all finite-size systems on which the generalized LSM theorem applies [33].
We will therefore call it the LSM-type theorem in the following.

Crucially, the LSM-type theorem for the IQHE relies on a nondegenerate ground
state such as a non-interacting band insulating state. In the case of ground
state degeneracy however, which commonly takes place at a phase transition due
to spontaneous symmetry breaking (SSB), the theorem’s implications have to be
reevaluated.

In this work we scrutinize the physical implications of many-body magnetic
translation invariance on the Hall conductivity in the light of SSB. In
particular, the role of the specific choice of a given cluster, on which the
validity of the LSM-type theorem and the mechanism of SSB depend in a subtle
way, will be examined. For this purpose, we perform exact diagonalization (ED)
calculations of the interacting Hofstadter model on a square lattice for a
wide range of different band fillings and flux quantum ratios. Such
calculations are useful since they capture qualitative differences between
different clusters. In the context of Hofstadter physics, ED allows us to
study the effect of the discrete lattice (high flux quantum ratio) but also
the parameter regime, where the single particle physics is approximately
described by Landau levels (low flux quantum ratio). As a particular example
for SSB, we investigate the translation symmetry breaking charge-density-wave
(CDW) state in the (long-range) Coulomb interaction regime. For the
identification of topological phase transitions we make use of symmetry
indicators [33].

This paper is organized as follows: In Sec. II we review basic properties of
lattice Hamiltonians in the presence of a commensurate flux (in particular the
Hofstadter model), the magnetic translation algebra and the constraint imposed
by the LSM-type theorem on the Hall conductivity. In Sec. III we provide an
intuitive explanation backed by numerical calculations, how this constraint is
circumvented by SSB. We use ED to discuss the possibility of topological phase
transitions from Chern insulator to a checkerboard-pattern CDW phase for a
half-integer flux quantum ratio. We expand the ED study by investigating all
numerically feasible flux quantum ratios and integer band fillings of the
interacting Hofstadter model on a square lattice in Sec. IV. We close our
analysis by giving a thorough and rigorous account of the role of SSB in Sec.
V that addresses the remaining formal and technical questions of the
applicability of the LSM-type theorem in integer as well as fractional quantum
Hall systems. Finally, in Sec. VI we conclude our findings and discuss future
directions.

##  II Lattice Hamiltonians in Commensurate Magnetic Flux

###  II.1 Hofstadter model

To understand how a tight-binding Hamiltonian is affected by an external
magnetic field, one can consider a particle of charge $-e$ moving around a
surface that is pierced by a flux $\Phi$. The wave function obtains an
Aharonov-Bohm phase of $-2\pi\Phi/\Phi_{0}$, where $\Phi_{0}=hc/e$ is the
magnetic flux quantum, which can be seen by considering the change in the
Lagrange function $L\to L-e\dot{\bm{r}}\bm{A}/c$ upon adding a flux $\Phi$
which leads to the phase

$\operatorname{e}^{-\mathrm{i}\frac{e}{\hbar
c}\int\mathop{}\\!\mathrm{d}\bm{r}\bm{A}(\bm{r})}$ (2)

due to the change in the action
$S[\bm{r}]=\int_{0}^{t}\mathop{}\\!\mathrm{d}t^{\prime}L[\dot{\bm{r}},\bm{r}]$
in the path weight $\operatorname{e}^{\mathrm{i}S[\bm{r}]/\hbar}$. On a
single-orbital lattice model an external magnetic field can be implemented by
including so-called Peierls phases to the hopping matrix elements for spinless
fermions [10, 35]. Such phases can be interpreted as Aharonov-Bohm phases
originating from a magnetic flux piercing the different plaquettes of the
given lattice:

$c_{\bm{R}^{\prime}}^{\dagger}c_{\bm{R}}^{\phantom{\dagger}}\to
c_{\bm{R}^{\prime}}^{\dagger}c_{\bm{R}}^{\phantom{\dagger}}\operatorname{e}^{-\mathrm{i}\frac{e}{\hbar
c}\int_{\bm{R}}^{\bm{R}^{\prime}}\bm{A}(\bm{r}^{\prime})\mathop{}\\!\mathrm{d}\bm{r}^{\prime}}=:c_{\bm{R}^{\prime}}^{\dagger}c_{\bm{R}}^{\phantom{\dagger}}\operatorname{e}^{-\mathrm{i}\phi_{\bm{R}^{\prime},\bm{R}}}.$
(3)

The flux through a set of plaquettes enclosed by the path
$\bm{R}_{1},\bm{R}_{2},\dots,\bm{R}_{n},\bm{R}_{1}$ is given by the sum
$\phi_{\bm{R}_{2},\bm{R}_{1}}+\phi_{\bm{R}_{3},\bm{R}_{2}}+\dots+\phi_{\bm{R}_{1},\bm{R}_{n}}$.
A single-orbital nearest-neighbor (NN) tight-binding Hamiltonian on a square
lattice with a flux quantum ratio $\Phi/\Phi_{0}=\varphi$ per plaquette
becomes

$\displaystyle\hat{H}=$
$\displaystyle-t\sum_{m,n}c_{m+1,n}^{\dagger}c_{m,n}^{\phantom{\dagger}}$ (4)
$\displaystyle-t\sum_{m,n}\operatorname{e}^{-2\pi\mathrm{i}\varphi
m}c_{m,n+1}^{\dagger}c_{m,n}^{\phantom{\dagger}}+\text{h.c.},$

which is the Hofstadter model in the Landau gauge [10]. Different choices for
the phases $\phi_{\bm{R}^{\prime},\bm{R}}$ are equivalent up to a unitary
gauge transformation $c_{m,n}^{\dagger}\to
e^{i\lambda_{m,n}}c_{m,n}^{\dagger}$ [36] as long as the flux piercing each of
the plaquettes is the same [37]. The Chern numbers of the band spectrum of Eq.
(4) are uniquely determined by a specific Diophantine equation for all
possible $\varphi$ [38, 4].

###  II.2 Magnetic translation algebra

Since hopping amplitudes and fluxes of the Hamiltonian in Eq. (4) are
translation invariant, one can identify two unitary translation operators in
$\bm{x}$ and $\bm{y}$ direction defined by

$\displaystyle\hat{T}_{\bm{x}}^{M}c_{m,n}^{\dagger}\left(\hat{T}_{\bm{x}}^{M}\right)^{\dagger}=$
$\displaystyle\operatorname{e}^{-2\pi\mathrm{i}\varphi n}c_{m+1,n}^{\dagger}$
(5)
$\displaystyle\hat{T}_{\bm{y}}^{M}c_{m,n}^{\dagger}\left(\hat{T}_{\bm{y}}^{M}\right)^{\dagger}=$
$\displaystyle c_{m,n+1}^{\dagger}.$

These operators, consisting of the action of a symmetry operation in real
space times a unitary and space dependent gauge transformation are commonly
denoted magnetic translation operators [8, 9]. They generally do not commute
with each other. Acting on a Hilbert space of $N_{e}$ particles they satisfy
111Eq. (6b) indicates that magnetic translation operators transform as a
projective representation. It is deduced in the continuum from the Baker-
Campbell-Hausdorff formula, whereas in a discrete Hilbert space it simply
serves as the definition of $\hat{T}_{\bm{a}}^{M}$ for arbitrary Bravais
lattice vectors $\bm{a}$. In particular it should be noted that for
$\varphi=2\pi$, Eq. (6b) does not realize a non-trivial projective rep, as the
magnetic translation operators are abelian according to Eq. (6a).

$\displaystyle\hat{T}_{\bm{x}}^{M}\hat{T}_{\bm{y}}^{M}=$
$\displaystyle\operatorname{e}^{-2\pi\mathrm{i}\varphi
N_{e}}\hat{T}_{\bm{y}}^{M}\hat{T}_{\bm{x}}^{M}$ (6a)
$\displaystyle\hat{T}_{\bm{x}}^{M}\hat{T}_{\bm{y}}^{M}=$
$\displaystyle\operatorname{e}^{-\pi\mathrm{i}\varphi
N_{e}}\hat{T}_{\bm{x}+\bm{y}}^{M},$ (6b)

which is referred to as the magnetic translation algebra or Girvin-MacDonald-
Platzman algebra [40]. According to Eq. (6) these operators form a higher-
dimensional projective representation [41, 42, 43] for $\varphi
N_{e}\notin\mathbb{Z}$, in contrast to the usual one-dimensional irreducible
representations for the abelian translation group. A consequence is a
symmetry-protected degeneracy of the energy spectrum: for example, the
macroscopic degeneracy of single-particle Landau levels stems from the
magnetic translation symmetry of free particles [8].

