# Parent Berry curvature and the ideal anomalous Hall crystal

Tixuan Tan  Department of Physics, Stanford University, Stanford, CA 94305,
USA  Trithep Devakul<EMAIL_ADDRESS>Department of Physics, Stanford
University, Stanford, CA 94305, USA

###### Abstract

We study a model of electrons moving in a parent band of uniform Berry
curvature. At sufficiently high parent Berry curvature, we show that strong
repulsive interactions generically leads to the formation of an anomalous Hall
crystal: a topological state with spontaneously broken continuous translation
symmetry. Our results are established via a mapping to a problem of Wigner
crystallization in a regular 2D electron gas. We further show that the
resulting quasiparticle bands are “ideal” for realizing fractionalized
topological states. Our theory provides a unified perspective for
understanding several aspects of the recently observed integer and fractional
quantum anomalous Hall effects in rhombohedral pentalayer graphene.

##  I Introduction

Electron interactions and topology can cooperate to create fascinating states
of matter. This has been made abundantly clear in the field of 2D materials,
where much exciting progress has been made in the study of topological flat
band systems [1, 2]. In materials like twisted bilayer graphene (TBG) [3, 4,
5] or twisted transition metal dichalcogenides [6, 7], a moiré superlattice
leads to the formation of topological flat minibands which, when partially
occupied, provides an ideal setting for interactions and topology to dominate.
This has led to the recent experimental realization of exotic correlated
topological states such as fractional Chern insulators [8, 9, 10, 11, 12, 13,
14] at zero magnetic field [15, 16, 17, 18, 19]. Very recent observations of
the fractional quantum spin Hall effect [20] is further evidence that novel
states of matter, never seen before, are now becoming reality in topological
flat band systems.

There is another setting, without flat bands (at the single-particle level),
in which electronic topology and interactions can be at the forefront.
Consider a system of electrons at a semiconductor band edge, which we will
refer to as the “parent” band. Here, the density of electrons is low enough
that the atomic Brillouin zone is irrelevant. When the parent band is
topologically trivial, the system is a usual 2D electron gas (2DEG). In the
presence of strong repulsive interactions, the 2DEG can spontaneously
crystallize to form a Wigner crystal state [21]. However, as will be made
clear, topology becomes unavoidable when the parent band carries a high
concentration of Berry curvature near its band edge, as illustrated in Fig 1.
In the presence of strong repulsive interactions, it is possible that
electrons in this topological parent band may spontaneously crystallize to
form an exotic “anomalous Hall crystal” (AHC), a topological version of the
Wigner crystal with an integer Chern number [22, 23, 24, 25] at zero magnetic
field, arising from a synergistic interplay of electronic interaction and
parent band topology (which can also be viewed as a supersolid [26, 27, 28]).

One class of materials with highly concentrated Berry curvature is
rhombohedral multilayer graphene [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40]. We are motivated by the recent experimental observation of the integer
and fractional quantum anomalous Hall (QAH) effects in rhombohedral pentalayer
graphene (R5G) aligned with hBN [19], and theoretical works following [41, 42,
43, 44, 45] (see [46, 47] for a summary). Unlike TBG or twisted MoTe2, the
single-particle miniband structure of R5G/hBN does not possess an isolated
flat band [48, 49, 50, 51]. In fact, robust correlated topological states are
only observed when the electrons are localized on the layer _furthest_ from
the BN-induced moiré, implying that a strong moiré potential might actually be
deleterious. Intriguingly, Hartree-Fock (HF) finds the integer QAH state even
without the moiré [42, 43, 45], implying that the state may be adiabatically
connected to an AHC. This suggests that the parent band perspective, without
moiré, is the natural starting point. Not only this, numerical diagonalization
shows that partially filling the HF bands lead to fractional Chern insulators,
suggesting that its quantum geometry [52] is sufficiently “ideal” for the
realization of such states [9, 53, 54, 55, 56, 57, 58, 59, 60]. However, the
complexity of the microscopic model and subtleties in the treatment of
electronic interactions [45] pose a serious obstacle to developing a general
theoretical understanding of the origin of this state. Like the chiral limit
[61] or the heavy fermion picture [62] of TBG, there is a need for simple,
controlled, and analytically tractable models that can capture the essence of
this system and reveal the universal physics beneath.