For a non-interacting lattice Hamiltonian such as the Hofstadter model and for
a flux quantum ratio

$\varphi=\frac{p}{q},$ (7)

where in the following $p$ and $q$ will always be chosen to be coprime, the
single-particle band structure obtains a symmetry protected $q$-fold
degeneracy. This can be seen as follows: Since
$(\hat{T}_{\bm{x}}^{M})^{q},\hat{T}_{\bm{y}}^{M}$ are commuting translation
operators according to Eq. (6a) for $N_{e}=1$, they can be used to define
Bloch states $|{k_{x},k_{y}}\rangle$ with eigenvalues
$\operatorname{e}^{-\mathrm{i}qk_{x}},\operatorname{e}^{-\mathrm{i}k_{y}}$
(where we have set the lattice constant to 1). Then, since

$\hat{T}_{\bm{y}}^{M}\left(\hat{T}_{\bm{x}}^{M}|{k_{x},k_{y}}\rangle\right)=\operatorname{e}^{-\mathrm{i}(k_{y}-2\pi\varphi)}\left(\hat{T}_{\bm{x}}^{M}|{k_{x},k_{y}}\rangle\right)$
(8)

we obtain that $\hat{T}_{\bm{x}}^{M}|{k_{x},k_{y}}\rangle$ is a new eigenstate
of $\hat{T}_{\bm{y}}^{M}$, hence

$\hat{T}_{\bm{x}}^{M}|{k_{x},k_{y}}\rangle\propto|{k_{x},k_{y}-2\pi\varphi}\rangle.$
(9)

As a consequence
$|{k_{x},k_{y}}\rangle,\dots,|{k_{x},k_{y}-2\pi\varphi(q-1)}\rangle$ span a
$q$-dimensional irreducible projective representation space with degenerate
energy eigenvalues. This also explains why according to Eq. (1), the Chern
number can only change modulo $q$ because the topological phase transition of
a band is accompanied with the simultaneous closure of $q$ band gaps. A
complete derivation of Eq. (1) is given in Appendix A.

For interacting systems ED calculations have to be restricted to finite
systems. The infinite plane becomes a cluster with its boundaries being glued
together forming a torus. Besides piercing each plaquette with a flux, it is
now possible to insert fluxes through the non-contractible loops of the torus
[22, 37], see Fig. 1. They can be varied by imposing twisted boundaries onto
the system, which keep the fluxes through the individual plaquettes invariant.
The Hamiltonian in Eq. (4) becomes

$\displaystyle\hat{H}(\theta_{x},\theta_{y})=$
$\displaystyle-t\sum_{m=1}^{N_{x}}\sum_{n=1}^{N_{y}}\operatorname{e}^{-\mathrm{i}\theta_{x}\delta_{m,N_{x}}}c_{m+1,n}^{\dagger}c_{m,n}^{\phantom{\dagger}}$
(10)
$\displaystyle-t\sum_{m=1}^{N_{x}}\sum_{n=1}^{N_{y}}\operatorname{e}^{-2\pi\mathrm{i}\varphi
m}\operatorname{e}^{-\mathrm{i}\theta_{y}\delta_{n,N_{y}}}c_{m,n+1}^{\dagger}c_{m,n}^{\phantom{\dagger}}$
$\displaystyle+\text{h.c.},$

with twisted boundary $\theta_{x}$ in $\bm{x}$-direction and $\theta_{y}$ in
$\bm{y}$-direction. $N_{x/y}$ is the number of sites in $x/y$-direction. We
will always consider $N_{x}/q\in\mathbb{N}$, unless stated otherwise. This
choice automatically satisfies the quantization condition that the total flux
through all of the plaquettes can only be zero modulo $2\pi$, as sources of
magnetic flux may only emit integer multiples of $\Phi_{0}$ [8, 44].

Figure 1: Toroidal lattice geometry of a finite cluster in real space with two
fluxes corresponding to twisted boundary conditions $\theta_{x}$ and
$\theta_{y}$ through the non-contractable loops of the torus. A translation in
$\bm{y}$-direction relates the loops $L_{1}$ and $L_{2}$ to each other. Their
Aharonov-Bohm phases differ by the flux $2\pi\varphi N_{x}$ through the
surface between the two loops.

Importantly, a tight-binding Hamiltonian of the form of Eq. (10) is not
translation invariant for arbitrary $N_{x},N_{y}$. Let the Aharonov-Bohm phase
around a non-contractable loop $L_{2}$ in $\bm{x}$-direction for fixed $n$ be

$\displaystyle\sum_{m=1}^{N_{x}}\phi_{(m+1,n),(m,n)}=\theta_{x}.$ (11)

Around a similar loop $L_{1}$ for fixed $n-1$ we identify

$\displaystyle\sum_{m=1}^{N_{x}}\phi_{(m+1,n-1),(m,n-1)}=\sum_{m=1}^{N_{x}}\left(\phi_{(m+1,n-1),(m,n-1)}\right.$
(12) $\displaystyle+$
$\displaystyle\left.\phi_{(m+1,n),(m+1,n-1)}+\phi_{(m,n),(m+1,n)}+\phi_{(m,n-1),(m,n)}\right)$
$\displaystyle+$
$\displaystyle\sum_{m=1}^{N_{x}}\phi_{(m+1,n),(m,n)}=2\pi\varphi
N_{x}+\theta_{x},$

where we have used
$\phi_{\bm{R}^{\prime},\bm{R}}=-\phi_{\bm{R},\bm{R}^{\prime}}$. Note that the
first four terms in between the equal signs generate a loop around a plaquette
with flux $\varphi$. We see that generically the magnetic field piercing the
torus also contributes to the phase, see Fig. 1. The loop defined in Eq.
(LABEL:Eq:different_loops) is only equal to $\theta_{x}$ modulo $2\pi$, if
$N_{x}$ is an integer multiple of $q$. For $N_{y}$ an analogous condition for
closed loops with fixed $m$ holds. Consequently, under translation the
Hamiltonian transforms as 222Here we have omitted the dependency of the
translation operators on the twisted boundary condition for simplicity, which
does not influence the validity of Eq. (14) [33]. In [31] it is pointed out
that the definition of the magnetic translation operator as it appears in Eq.
(5) cannot be used anymore if the Hamiltonian does not commute with the
translation operator. Instead, a gauge of $\hat{H}(\bm{\theta})$ for all
$\bm{\theta}$ needs to be defined so that the magnetic translation operator
maps the Hamiltonians of this gauge to each other.

$\displaystyle\hat{T}_{\bm{x}}^{M}\hat{H}(\theta_{x},\theta_{y})\left(\hat{T}_{\bm{x}}^{M}\right)^{\dagger}=$
$\displaystyle\hat{H}(\theta_{x},\theta_{y}-2\pi\varphi N_{y})$ (13)
$\displaystyle\hat{T}_{\bm{y}}^{M}\hat{H}(\theta_{x},\theta_{y})\left(\hat{T}_{\bm{y}}^{M}\right)^{\dagger}=$
$\displaystyle\hat{H}(\theta_{x}+2\pi\varphi N_{x},\theta_{y}),$

so the system is only translation invariant if $N_{x}$ and $N_{y}$ are integer
multiples of $q$ [8, 9].

###  II.3 Constraints imposed by the LSM-type theorem on the Hall conductivity

It is possible to define a many-body Chern number of the ground state
$|{\Psi(\theta_{x},\theta_{y})}\rangle$ with the Berry connection
$\bm{A}=\mathrm{i}\langle{\Psi(\theta_{x},\theta_{y})|\bm{\nabla}_{\bm{\theta}}|\Psi(\theta_{x},\theta_{y})}\rangle$),
which is related to the Hall-conductivity by Laughlin’s gauge invariance
argument [22]. This Chern number is also well defined upon adding arbitrary
interactions to the Hamiltonian as long as an energy gap separates the ground
state manifold from excitations for all twist angles. A natural question that
arises is whether it is possible to generalize Eq. (1) to many-body systems.
In the context of the generalized LSM theorem it can be shown that [31, 33,
32]

$\operatorname{e}^{2\pi\mathrm{i}(\frac{p}{q}C_{\text{many-body}}-\rho)}=1$
(14)

with the many-body Chern number $C_{\text{many-body}}$. This holds provided
that a given twist angle combination $(\theta_{x},\theta_{y})$ is mapped to
$q$ different twist angles under translation and the system has a unique
ground state for each $\bm{\theta}$ (see Appendix A). In the following we will
refer to Eq. (14) as the LSM-type theorem. Clusters, where the LSM-type
theorem strictly holds are those where the Hamiltonian
$\hat{H}(\theta_{x},\theta_{y})$ is transformed into $q$ different
$\hat{H}(\theta_{x}^{\prime},\theta_{y}^{\prime})$ under translation. We will
call these clusters LSM-clusters. On the other hand, if the Hamiltonian is
left invariant under translation, we will speak of a non-LSM cluster.
Naturally, every cluster containing $c_{x}q\times c_{y}q$ lattice sites
($c_{x},c_{y}\in\mathbb{N}$) is a non-LSM cluster as a consequence of Eq.
(13), but can be made into a LSM cluster by adding one row either in
$\bm{x}$\- or $\bm{y}$-direction. It is important to realize that Eq. (14)
cannot be proven for non-LSM clusters, even though it is to be expected that
observables such as the Hall conductivity should not change in a significant
way by adding a single row in $\bm{x}$\- or $\bm{y}$-direction when
approaching the thermodynamic limit.

Interaction terms, such as nearest-neighbor or long-range Coulomb interaction,
do not affect the translation invariance of the Hamiltonian. Therefore, one
may not anticipate interaction-driven topological phase transitions from a
non-trivial Chern insulating to a trivial insulating phase because of the LSM-
type theorem Eq. (14). On the other hand, sufficiently large Coulomb
interaction is expected to induce localized charge ordered phases such as
Wigner-crystal or CDW states [46, 47, 23] that have zero Hall conductivity. In
Sec. III and IV we will provide numerical evidence for the intuitive claim
that long-range interaction drives the system into a topologically trivial CDW
state, before rigorously discussing in Sec. V how SSB of translation
invariance leads to a breakdown of the constraint on the Hall conductivity
given by the LSM-type theorem.

##  III Topological Phase Diagram of the interacting Hofstadter Model for
Half-Integer Flux Quanta

Figure 2: Band spectrum of the $t$-$t^{\prime}$–Hofstadter model for
$\varphi=1/2$ along the high symmetry path $\Gamma(0,0)\to X(\pi/q,0)\to
M(\pi/q,\pi/q)\to\Gamma(0,0)$ of the magnetic wallpaper group
$p4m^{\prime}m^{\prime}$. The choice of hopping parameters of
$t=1=2t^{\prime}$ results in a relatively flat band structure, mimicking the
physics of Landau levels. The single-particle energy gap is exactly equal to
$4t$.

The non-interacting nearest-neighbor (NN) Hofstadter model with $\varphi=1/2$
has symmetry-protected Dirac cones due to time reversal invariance (TRI) [48,
49] because every closed loop contains an integer multiple of $\pi$ fluxes.

We include next-nearest-neighbor (NNN) hoppings to break TRI,

$\displaystyle\hat{H}=$
$\displaystyle\sum_{m=1}^{N_{x}}\sum_{n=1}^{N_{y}}\left\\{-t\operatorname{e}^{-\mathrm{i}\theta_{x}\delta_{m,N_{x}}}c_{m+1,n}^{\dagger}c_{m,n}^{\phantom{\dagger}}\right.$
(15)
$\displaystyle-t\operatorname{e}^{-\mathrm{i}\theta_{y}\delta_{n,N_{y}}}\operatorname{e}^{-\pi
m\mathrm{i}}c_{m,n+1}^{\dagger}c_{m,n}^{\phantom{\dagger}}$
$\displaystyle-t^{\prime}\operatorname{e}^{-\mathrm{i}\theta_{x}\delta_{m,N_{x}}}\operatorname{e}^{-\mathrm{i}\theta_{y}\delta_{n,N_{y}}}\operatorname{e}^{-\pi(m+1/2)\mathrm{i}}c_{m+1,n+1}^{\dagger}c_{m,n}^{\phantom{\dagger}}$
$\displaystyle\left.-t^{\prime}\operatorname{e}^{-\mathrm{i}\theta_{x}\delta_{m,N_{x}}}\operatorname{e}^{\mathrm{i}\theta_{y}\delta_{1,N_{x}}}\operatorname{e}^{\pi(m+1/2)\mathrm{i}}c_{m+1,n-1}^{\dagger}c_{m,n}^{\phantom{\dagger}}\right\\}$
$\displaystyle+\text{h.c.}$

where the Peierls phases are again determined with Eq. (2) in Landau gauge. We
set $t=1=2t^{\prime}$. The band structure and Brillouin zone are shown in Fig.
2. The energetically lower band of the single-particle band structure has a
Chern number of $C=1$, in agreement with Eq. (1). We include nearest-neighbor
Coulomb interactions:

$\hat{H}_{\text{int}}(V)=V\displaystyle\sum_{m=1}^{N_{x}}\displaystyle\sum_{n=1}^{N_{y}}(\hat{n}_{m,n}\hat{n}_{m+1,n}+\hat{n}_{m,n}\hat{n}_{m,n+1}).$
(16)

that drive the Chern insulating phase into a trivial insulating phase. In the
limit $V\gg t$ we expect at half-filling a CDW-phase. If we imagine the
cluster to be in a checkerboard structure, then the two ground state wave
functions will be a product state of all particles being localized at “black”
or “white” squares, instead of the delocalized Wannier functions of a Chern
insulator. The charge order is indicated by the static structure factor

$S(\bm{K})=\frac{1}{(N_{x}N_{y})^{2}}\displaystyle\sum_{\bm{r},\bm{r}^{\prime}}\operatorname{e}^{-\mathrm{i}\bm{K}(\bm{r}-\bm{r}^{\prime})}\langle{\hat{n}_{\bm{r}}\hat{n}_{\bm{r}^{\prime}}}\rangle$
(17)

which for a checkerboard pattern will have a peak at $\bm{K}=(\pi,\pi)$,
whereas $S(\bm{K})$ approaches zero at other $\bm{K}$. In addition, quantum
phase transitions of interacting systems are indicated by the fidelity metric
[50, 51, 52]

$g(V,\delta
V)=\frac{2}{N_{x}N_{y}}\frac{1-|\langle{\Psi_{0}(V)|\Psi_{0}(V+\delta
V)}\rangle|}{(\delta V)^{2}},$ (18)

where in our case $|{\Psi(V)}\rangle$ is the ground state of
$\hat{H}+\hat{H}_{\text{int}}(V)$. In the vicinity of a critical point one
expects a peak of the fidelity metric, irrespective of whether a possible
order parameter is commensurable with the chosen cluster. Instead of the
fidelity metric given in Eq. (18) we evaluate the negative second derivative
of the ground state energy with respect to $V$, which is closely related to
the fidelity metric [50, 51] and provides qualitatively the same information,
but is numerically less costly to evaluate:

$\chi_{E}(V)=-\frac{d^{2}E_{0}(V)}{dV^{2}}.$ (19)

The topology of the ground state(s) can also change concomitantly with a CDW
phase transition [23]. Whereas the phase transition to the CDW-phase can occur
due to an avoided crossing of ground states with a different order parameter,
a topological phase transition must have a level crossing at some twist angle
$\theta$. The Chern number is expected to change from 1 in the Chern
insulating phase to 0 in the CDW-phase at high $V$. It is then possible to
focus on the rotation eigenvalues at the three high symmetry points
$(\theta_{x},\theta_{y})=(0,0),(\pi,\pi)$ and $(0,\pi)$ (which by $C_{4}$
symmetry is related to the point at $(\pi,0)$), where the level-crossing must
occur [23]. This is the case because the Chern number is given modulo 4 by the
eigenvalues of the rotation operators [33]:

$\operatorname{e}^{-\frac{\pi\mathrm{i}C}{2}}=W_{C_{4}^{+}}^{(0,0)}W_{C_{4}^{+}}^{(\pi,\pi)}W_{C^{\phantom{+}}_{2}}^{(0,\pi)}$
(20)

with

$W_{C_{n}}^{(\theta_{x},\theta_{y})}=\langle{\Psi(\theta_{x},\theta_{y})|C_{n}|\Psi(\theta_{x},\theta_{y})}\rangle$
(21)

and the rotation operators (for arbitrary flux $\varphi$) defined by

$\displaystyle
C_{4}^{+}c_{m,n}^{\dagger}\left(C_{4}^{+}\right)^{\dagger}=\operatorname{e}^{\mathrm{i}\left(2\pi\varphi
mn-2\pi\varphi(N_{x}+1)m+\frac{\pi\varphi}{2}(N_{x}+1)^{2}\right)}$ (22)
$\displaystyle c_{-n+N_{x}+1,m}^{\dagger}$

(with $N_{x}=N_{y}$) and

$\displaystyle C_{2}c_{m,n}^{\dagger}C_{2}^{\dagger}$
$\displaystyle=\operatorname{e}^{\mathrm{i}\left(2\pi\varphi
n(N_{x}+1)-\pi\varphi(N_{x}+1)(N_{y}+1)\right)}$ (23)
$\displaystyle\qquad\qquad\qquad\qquad c_{-m+N_{x}+1,-n+N_{y}+1}^{\dagger}.$

Note that in this form $(C_{4}^{+})^{4}=\openone=C_{2}^{2}$. A gauge
transformation has to be added to the pure rotation because we do not use a
gauge with a rotation-symmetric vector potential for the Hamiltonian: it
cannot be defined for certain cluster geometries such as $q\times q$ clusters.
For the following calculations for non-LSM clusters the Hamiltonian
$\hat{H}(\bm{\theta})$, Eq. (15), will be considered, however exclusively in
the regime of twist angles $\bm{\theta}$ where the change of the symmetry
eigenvalue and hence the level crossing is identified.

Figure 3: ED of a $4\times 4$ cluster. (a) Static structure factor
$S_{\text{CDW}}$ at $\bm{K}=(\pi,\pi)$ corresponding to a checkerboard
pattern. (b) Energy of the ground state $E_{0}$ and first two excited states
($E_{1},E_{2}$). (c) Energy gap between $E_{0}$ and ($E_{1},E_{2}$). (d)
Second derivative of $E_{0}$ as a function of the interaction strength $V/t$.
The interaction strength is varied in steps $\delta V=4/1000$. The IQHE and
CDW phases are displayed in white and blue, respectively.

We first display calculations for $N_{x}\times N_{y}$ clusters of the
Hamiltonian in Eq. (15), where both $N_{x}$ and $N_{y}$ are multiples of
$q=2$. These clusters are non-LSM clusters, so they are technically not
constrained by Eq. (14) and the $\bm{\theta}$-dependence of an eigenstate can
yield a trivial Chern number. Note that these clusters are also compatible
with the checkerboard pattern CDW. In Fig. 3 we display calculations of a
$C_{4}$-symmetric cluster with $4\times 4$ sites at $\bm{\theta}=(0,0)$ using
an implicit restarted Arnoldi method implemented in Arpackpp. The phase
transition from the Hall insulating to the CDW-state is clearly indicated by a
jump in the CDW structure factor and a peak in $\chi_{E}(V)$ at $V=2.068\pm
0.04$. In the CDW-phase, the system has a quasi-degenerate (= degenerate up to
some finite size splitting) twofold ground state, corresponding to the two
possible charge-density-wave patterns with a small finite size gap. The
symmetry eigenvalue $W_{C_{4}}^{(0,0)}$ of the lower energy ground state
changes from $-\mathrm{i}$ to $1$, which indicates a Chern-trivial phase of
the ground state according to Eq. (20). We verified the absence of accidental
crossings for different twist angles in the CDW phase, which ensures that the
many-body Chern number of the lower energy ground state is well defined.