Our theory fulfills this role. We explain all these puzzling observations as
natural consequences of strong electronic interactions in the presence of high
parent Berry curvature.

Figure 1: Summary of results and methodology. A topological parent band with a
high concentration of Berry curvature $\mathcal{B}>0$ is driven into a $C>0$
anomalous Hall crystal (AHC) by strong interactions, or a $C<0$ Chern
insulator (CI) by a periodic potential. These results are obtained by
utilizing a unitary mapping to a model with a trivial parent band. The trivial
parent band is driven to a Wigner crystal (WC) by interactions or a band
insulator (BI) by a periodic potential. The AHC is mapped to the WC by a
unitary transformation $\hat{Q}$, while the CI is mapped to the BI by a
different unitary $\hat{W}$.

We first introduce and study a model of an idealized parent band with a
uniform, continuously tunable, parent Berry curvature. This model allows us to
isolate the effect of parent Berry curvature from all other extraneous
details. In this model, a strong repulsive interactions and/or periodic scalar
potential leads to gapped states with interesting properties and enough
structure that exact statements can be made about their topology in certain
limits.

We show that in the presence of a high parent Berry curvature $\mathcal{B}$,
both strong interactions or a periodic potential generically drive the system
into a topological state with non-zero Chern number. Consider a system of
spinless electrons with uniform parent Berry curvature $\mathcal{B}\approx
2\pi N/\Omega_{\mathrm{BZ}}$, where $N$ is a positive integer and
$\Omega_{\mathrm{BZ}}=4\pi^{2}n$ is the area of a Brillouin zone (chosen such
that the electron density $n$ corresponds to unit filling). We demonstrate
that repulsive interactions will drive the system into an AHC with Chern
number $C=N$, while a commensurate periodic potential will instead lead to a
Chern insulator with $C=-N$. Thus, there is very different behavior depending
on whether the gap is interaction or potential-induced! Our results reveal
that the two effects, repulsive interactions and a periodic potential,
generically compete and drive the system to two fixed points with different
topology.

These statements are established by utilizing a unitary mapping of the second-
quantized Hamiltonian of a topological parent band model with
$\mathcal{B}=2\pi N/\Omega_{\mathrm{BZ}}$ to a representative model with a
trivial parent band $\mathcal{B}=0$. The general methodology is illustrated in
Fig 1. The mapping makes use of the $2\pi$ flux periodicity of an effective
momentum space Hofstadter model. Since the $\mathcal{B}=0$ state forms a
trivial Wigner crystal state in the strongly interacting limit, or a trivial
band insulator in a periodic potential, the properties of the
$\mathcal{B}=2\pi N/\Omega_{\mathrm{BZ}}$ can be deduced. The mapping
transforms the Chern number in a simple way, giving rise to the $C=\pm N$
states.

Surprisingly, our results imply a direct relation between a Wigner crystal,
which is adiabatically connected to a classical state of point-like electrons,
and the AHC, which is a quantum strongly interacting topological state. We
exploit this mapping to write down an explicit wavefunction ansatz for the AHC
state that is accurate in the limit of strong repulsive interactions. Using
our wavefunction, we demonstrate that the quantum geometry of the
quasiparticle bands of the AHC becomes perfectly ideal in this limit.

Finally, although we study an idealized model, we expect that our results
should hold quite generally. If the parent band feature a concentrated Berry
curvature near the band edge, our analysis will apply as long as the relevant
low-energy states involved are within the region of high Berry curvature. We
expect that our model, and the methodology developed in this work, will be
useful for future studies into the role of Berry curvature in many-body
physics.