To take into account finite-size effects, calculations on a $4\times 6$
cluster have also been performed. Note that this cluster breaks $C_{4}$
rotation symmetry, so instead of Eq. (20) the many-body Chern number is
determined with

$\operatorname{e}^{\pi\mathrm{i}C}=W_{C_{2}}^{(0,0)}W_{C_{2}}^{(0,\pi)}W_{C_{2}}^{(\pi,0)}W_{C_{2}}^{(\pi,\pi)}.$
(24)

There is no qualitative difference in the results of these calculations
compared to those shown in Fig. 3. The phase transition occurs at $V=2.078\pm
0.003$ and is again accompanied by a change of the Chern number of the ground
state, given by the symmetry indicator in Eq. (24).

Next we consider a $4\times 5$ cluster, which is an LSM cluster as $N_{y}$ is
an odd number. The Hamiltonian now transforms under translation, according to
Eq. (13), as

$\hat{T}_{\bm{x}}^{M}\hat{H}(\theta_{x},\theta_{y})\left(\hat{T}_{\bm{x}}^{M}\right)^{\dagger}=\hat{H}(\theta_{x},\theta_{y}+\pi).$
(25)

As a result, any single state with an energy gap for all twist angles must
have a non-trivial Chern number according to Eq. (14). In addition, the ground
state degeneracy of the CDW-phase increases, as a checkerboard pattern is not
commensurable with the lattice dimensions anymore. In the limit of $t\to 0$,
$V=1$ there is a tenfold ground state degeneracy with an excitation gap of
$V$. This can be understood from the fact that the reciprocal lattice vector
$\bm{K}=(\pi,\pi)$ is not in the reciprocal lattice of clusters for odd
$N_{y}$.

Figure 4: Ground state and ground state quasidegeneracy of a (a) Hall
insulating versus (b) trivial insulating state. In the trivial insulating
state, each gapped quasidegenerate ground state possesses a nontrivial Chern
number. In both cases the ground state(s) have to be separated by some energy
gap $\Delta E$ from excitations.

How does the phase transition occur and can we still speak of a topological
phase transition? The system is driven into the CDW by an avoided crossing,
where the nondegenerate Hall insulating state turns into ten quasidegenerate
CDW states. The system undergoing an avoided crossing instead of a level
crossing is symmetry protected by the LSM-type theorem. For larger LSM
clusters, the discontinuous nature of the phase transition that is evident for
the non-LSM clusters studied before will become more apparent. While it is
true that the lowest energy ground state and the other quasidegenerate states
(assuming that they are all gapped with respect to each other) are
individually topologically non-trivial, the sum of their Chern numbers must be
equal to zero as they are adiabatically connected to Chern-trivial states at
$t\to 0$, $V=1$. This situation is displayed in Fig. 4. In this sense the
phase transition is still topological, which will be discussed much more
carefully in Sec. V.

Figure 5: ED of a $4\times 5$ cluster. (a) Energy of the ground state $E_{0}$
and first two excited states ($E_{1},E_{2}$). (b) Energy gap between $E_{0}$
and ($E_{1},E_{2}$). (c) Second derivative of $E_{0}$ as a function of the
interaction strength $V/t$. The interaction strength is varied in steps
$\delta V=4/1000$. The IQHE and CDW phases are displayed in white and blue,
respectively. The two images correspond to two orthogonal states that minimize
$\hat{H}_{\text{int}}$ and that are related to each other by two virtual
hoppings.

Calculations are shown in Fig. 5. Since the LSM-type theorem prohibits a
topological phase transition of the lowest energy ground state, there is no
obvious choice for a particular twist angle $\bm{\theta}$ anymore. We pick
$\bm{\theta}=(0,-\pi\varphi N_{y})$, where the Hamiltonian is at least $C_{2}$
invariant. The phase transition is still indicated by a peak of $\chi_{E}(V)$
at $V=2.71\pm 0.01$, although due to strong finite size effects and the
cluster not being commensurable to the CDW order, it appears broad on the
logarithmic scale. It should be noted that even at $V=10$ there are still
considerable energy gaps between the three quasi-degenerate ground states.
This can be understood by considering perturbation theory. The two ground
states of a commensurate cluster are related by $N_{e}$ virtual hoppings with
an amplitude of $t/V$, whereas in a non-commensurate cluster hybridization
between some of the ground states will approximately take $\sqrt{\rho N_{e}}$
virtual hoppings, as the CDW order is incommensurable with the lattice
geometry by one row, leading to a much larger splitting between the lowest
energy levels.

(a)

(b)

(c)

Figure 6: Static structure factor of the Hofstadter Model on a $4\times 5$
cluster at (a) $V=2$, (b) $V=4$ and (c) $V=\infty$. Corresponding to the
cluster dimensions, the wave vectors of the reciprocal lattice are given by
$K_{x}=-\pi,-\pi/2,\dots,\pi/2$ and $K_{y}=-4\pi/5,-2\pi/5,\dots,4\pi/5$.

In Fig. 6 we display calculations of the CDW structure factor at all $\bm{K}$
points corresponding to the $4\times 5$ cluster. Since $\bm{K}=(\pi,\pi)$ is
no longer contained, the CDW structure factor for $V=\infty$ shows a peak at
$K_{x}=\pi$, $K_{y}=\pm 4\pi/5$ of $S_{\text{CDW}}(\bm{K})=0.082(6)$. While
charge order is clearly indicated at large $V$, $S_{\text{CDW}}(\bm{K})$ grows
muchs slower than in the commensurate non-LSM clusters studied before. Some
order does emerge after the phase transition at $V=4$ indicated by
$S_{\text{CDW}}(\bm{K})=0.045(9)$ compared to $V=2$ with
$S_{\text{CDW}}(\bm{K})=0.025(4)$, however even at $V=20$ one finds
$S_{\text{CDW}}(\bm{K})=0.053(6)$, which is still far away from the value at
$t=0$, $V=1$.

Comparing our results from the non-LSM clusters with the LSM cluster, the
discontinuous phase transition into a topologically trivial CDW phase is less
clear indicated in the LSM cluster, as the level crossings in the non-LSM
clusters lead to a jump of the CDW structure factor and a change of the lowest
energy ground state topology. In comparison, the avoided level crossing with
continuous $S(\bm{K})$ and $\chi_{E}$ of the LSM cluster makes the
identification of phase boundaries more challenging, as well as the fact that
the change in the topology of the system can only be described by considering
all quasidegenerate ground states together.

We close this section by appending a comment on the interplay between the
topological phase transition of the ground state wave function, the CDW
transition and the symmetries of the cluster. Using ED calculations, it is
possible to map out the phase boundaries at zero temperature of the
Hamiltonian with respect to some parameter using the lowest energy ground
state. The topological phase transition may then not occur at the same
parameter as the CDW transition transition solely due to finite size effects;
in particular this is often caused by the cluster not having the same
symmetries as the infinite plane [23, 24]. In our case there will be no
topological phase transition of the lowest energy ground state wave function
at all for LSM-clusters, where the Hamiltonian does not commute with the
translation operator, while we found perfect overlap between both transitions
for non-LSM clusters. We have already argued that for the characterization of
the topological phase transition one has to consider the topology of all
quasidegenerate ground states, instead of the lowest energy one. On non-LSM
clusters however, the topological phase transition is also accompanied, by a
topological phase transition of the lowest energy ground state by a level
crossing. One arrives therefore at the seemingly paradoxical but by no means
contradictory result that in the case of actual translation invariance, the
Chern number of the ground state is not constrained by the LSM-type theorem,
whereas in the other cases the LSM-type theorem is strictly valid and one
finds a topological phase—that however is protected only by a finite size gap.

##  IV Topological Phase Transitions for Various Flux Quantum Ratios

We want to complete the numerical study of Sec. III and inquire the
possibility of topological phase transitions of the Hofstadter model for flux
quantum ratios $\varphi\neq 1/2$ and particle densities $\rho\neq 1/2$ that
can still be explored within the limits of ED. In contrast to other methods,
ED has the advantage that the aforementioned influence of the cluster geometry
on the validity of the LSM constraint can be inquired; in addition, a mean-
field ansatz is difficult to implement for band fillings other than $1/2$,
where the order parameter is unknown. This is already an issue on the
honeycomb lattice [53], but especially problematic on the square lattice.

In the following, the Hamiltonian is always of the form

$\hat{H}=\hat{H}(\bm{\theta})+\hat{H}_{\text{int}}(V),$ (26)

where $\hat{H}(\bm{\theta})$ has been defined in Eq. (10). NNN-hopping is not
included as NN-hopping is sufficient to break TRI. Here
$\hat{H}_{\text{int}}(V)$ provides the long-range Coulomb interaction

$\hat{H}_{\text{int}}(V)=\displaystyle\sum_{m\leq
m^{\prime}=1}^{N_{x}}\displaystyle\sum_{n\leq
n^{\prime}=1}^{N_{y}}\frac{V}{r_{\text{min}}}\hat{n}_{m,n}\hat{n}_{m^{\prime},n^{\prime}}$
(27)

where $r_{\text{min}}$ is taken to be the smallest distance on the torus
geometry. The interaction range is now truly long range and decays like $1/r$,
in contrast to Eq. (16), where nearest neighbor Coulomb interaction was
considered to be sufficient to obtain a CDW at half filling.

We perform calculations for the flux quantum ratios $p/q=1/5$ and $p/q=2/5$
with a particle density corresponding to both one or two occupied Hofstadter
bands. The critical interaction strengths for each flux quantum ratio and
number of particles as well as the single particle gap $\Delta E$ of the non-
interacting band structure is displayed in Tab. 1. As a general trend, smaller
single particle bandwidths correspond to smaller $V_{\text{crit}}$. Plots of
the single particle band structures are presented in Appendix B.

Plots for the static structure factors are provided in Appendix C for the
closest numerically evaluated interaction strengths before and after the phase
transition into the CDW regime. The onset of charge order is clearly indicated
by the jump of $S_{\text{CDW}}$ as several $\bm{K}$ values. In addition, for
clusters with $\rho=2/5$ we observe a breaking of $C_{4}$ rotation symmetry.

$p/q$ | $N_{e}$ | $N_{x}\times N_{y}$ | $\Delta E$ | $V_{\text{crit}}$
---|---|---|---|---
1/5 | 5 | $5\times 5$ | 1.553699 | $22.652\pm 0.002$
1/5 | 10 | $5\times 5$ | 0.520147 | $4.7875\pm 0.0125$
2/5 | 5 | $5\times 5$ | 0.157179 | $4.918\pm 0.002$
2/5 | 10 | $5\times 5$ | 1.539145 | $17.775\pm 0.025$
Table 1: Critical interaction strengths $V_{\text{crit}}$ and single-particle
band gaps $\Delta E$ for varying flux quanta, particle numbers and lattice
dimensions. The particle number $N_{e}$ always corresponds to an integer
filling of non-interacting bands.

##  V Spontaneous translation invariance breaking

In view of the previous numerical results, in this section we want to inquire
the relation between spontaneous translation symmetry breaking and the
topological phase transition from Chern to trivial insulators with magnetic
translation symmetry more rigorously, aware that the LSM-type theorem is a
non-perturbative result. It is clear that for macroscopically large systems,
the physics behind the phase transition cannot change in a meaningful way
depending on the specific cluster geometry. Nevertheless, in the following
argumentation we will have to carefully distinguish between LSM and non-LSM
clusters, by virtue of the fact that some properties are only mathematically
well-defined in one or the other. An example is the LSM-type theorem itself,
which only strictly holds for LSM geometries. Nonetheless, the physical
relevance of these properties has to be the same in the thermodynamic limit,
independent of the lattice geometry.

We start by considering integer fillings—that is an integer number of
Hofstadter bands are occupied. These cases have been investigated numerically
in this work. Then we continue with the consideration of fractional fillings
and the differences between lattice and continuum models in this context. For
convenience, when translation symmetry is addressed, we always have the
magnetic translation operator $\hat{T}_{\bm{a}}^{M}$ in mind unless stated
otherwise.

###  V.1 Integer quantum Hall effect

The formation of a CDW phase for a translation invariant Hamiltonian requires
spontaneous symmetry breaking of translation invariance. In the thermodynamic
limit, the Hamiltonian will obtain a set of degenerate ground states. The wave
function that describes a real sample (at zero temperature) will be a
superposition of these ground states that breaks translation symmetry. In
compliance with our numerical calculations we want to understand this
mechanism and its physical implications on the LSM-type theorem now for
possibly macroscopic systems, but still of finite size. We need to understand
why quasidegeneracy is essential for the phase transition into a topologically
trivial insulating state and why its origin must be given by the spontaneous
breaking of translation symmetry as opposed to some other cause.

On non-LSM clusters, i.e. clusters where the magnetic translation operators
commute with the Hamiltonian, spontaneous breaking of translation symmetry
must be accompanied by a set of quasi-degenerate ground states that are
eigenstates of the translation operators with different crystalline momenta
$\bm{K}$. While the different non-degenerate eigenstates (in particular the
ground states) of the Hamiltonian are naturally eigenstates of the translation
operator for these finite size clusters, a superposition of states belonging
to different $\bm{K}$ breaks translation invariance.

Why does the ground state wave function of a real sample break translation
symmetry? In the following we will assume that the symmetry breaking phase
also has a local order parameter, such as the varying charge density. Then the
macroscopic wave function will collapse upon any measurement of the local
order parameter into one of the symmetry broken states. The system is trapped
in the symmetry broken state due to ergodicity breaking [54]. A flux insertion
to measure the Hall conductivity within the linear response regime is too
small a perturbation to let the system tunnel into another ground state, in
particular if the broken symmetry is discrete. Note that this argument relies
on the existence of a local order parameter, a point to which we will come
back to in the next subsection.

We have seen that the CDW phase comes with a quasidegenerate ground state
spectrum and a translation symmetry breaking state best describes the real
system. How is this connected to topology? On non-LSM clusters Eq. (14) does
not strictly hold anymore. This manifests itself in the fact that in that case
it is possible to have gapped eigenstates of $\hat{H}(\bm{\theta})$ for all
twist angles $\bm{\theta}$ with a trivial many-body Chern number, which
justified the use of symmetry-indicators to detect the topological phase
transition in Sec. III. Is a ground state quasi-degeneracy then required to
set up a topologically trivial phase and does it have to be related to
spontaneous symmetry breaking?

Let us first address the question of degeneracy. In the case of LSM clusters
they must have a quasi-degeneracy of a multiple of $q$ according to Eq. (14),
if they are supposed to belong to a trivial insulating phase, otherwise the
sum of the Chern numbers of the ground states could not be zero and the ground
states would not be adiabatically connected to localized, hence topologically
trivial CDW states. On non-LSM clusters that have no long-range structure of
period $q$ 333In contrast, a long-range structure of period $q$ could be
provided for example by a stripe pattern of impurities that would only break
translation symmetry in one direction and would in fact result in a
nondegenerate and topologically trivial ground state if the stripe pattern is
commensurable with the given cluster. one expects to find the same quasi-
degeneracy [29]. If the physical origin of the quasi-degeneracy is the
formation of a CDW, then the number of ground states may only change depending
on whether the CDW pattern is commensurable with the given cluster as we have
seen in Sec. III. However, the very presence of many ground states must
pertain to all possible impurity-free clusters.

Why is the degeneracy a multiple of $q$ for non-LSM clusters? Consider mean-
field theory: if the formation of a CDW is modeled by an on-site potential of
an effective single-particle Hamiltonian, then the unit cell of the effective
Hamiltonian must contain an integer multiple of a flux quantum for the single
particle band structure to be topologically trivial according to Eq. (1). This
is only possible if it contains a multiple of $q$ original unit cells
plaquettes resulting in an integer multiple of $q$ different possible
positions of the on-site potential and hence an integer multiple of $q$
different ground states. Note that one considers different mean-field
Hamiltonians for a single Hamiltonian of the many-body system. The ground
states are degenerate, as the different mean-field Hamiltonians are equivalent
to each other under translation on non-LSM clusters.

We have argued, why a quasi-degeneracy of ground states in a translation
invariant Hamiltonian in an external magnetic field is essential to have a
trivial Hall conductivity. Why is the spontaneous breaking of the translation
symmetry the origin of the quasi-degeneracy in a topologically trivial phase?
An example for a different discrete symmetry that may be broken spontaneously
is TRI, which can in principle occur at a flux quantum ratio of $p/q=1/2$,
where the Hamiltonian commutes with the TR operation if one only includes
nearest neighbor hoppings in the Hamiltonian. In case of spontaneous TRI
breaking, the Hamiltonian would have two time reversal partners as ground
states that do not have to break translation symmetry. In Ref. [31], Eq. (14)
is argued to hold in the thermodynamic limit if the ground state is an
eigenstate of the magnetic translation operator. If translation maps a flux
(associated to some twist angle) to a different flux according to Eq. (13),
then ground states possess an effective translation symmetry, expressed
through the modified translation operator

$\tilde{\hat{T}}^{M}_{\bm{x}}=\hat{F}_{y}(2\pi\varphi
N_{y})\hat{T}^{M}_{\bm{x}},$ (28)

where $\hat{F}_{y}(2\pi\varphi N_{y})$ is a flux insertion to exactly
compensate the change of Aharonov-Bohm phases around non-contractable loops of
the given cluster. Let us pick one of the two time reversal partners. Under
the tacit assumption that this state cannot transform into its partner by
translation or adiabatic flux insertion, the Hall-conductivity would then
still be quantized to non-zero values, as the effective translation symmetry
maps the state onto itself. A topologically trivial phase hence requires SSB
of translation symmetry.

We want to provide a final proof, why the Hall conductivity for sufficiently
strong long-range interaction has to be trivial, independent of the filling
factor, lattice geometry and underlying tight-binding Hamiltonian. In the
limit of infinitely strong interaction strength, ground states must minimize
$\langle{\hat{H}_{\text{int}}}\rangle$. The set of ground states will then
only consist of charge-ordered states $|{\tilde{\Psi}_{i}}\rangle$ that do not
hybridize with respect to the tight-binding Hamiltonian, i.e.
$\langle{\tilde{\Psi}_{i}|\hat{H}|\tilde{\Psi}_{j}}\rangle=0$, as going from
one state to the other would require multiple fermions to hop to a different
lattice site, provided the interaction is long ranged. As a consequence,
ground states will not be affected by a flux insertion so that only zero Hall
transport is possible. Finite $t/V\ll 1$ leads to negligible corrections; in
the case of $\varphi=1/2=\rho$ and non-LSM clusters, for example, these are of
the order of $\mathcal{O}(t/V)^{N_{e}}$.