##  II Model

###  II.1 General Hamiltonian

We consider general Hamiltonian describing electrons projected to a single
parent band, potentially in the presence of a periodic electrostatic potential
$U(\bm{r})$ and density-density interactions $V(\bm{r})$,

$\hat{\mathcal{H}}=\hat{\mathcal{H}_{0}}+\hat{\mathcal{H}}_{\mathrm{pot}}+\hat{\mathcal{H}}_{\mathrm{int}}$
(1)

In second quantized notation, the kinetic term is

$\hat{\mathcal{H}_{0}}=\sum_{\bm{k}}{c}_{\bm{k}}^{\dagger}\mathcal{E}(\bm{k}){c}_{\bm{k}}$
(2)

where $c_{\bm{k}}^{\dagger}$ creates an electron at momentum $\bm{k}$ in the
parent band, and $\mathcal{E}(\bm{k})$ is the parent band dispersion. The
effect of a periodic potential
$U(\bm{r})=\sum_{\bm{g}}U_{\bm{g}}e^{i\bm{g}\cdot\bm{r}}$ gives rise to the
term

$\hat{\mathcal{H}}_{\mathrm{pot}}=\sum_{\bm{k},\bm{k}^{\prime}}{c}_{\bm{k}^{\prime}}^{\dagger}U_{\bm{k}^{\prime}-\bm{k}}\mathcal{F}(\bm{k}^{\prime},\bm{k}){c}_{\bm{k}}$
(3)

where
$\mathcal{F}(\bm{k^{\prime}},\bm{k})=\innerproduct{u_{\bm{k}^{\prime}}}{u_{\bm{k}}}$
is the form factor, and $\ket{u_{\bm{k}}}$ are the periodic parts of the
parent band Bloch wavefunctions. We consider a density-density interaction

$\hat{\mathcal{H}}_{\mathrm{int}}=\frac{1}{2A}\sum_{\bm{k}_{1}\bm{k}_{2}\bm{k}_{3}\bm{k}_{4}}\tilde{V}_{\bm{k}_{1}\bm{k}_{2}\bm{k}_{3}\bm{k}_{4}}{c}_{\bm{k}_{1}}^{\dagger}{c}_{\bm{k}_{2}}^{\dagger}{c}_{\bm{k}_{3}}c_{\bm{k}_{4}}$
(4)

where $A$ is the total area of the system, and

$\begin{split}\tilde{V}_{\bm{k}_{1}\bm{k}_{2}\bm{k}_{3}\bm{k}_{4}}=&V(\bm{k}_{1}-\bm{k}_{4})\mathcal{F}(\bm{k}_{1},\bm{k}_{4})\mathcal{F}(\bm{k}_{2},\bm{k}_{3})\\\
&\times\delta_{\bm{k}_{1}+\bm{k}_{2}-\bm{k}_{3}-\bm{k}_{4}}\end{split}$ (5)

where $V(\bm{q})=\int V(\bm{r})e^{-i\bm{q}\cdot\bm{r}}d\bm{r}$ is the Fourier
transform of the real-space density-density interaction. This model thus far
is entirely general. The Berry curvature and quantum geometry of the parent
band are encoded in the form factors $\mathcal{F}$. The remainder of this
section will discuss a particular microscopic model giving rise to the parent
band.

###  II.2 Multifold band inversion model

We are interested in a special family of form factors $\mathcal{F}$. We
introduce the “multifold band inversion” model,

$\hat{\mathcal{H}}^{\mathrm{MBI}}=\sum_{\bm{k}}{a}^{\dagger}_{\bm{k}\sigma^{\prime}}H^{\mathrm{MBI}}_{\sigma^{\prime}\sigma}(\bm{k}){a}_{\bm{k}\sigma}$
(6)

where $a^{\dagger}_{\bm{k}\sigma}$ creates an electron at momentum $k$ with an
internal degree of freedom $\sigma=-S,\dots,S$ (thus describing an effective
spin-$S$ fermion, although the microscopic origin is unimportant). The Bloch
Hamiltonian is