###  V.2 Fractional quantum Hall effect

We now discuss the more general case of fractional fillings, in particular the
fractional quantum Hall effect (FQHE). Here, a genuine many-body treatment is
inevitable as the presence of quasiparticles in a topologically ordered system
does not permit the usual mean-field ansatz 444This does not rule out the
possibility of a mean-field ansatz in terms of an effective quasiparticle
Hamiltonian [75]. In contrast to integer fillings, ground state degeneracy is
enforced by the generalized LSM theorem in the fractional Hall insulating
phase and the LSM-type theorem becomes [33]

$\operatorname{e}^{2\pi\mathrm{i}(\frac{p}{q}C-\rho)D}=1,$ (29)

with $q\rho=\bar{m}/D$ and coprime $\bar{m},D$, where $D$ is equal to the
ground state degeneracy and $C$, which is now to be interpreted as the Chern
number per ground state, is a rational fraction of $D$. We will argue why the
characteristic topological properties of these ground states, among others the
LSM-type theorem Eq. (29), are protected by magnetic translation symmetry
before arguing why CDW states are trivial nevertheless.

One of the hallmarks of the FQHE is topological order that manifests itself on
a torus geometry as ground state degeneracy due to the presence of
quasiparticles in the thermodynamic limit [57]. The number of ground states is
independent of the chosen cluster as its origin lies in the quasiparticle
statistics and not in some long-range structure in real space. For the
calculation of the Hall conductivity, the joint Chern number of all ground
states has to be computed, in contrast to the integer filling states in Sec.
V.1, where we argued that a flux insertion should be seen as a small
perturbation that does not allow a ground state to tunnel into a different
ground state. However, in the case of the FQHE, the different ground states
have to transform into each other under flux insertion as argued in [58, 57]
from a field-theoretic perspective. Let

$\tilde{\hat{F}}_{x/y}(\Phi_{0})$ (30)

be a combination of a flux insertion operation of $\Phi_{0}$ and a gauge
transformation (see Eq. (46)) that maps a ground state to another ground
state. Then

$\tilde{\hat{F}}_{x}(\Phi_{0})\tilde{\hat{F}}_{y}(\Phi_{0})\cong\operatorname{e}^{-2\pi\mathrm{i}/D}\tilde{\hat{F}}_{y}(\Phi_{0})\tilde{\hat{F}}_{x}(\Phi_{0}).$
(31)

The symbol $\cong$ restricts Eq. (31) to the ground state subspace. Eq. (31)
implies at least $D$ ground states as the flux insertion operators obey the
same algebra as magnetic translation operators. This transformation behavior
as well as the ground state degeneracy [59] can be proven exactly for some
non-LSM clusters as demonstrated in Appendix D, even though, similarly to the
generalized LSM theorem and LSM-type theorem, there are some technicalities
depending on the chosen cluster. In particular, as any stable exact degeneracy
has to be symmetry protected, Eq. (31) strictly holds only if the
topologically degenerate ground states (that is degenerate in the
thermodynamic limit even in the presence of impurities or lattice potential
[58]) are related by magnetic translation symmetry (see Eq. (43) and Eq.
(44)). Otherwise finite size splittings lead to a change of the algebra in Eq.
(31), provided the adiabatic flux insertion is slow compared to finite size
gaps, which is not an issue in the thermodynamic limit. Since the different
states transform into each other, when inserting flux to measure the Hall
conductivity, the contribution of all ground states has to be summed.

What happens in the CDW phase regime? In the limit $t\to 0$, $V=1$ it is clear
that CDW states don’t transform under a flux insertion and should hence be
eigenstates of both flux insertion operators, implying that the flux insertion
operators in Eq. (30) now commute. This already implies at least $D^{2}$
quasi-degenerate ground states, compared to the $D$ ground states of the FQHE,
see Appendix D. In addition, the number of ground states also has to be an
integer multiple of $q$, according to Eq. (29). The quasi-degeneracy turns
into sets of $D$ exactly degenerate ground states if they are related by
translation, as then they transform under irreducible projective
representations $D_{M}^{\bm{K}}$ of the translation group [8]. It should be
noted that these degenerate states possess different crystalline momenta, when
expressed in the eigenbasis of either $\hat{T}_{\bm{x}}^{M}$ or
$\hat{T}_{\bm{y}}^{M}$. This alone, however, does not indicate spontaneous
symmetry breaking. In the FQHE regime this difference in momentum is carried
by quasiparticles known as visons in the context of quantum Hall physics [31,
60]. Instead, in the CDW phase regime there have to be $D$ copies of each
$D_{M}^{\bm{K}}$. In the case that the many-body translation operators commute
and states transform under the usual 1D irreducible representations of the
translation group, the symmetry protection of the ground state degeneracy is
lifted and only maintained in the thermodynamic limit. The ground states of
the FQHE then all carry the same momentum, whereas the CDW are again
characterized by different momenta [60].

###  V.3 Continuous space

So far only lattice Hamiltonians have been considered. We close this section
by touching briefly on free interacting particles. The absence of a periodic
lattice potential implies continuous translation invariance in the infinite
plane. On a torus, following the argumentation leading to Eq. (13), a twist
angle $\bm{\theta}$ can be transformed into any other twist angle by a
suitable translation. This property is independent of the chosen sample
dimensions so that the LSM-type theorem always applies exactly. In fact,
translation invariance implies that the Chern numbers of each Landau level
have to be equal to one, which can be seen in the following way. It is
possible to divide a torus, being pierced by $N_{\Phi_{0}}$ unit flux quanta,
into a grid of $N_{c}$ virtual plaquettes with a flux of $p/q$ flux quanta per
plaquette, so that $N_{\Phi_{0}}=N_{c}p/q$. A filling factor of $n_{LL}$
occupied Landau levels leads to an average particle density of

$\rho=\frac{N_{\Phi_{0}}n_{LL}}{N_{c}}=\frac{p}{q}n_{LL}.$ (32)

particles per virtual plaquette With coprime $p,q$ and according to Eq. (14)
this immediately implies that

$C-n_{LL}=0\text{ mod }q.$ (33)

Since this argument is independent of the partition of the torus, Eq. (33) has
to hold for all $q$, leading to $C=n_{LL}$.

At sufficiently large long-range interactions, a quantum Hall system can be
expected to be driven into a Wigner crystal phase with zero Hall conductance
[61]. In case the classical Wigner lattice does not contain an integer number
of flux quanta, even in the limit of strong coulomb interaction, finite
kinetic energy may cause the electrons to condense into stripe or bubble
patterns that may lead to further reduction of translation symmetry [62, 63].
From the previous paragraph it follows that either the number of ground states
would have to be infinite which can be ruled out for finite clusters or there
has to be a gapless excitation spectrum. This is realized, at least in the low
energy sector, by Goldstone modes as a consequence of the breakdown of
continuous translation symmetry: in a Wigner crystal electrons order
periodically and hence possess gapless phonon-like excitations [64]. The
stability of such a charge ordering in the case of free particles depends on
the electrons being subject to a long-range interaction (especially at finite
temperature, where long range interaction provides a loophole of the
Hohenberg-Mermin-Wagner theorem [65]). Indeed, in the case of the FQHE
sufficiently short ranged interaction actually stabilizes the incompressible
quantum Hall liquid state [66, 67].

It should also be noted that the breakdown of translation invariance is a
necessary but no sufficient criterium for a trivial Hall conducting phase.
Nonzero Hall conductivity may coexist with electrons forming crystalline order
[61] and can be caused by collective excitations, which is backed by
experimental data [68, 69, 70].

##  VI Conclusion and Outlook

In this paper we have studied topological phase transitions of the interacting
Hofstadter model from a Chern insulating to a trivially insulating charge-
density-wave phase for various flux quantum ratios and particle numbers. The
constraint on the quantization of the Hall conductivity due to Lieb-Schultz-
Mattis (LSM) type theorem, depending on translation symmetry that in
particular rule out a Chern-trivial phase, is circumvented via spontaneous
breaking of translation symmetry driven by long-range interaction. Qualitative
differences between the inquired cluster geometries emerge depending on
whether they have to strictly obey the LSM type theorem. On a large class of
clusters the topological phase transition occurs via a symmetry protected
level crossing with a jump of the charge density wave structure factor
indicating a discontinuous phase transition into a topologically trivial CDW
phase. Only on a specific set of cluster geometries it is seen that a
topologically non-trivial phase is protected by a finite size gap which
however quickly goes to zero in the thermodynamic limit.

For future studies, it would be interesting to inquire interacting quantum
spin Hall phases subject to half integer flux quanta, where LSM-type theorems
also impose a constraint on the $Z_{2}$ invariant [71], in particular it is
far less clear, whether a finite size gap can stabilize a genuine topological
many-body phase, as the definition of the $Z_{2}$ invariant often relies on
field-theoretic arguments or a Schmidt decomposition which can be problematic
in the case of quasi-degeneracy. A tool that may be more suited to study SSB
for larger systems, even though the subtleties of the lattice geometry may be
lost, is DMRG, where the obtained ground state usually spontaneously breaks
symmetries, if the true ground state is quasi-degenerate [72]. Finally, while
the present study focuses on the Hofstadter Hamiltonian as a popular toy
model, in particular in the study of cold-atom gases [16], our results may
also be of value for quantum Hall phases in Moiré superlattices, where the
flux quantum per unit cell can be of the order of a unit flux quantum [73,
74].

###### Acknowledgements.

We wish to thank Elias Lettl and Alexander Wietek for stimulating discussions.
This research was supported by the Deutsche Forschungsgemeinschaft (DFG)
through QUAST-FOR5249 - 449872909 (project TP4) and it was supported in part
by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics
(KITP).

##  Appendix A Proof of the Hall conductivity constraint

Here we outline how to deduce Eq. (1) and its many-body generalization. The
Chern number of Bloch states $|{\Psi^{n}(\bm{k})}\rangle$ occupying a gapped
band is given by

$C_{n}=-\frac{1}{2\pi}\int_{\text{BZ}}\mathop{}\\!\mathrm{d}\bm{S}\bm{\nabla}_{\bm{k}}\times\bm{A}^{n}(\bm{k})$
(34)

with the Berry connection

$\bm{A}^{n}(\bm{k})=\mathrm{i}\langle{\Psi^{n}(\bm{k})|\bm{\nabla}_{\bm{k}}|\Psi^{n}(\bm{k})}\rangle.$
(35)

It is possible to rewrite Eq. (34) using Stokes’ theorem as a line integral
over the boundary of the Brillouin zone [38]. Then the wave function must be
chosen to be smooth over the entire Brillouin zone but cannot satisfy the
boundary condition of the Brillouin zone, if it has a non-zero Chern number.
For the Bloch states defined in the leadup to Eq. (8) we can define a smooth
gauge with the boundary conditions

$\begin{gathered}|{k_{x}+2\pi/q,k_{y}}\rangle=|{k_{x},k_{y}}\rangle\\\
|{k_{x},k_{y}+2\pi}\rangle=\operatorname{e}^{-\mathrm{i}qCk_{x}}|{k_{x},k_{y}}\rangle.\end{gathered}$
(36)

Furthermore, we specify the wave functions to satisfy [7]

$\hat{T}_{\bm{x}}^{M}|{k_{x},k_{y}}\rangle=\operatorname{e}^{-\mathrm{i}tk_{x}q}|{k_{x},k_{y}-2\pi\varphi}\rangle,$
(37)

where the integer $t$ will be model specific (in the case of the Hofstadter
model with only nearest neighbor hopping, they are given by a specific
Diophantine equation [38] as a special case of Eq. (1)). We then identify

$\displaystyle\operatorname{e}^{-\mathrm{i}qk_{x}}|{k_{x},k_{y}}\rangle=\left(T_{\bm{x}}^{M}\right)^{q}|{k_{x},k_{y}}\rangle=\operatorname{e}^{-\mathrm{i}tk_{x}q^{2}}|{k_{x},k_{y}-2\pi
p}\rangle$ (38)
$\displaystyle=\operatorname{e}^{-\mathrm{i}tk_{x}q^{2}-\mathrm{i}Cpqk_{x}}|{k_{x},k_{y}}\rangle,$

leading to

$C\varphi-\frac{1}{q}=0\text{ mod 1}.$ (39)

If we don’t occupy a single band, but many bands with a number of particles
per unit cell $\rho$, then Eq. (1) follows.

For many-body systems, Eq. (14) can be shown analogously [33]. The only major
difference is that the many-body translation operators technically depend on
twist angles resulting in some additional gauge transformations in the proof.
For example,

$\displaystyle\hat{T}_{\bm{x}}^{M,\theta_{x}}c_{m,n}^{\dagger}\left(\hat{T}_{\bm{x}}^{M,\theta_{x}}\right)^{\dagger}=$
$\displaystyle\operatorname{e}^{-\mathrm{i}\theta_{x}\delta_{m,N_{x}}}\operatorname{e}^{-2\pi\mathrm{i}\varphi
n}c_{m+1,n}^{\dagger}$ (40)
$\displaystyle\hat{T}_{\bm{y}}^{M,\theta_{y}}c_{m,n}^{\dagger}\left(\hat{T}_{\bm{y}}^{M,\theta_{y}}\right)^{\dagger}=$
$\displaystyle\operatorname{e}^{-\mathrm{i}\theta_{y}\delta_{n,N_{y}}}c_{m,n+1}^{\dagger}$

commute with the Hofstadter Hamiltonian in Eq. (10), provided $N_{x/y}$ are
integer multiples of $q$. In addition, while for single particles on the
infinite plane one can pick any

$\left(\hat{T}_{\bm{x}}^{M}\right)^{\alpha},\left(\hat{T}_{\bm{y}}^{M}\right)^{\beta}$
(41)

with $\alpha\beta=q$ to define a set of Bloch vectors (where for convenience
one choses $\alpha=q$, $\beta=1$), in the many-body case the translations that
map one twist angle to another one are determined by the cluster geometry. If
translations in the $\bm{x}$-direction result according to Eq. (13) in
$\alpha^{\prime}$ different twist angles and translations in the
$\bm{y}$-direction result in $\beta^{\prime}$ different twist angles, then Eq.
(14) follows if $\alpha^{\prime}\beta^{\prime}=q$. Otherwise for
$\alpha^{\prime}\beta^{\prime}<q$ one obtains a weaker quantization condition
or none in the case of $\alpha^{\prime}=\beta^{\prime}=1$.

##  Appendix B Single-particle band structures

The single particle band structures of the Hofstadter model are displayed in
Fig. 7 along the high symmetry path $\Gamma(0,0)\to X(\pi/q,0)\to
M(\pi/q,\pi/q)\to\Gamma(0,0)$ of the magnetic wallpaper group
$p4m^{\prime}m^{\prime}$.

(a)  (b)

Figure 7: Band spectra of the Hofstadter model for (a) $\varphi=1/3$, (b)
$\varphi=1/4$, (c) $\varphi=1/5$, (d) $\varphi=2/5$, (e) $\varphi=1/6$.

##  Appendix C Static structure factors

Static structure factors of the Hofstadter model for $\varphi,\rho=1/5,2/5$
are displayed in Fig. 8.

(a)  (b)  (c)  (d)

(e)  (f)  (g)  (h)

Figure 8: Static structure factors of the Hofstadter Model on $5\times 5$
clusters shortly before (left panels) and after (right panels) the phase
transition. $N_{e}=5$ and $\varphi=1/5$ (a),(b), $N_{e}=5$ and $\varphi=2/5$
(c),(d), $N_{e}=10$ and $\varphi=1/5$ (e),(f), $N_{e}=10$ and $\varphi=2/5$
(g),(h).

##  Appendix D Ground States Under Flux Insertion

In this appendix we want to deduce Eq. (31) in the fractional quantum Hall
regime and discuss the consequences for the CDW phase.

On a torus, a FQH phase has a $D$-fold quasi-degenerate ground state. Let the
particle density be

$\rho=\frac{n}{qD},$ (42)

where $n$ and $D$ are coprime. We consider in the following only non-LSM
clusters with $N_{x}=n_{x}q$ and $N_{y}=n_{y}q$. The algebra of magnetic
translation operators obeys

$\hat{T}_{\bm{x}}^{M}\hat{T}_{\bm{y}}^{M}=\operatorname{e}^{-2\pi\varphi\mathrm{i}N_{e}}\hat{T}_{\bm{y}}^{M}\hat{T}_{\bm{x}}^{M},$
(43)

where

$\operatorname{e}^{-2\pi\varphi\mathrm{i}N_{e}}=\operatorname{e}^{-2\pi\frac{p}{q}\mathrm{i}N_{x}N_{y}\rho}=\operatorname{e}^{-2\pi\mathrm{i}\frac{pnn_{x}n_{y}}{D}}=\operatorname{e}^{-2\pi\mathrm{i}\frac{m}{D}}$
(44)

and we stipulate coprime $m$ and $D$ (note that this may not always be
possible depending on the flux quantum ratio and particle number per unit
cell, e.g. in the case of $n=1$, $q=7$ and $D=3$). As a consequence, the
ground states are exactly degenerate. Under continuous flux insertion of a
flux $\Phi$ during a time $T$, say, without loss of generality in
$\bm{y}$-direction, the hoppings of the Hamiltonian change as

$c_{m+1,n}^{\dagger}c_{m,n}\to
c_{m+1,n}^{\dagger}c_{m,n}\operatorname{e}^{-\mathrm{i}\frac{\Phi}{\Phi_{0}}\frac{1}{N_{x}}\frac{t}{T}},$
(45)

which is unitarily equivalent to a twisted boundary of $\Phi t/\Phi_{0}T$. The
form of Eq. (45) has the advantage that it leaves the Hamiltonian translation
invariant, i.e.

$[\hat{T}_{\alpha}^{M},\hat{H}(\Phi t/T)]=0,$ (46)

where here and in the following $\alpha$ will denote either $x$ or $y$. Note
that the insertion of a flux quantum $\Phi_{0}$ does not produce a measurable
Aharonov-Bohm phase, as it is periodic in $\Phi_{0}$. Consequently,
$\hat{H}(\Phi_{0})$ and $\hat{H}(0)$ are equivalent up to a gauge
transformation.