$H^{\mathrm{MBI}}_{\sigma^{\prime}\sigma}(\bm{k})=h_{0}(\bm{k})\delta_{\sigma^{\prime}\sigma}+[\bm{h}(\bm{k})\cdot\bm{S}]_{\sigma^{\prime}\sigma}$
(7)

where

$\bm{h}(\bm{k})=\left(vk_{x},vk_{y},\frac{|\bm{k}|^{2}}{m}+\delta\right)$ (8)

and $h_{0}(\bm{k})=(\frac{1}{2}-S)|\bm{h}(\bm{k})|$. Here,
$\bm{S}=(S^{x},S^{y},S^{z})$ are the standard spin-$S$ matrices satisfying
$[S^{x},S^{y}]=iS^{z}$ and cyclic permutations thereof, and
$\bm{S}^{2}=S(S+1)$. For $S=\frac{1}{2}$, $H^{\mathrm{MBI}}(\bm{k})$ is the
two-band low-energy model for a topological band inversion [63].

At each momentum $\bm{k}$, $H^{\mathrm{MBI}}(\bm{k})$ simply acts as a spin
Zeeman field $\bm{h}(\bm{k})$, thus the eigenvalues are given by

$E^{\mathrm{MBI}}_{n}(\bm{k})=h_{0}(\bm{k})+(S-n)|\bm{h}(\bm{k})|=\left(\frac{1}{2}-n\right)|\bm{h}(\bm{k})|.$
(9)

where $n=0,\dots,2S$ is the band index. We shall focus on the highest energy
band as our parent band, $E^{\mathrm{MBI}}_{0}(\bm{k})$. The purpose of the
$h_{0}(\bm{k})$ term is to ensure that all other bands have negative energy,
and can be assumed to be fully filled and inert. The $n=0$ band is
topologically trivial for $\delta>0$, and becomes topological for $\delta<0$
with a Chern number of $C=2S$. The point $\delta=0$ describes a multifold
fermion critical point in which all $2S$ bands touch at $\bm{k}=0$.

###  II.3 Ideal point and large $S$ limit

We now consider a special “ideal” point by setting
$\delta=\delta^{*}\equiv-{mv^{2}/4}$. At this point, $H^{\mathrm{MBI}}$ has
several desirable properties. The energy of the highest band becomes
parabolic, $E_{0}(\bm{k})=\frac{k^{2}}{2m}+\frac{mv^{2}}{8}$. The
corresponding eigenvector of $H^{\mathrm{MBI}}(\bm{k})$ in the eigenbasis of
$S^{z}$ is

$v_{\sigma}(\bm{k})=\sqrt{\frac{(2S)!}{(S-\sigma)!(S+\sigma)!}}\frac{(k_{x}-ik_{y})^{S+\sigma}M^{S-\sigma}}{(|\bm{k}|^{2}+M^{2})^{S}}$
(10)

where $M=mv/2$. The eigenstates
$\ket{u_{\bm{k}}}=\sum_{\sigma=-S}^{S}v_{\sigma}(\bm{k})\ket{\sigma}$ give
rise to the form factors

$\mathcal{F}(\bm{k}^{\prime},\bm{k})=\frac{(M^{2}+\bm{k}^{\prime}\cdot\bm{k}-i\bm{k^{\prime}}\crossproduct\bm{k})^{2S}}{(M^{2}+|\bm{k}^{\prime}|^{2})^{S}(M^{2}+|\bm{k}|^{2})^{S}}$
(11)

where $\bm{k}\crossproduct\bm{q}\equiv k_{x}q_{y}-k_{y}q_{x}$, and the Berry
curvature distribution