We define the so-called “large gauge transformation” [31]

$\hat{U}_{\alpha}=\exp\left(\frac{-2\pi\mathrm{i}}{L_{\alpha}}\displaystyle\sum_{m,n}r_{\alpha}c_{m,n}^{\dagger}c_{m,n}\right),$
(47)

where $r_{\alpha}$ is the $\alpha$-component of $(m,n)$. It is easy to verify
that $\hat{H}(\Phi_{0})=\hat{U}_{\alpha}\hat{H}(0)\hat{U}_{\alpha}^{\dagger}$.
We can now define the operator

$\tilde{\hat{F}}_{\alpha}(\Phi_{0})=\hat{U}_{\alpha}^{\dagger}\hat{F}_{\alpha}(\Phi_{0})$
(48)

with the usual flux insertion operator

$\hat{F}_{\alpha}(\Phi)=\hat{T}\exp\left(-\frac{\mathrm{i}}{\hbar}\int_{0}^{T}\mathop{}\\!\mathrm{d}t\hat{H}(\Phi
t/T)\right)$ (49)

and time ordering operator $\hat{T}$. Consequently,
$\tilde{\hat{F}}_{\alpha}(\Phi_{0})$ is to be interpreted as an operator that
introduces a flux quantum $\Phi_{0}$ adiabatically through the action of
$\hat{F}_{\alpha}(\Phi_{0})$ for large enough $T$, and then the gauge
transformation with $\hat{U}_{\alpha}^{\dagger}$ converts $\hat{H}(\Phi_{0})$
to the original Hamiltonian $\hat{H}(0)$. Summing up,
$\tilde{\hat{F}}_{\alpha}(\Phi_{0})$ maps a ground state of the Hamiltonian to
a ground state of the Hamiltonian in the same gauge.

In the presence of translation invariance we can say more about the algebra of
$\tilde{\hat{F}}_{\alpha}(\Phi_{0})$. If translations commute with the
Hamiltonian, then any translation will commute with
$\hat{F}_{\alpha}(\Phi_{0})$; on the other hand, by defining $\bar{\alpha}=y$
if $\alpha=x$ and $\bar{\alpha}=x$ if $\alpha=y$

$\hat{T}_{\alpha}^{M}\hat{U}_{\alpha}=\hat{U}_{\alpha}\hat{T}_{\alpha}^{M}\exp\left(2\pi\mathrm{i}\rho
N_{\bar{\alpha}}\right)=\hat{U}_{\alpha}\hat{T}_{\alpha}^{M}\exp\left(2\pi\mathrm{i}\frac{nn_{\bar{\alpha}}}{D}\right)$
(50)

with coprime $nn_{\bar{\alpha}}$ and $D$ according to Eq. (44) and

$\hat{T}^{M}_{\alpha}U_{\bar{\alpha}}=U_{\bar{\alpha}}\hat{T}^{M}_{\alpha}.$
(51)

This in turn implies

$\hat{T}^{M}_{\alpha}\tilde{\hat{F}}_{\alpha}(\Phi_{0})=\tilde{\hat{F}}_{\alpha}(\Phi_{0})\hat{T}^{M}_{\alpha}\exp\left(-2\pi\mathrm{i}\frac{nn_{\alpha}}{D}\right),$
(52)

whereas $\hat{T}^{M}_{\alpha}$ and $\tilde{\hat{F}}_{\bar{\alpha}}(\Phi_{0})$
commute.

Now, let $|{\Psi}\rangle$ be an eigenstate of $T_{\bm{y}}^{M}$,

$\hat{T}_{\bm{y}}^{M}|{\Psi}\rangle=\operatorname{e}^{-\mathrm{i}K_{y}}|{\Psi}\rangle.$
(53)

Then

$\hat{T}^{M}_{\bm{y}}(\hat{T}_{\bm{x}}^{M}|{\Psi}\rangle)=\operatorname{e}^{2\pi\mathrm{i}\frac{m}{D}}\operatorname{e}^{-\mathrm{i}K_{y}}(\hat{T}_{\bm{x}}^{M}|{\Psi}\rangle),$
(54)

but also

$\hat{T}^{M}_{\bm{y}}(\tilde{\hat{F}}_{\bm{y}}(\Phi_{0})^{-pn_{y}}|{\Psi}\rangle)=\operatorname{e}^{2\pi\mathrm{i}\frac{m}{D}}\operatorname{e}^{-\mathrm{i}K_{y}}(\tilde{\hat{F}}_{\bm{y}}(\Phi_{0})^{-pn_{y}}|{\Psi}\rangle).$
(55)

Since all $D$ states carry a unique $K_{y}$ momentum, and an adiabatic flux
insertion can only map a ground state to another ground state, it follows that

$\tilde{\hat{F}}_{\bm{y}}(\Phi_{0})^{-pn_{y}}|{\Psi}\rangle\propto\hat{T}_{\bm{x}}^{M}|{\Psi}\rangle$
(56)

and

$\tilde{\hat{F}}_{\bm{x}}(\Phi_{0})|{\Psi}\rangle\propto|{\Psi}\rangle.$ (57)

Vice versa one finds in the eigenbasis of $\hat{T}_{\bm{x}}^{M}$

$\hat{T}^{M}_{\bm{x}}(\tilde{\hat{F}}_{\bm{x}}(\Phi_{0})^{-pn_{x}}|{\Psi}\rangle)=\operatorname{e}^{2\pi\mathrm{i}\frac{m}{D}}\operatorname{e}^{-\mathrm{i}K_{x}}(\tilde{\hat{F}}_{\bm{x}}(\Phi_{0})^{-pn_{x}}|{\Psi}\rangle)$
(58)

and $\tilde{\hat{F}}_{\bm{x}}(\Phi_{0})$ is now diagonal. This implies a
similar transformation behavior between translations and flux insertion
according to

$\displaystyle\tilde{\hat{F}}_{\bm{y}}(\Phi_{0})^{-pn_{y}}\cong$
$\displaystyle\hat{T}_{\bm{x}}^{M}$ (59)
$\displaystyle\tilde{\hat{F}}_{\bm{x}}(\Phi_{0})^{-pn_{x}}\cong$
$\displaystyle\left(\hat{T}_{\bm{y}}^{M}\right)^{\dagger}.$

As a consequence we find for the commutation relation between the two flux
insertion operators

$\tilde{\hat{F}}_{\bm{x}}(\Phi_{0})^{pn_{x}}\tilde{\hat{F}}_{\bm{y}}(\Phi_{0})^{pn_{y}}=\operatorname{e}^{-2\pi\mathrm{i}\frac{m}{D}}\tilde{\hat{F}}_{\bm{y}}(\Phi_{0})^{pn_{y}}\tilde{\hat{F}}_{\bm{x}}(\Phi_{0})^{pn_{x}}$
(60)

or

$\tilde{\hat{F}}_{\bm{x}}(\Phi_{0})\tilde{\hat{F}}_{\bm{y}}(\Phi_{0})=\operatorname{e}^{2\pi\mathrm{i}\frac{nf}{D}}\tilde{\hat{F}}_{\bm{y}}(\Phi_{0})\tilde{\hat{F}}_{\bm{x}}(\Phi_{0}),$
(61)

where $f$ is some natural number coprime to $D$, so that $fp=1\text{ mod }D$,
which exists according to Bézout’s identity.

How do the above results change in a CDW phase? In that case, the symmetry
broken states should be eigenstates of flux insertion operations as argued in
Sec. V.1, hence

$\tilde{\hat{F}}_{\alpha}(\Phi_{0})|{\Psi}\rangle=\operatorname{e}^{\mathrm{i}\gamma_{\alpha}}|{\Psi}\rangle.$
(62)

We can transform $|{\Psi}\rangle$ into a state orthogonal to it by
translation:

$\displaystyle\tilde{\hat{F}}_{\alpha}(\Phi_{0})\left(\hat{T}_{\bm{x}}^{M}\right)^{l_{x}}\left(\hat{T}_{\bm{y}}^{M}\right)^{l_{y}}|{\Psi}\rangle$
(63) $\displaystyle=$
$\displaystyle\exp\left(2\pi\mathrm{i}\frac{nn_{\bar{\alpha}}l_{\alpha}}{D}\right)\operatorname{e}^{\mathrm{i}\gamma_{\alpha}}\left(\hat{T}_{\bm{x}}^{M}\right)^{l_{x}}\left(\hat{T}_{\bm{y}}^{M}\right)^{l_{y}}|{\Psi}\rangle$

Therefore, there must be at least $D^{2}$ orthogonal quasi-degenerate CDW
states. To be more precise, if

$\left(\hat{T}_{\alpha}^{M}\right)^{D}|{\Psi}\rangle\propto|{\Psi}\rangle,$
(64)

then the CDW states transform exactly like single particles in a $D\times D$
cluster with periodic boundary conditions and $m/D$ flux quanta per unit cell.
It is well known that these states belong to $D_{M}^{\bm{K}=0}$, the
irreducible projective representation of the translation group at $\bm{K}=0$.
In general $\bm{K}$ distinguishes all different possible projective
representations of the translation group belonging to a cluster, just like the
well-known one-dimensional irreducible representations of the translation
group [8]. Assume

$\left(\hat{T}_{\alpha}^{M}\right)^{D\delta_{\alpha}}|{\Psi}\rangle\propto|{\Psi}\rangle$
(65)

with the smallest possible $\delta_{\alpha}$, then all of the CDW states
belong to $D$ copies of $D_{M}^{\bm{K}}$ for each
$K_{\alpha}=0,2\pi/\delta_{\alpha},\dots
2\pi(\delta_{\alpha}-1)/\delta_{\alpha}$. Finally, we want to make two
remarks. In the case $\delta_{\alpha}=1$ (this could for example be realized
for $q=D=3$), all CDW states transform under the same projective irrep
$D_{M}^{\bm{K}=0}$ which does not contradict the notion of symmetry breaking,
as the projective irreducible representations of the translation group are
higher dimensional. Secondly, the minimal required number of quasi-degenerate
ground states in the CDW regime is higher than would be the case for ordinary
translation symmetry (without an external magnetic field), where both the CDW
and the topologically ordered phase can have the same number of ground states,
providing no clear sign for topological order.

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