$\mathcal{B}(\bm{k})=-\imaginary\sum_{\mu,\nu}[\epsilon_{\mu\nu}\innerproduct{\partial_{k_{\mu}}u_{\bm{k}}}{\partial_{k_{\nu}}u_{\bm{k}}}]=\frac{4SM^{2}}{(|\bm{k}|^{2}+M^{2})^{2}}$
(12)

which integrates to a total Chern number of
$C=\frac{1}{2\pi}\int\mathcal{B}(\bm{k})d\bm{k}=2S$ as advertised, where
$\mu,\nu\in\\{x,y\\}$ and $\epsilon_{xy}=-\epsilon_{yx}=1$. Furthermore, the
Fubini-Study metric

$\begin{split}g^{\mathrm{FS}}_{\mu\nu}(\bm{k})&=\real\left[\innerproduct{\partial_{k_{\mu}}u_{\bm{k}}}{\partial_{k_{\nu}}u_{\bm{k}}}-\innerproduct{\partial_{k_{\mu}}u_{\bm{k}}}{u_{\bm{k}}}\innerproduct{u_{\bm{k}}}{\partial_{k_{\nu}}u_{\bm{k}}}\right]\\\
&=\frac{2SM^{2}}{(|\bm{k}|^{2}+M^{2})^{2}}\delta_{\mu\nu}=\frac{1}{2}\mathcal{B}(\bm{k})\delta_{\mu\nu}\end{split}$
(13)

exactly saturates both fundamental bounds: the trace
$\Tr[g^{\mathrm{FS}}_{\mu\nu}(\bm{k})]\geq|\mathcal{B}(\bm{k})|$ and
determinant
$\det[g^{\mathrm{FS}}_{\mu\nu}(\bm{k})]\geq\frac{1}{4}|\mathcal{B}(\bm{k})|^{2}$
conditions, at all $\bm{k}$. In this sense, the quantum metric is the
“minimal” given the Berry curvature distribution $\mathcal{B}(\bm{k})$. This
model therefore allows for the systematic study of the effect of parent band
Berry curvature, with minimal contributions from other quantum geometric
effects. The total magnitude of $\mathcal{B}(\bm{k})$ can be tuned by $S$, and
the distribution can be tuned by $M$.

The utility of this model is that it allows for an arbitrarily high total
Berry curvature, which can be made arbitrarily uniform or concentrated. This
should be contrasted with, say, a massive Dirac fermion in which the total
Berry flux is limited to $\pi$. More generally, in the effective low-energy
two-band model of rhombohedral $N$-layer graphene in a displacement field
[64], $H(k)=\Delta\sigma^{z}+t(\hbar
v/t)^{N}\left[(k_{x}-ik_{y})^{N}\sigma^{+}+h.c.\right]$, the total Berry flux
is $N\pi$. While the total Berry flux can be made arbitrarily high, the
distribution becomes highly non-uniform (it takes a ring-like shape) and the
quantum metric is not minimal. Hence, $H^{\mathrm{MBI}}$ at the ideal point
provides a controlled model for studying the effect of Berry curvature while
minimizing other details.

In this work, we consider the case of a uniform positive Berry curvature
distribution $\mathcal{B}(\bm{k})=\mathcal{B}>0$. This can be achieved by
setting $M=\sqrt{4S/\mathcal{B}}$ and then taking the limit
$S\rightarrow\infty$ (keeping $m$ fixed). In this limit, the form factors are
given by

$\mathcal{F}_{\mathcal{B}}(\bm{k}^{\prime},\bm{k})=\exp{-\frac{\mathcal{B}}{4}(|\bm{k}^{\prime}-\bm{k}|^{2}+2i\bm{k}^{\prime}\crossproduct\bm{k})}$
(14)

which consists of a geometric factor
$e^{-\frac{\mathcal{B}}{4}|\bm{k}^{\prime}-\bm{k}|^{2}}$ arising due to the
“quantum distance” between the two states, and a phase factor which can be
interpreted as a momentum space Aharonov-Bohm factor
$\arg\mathcal{F}_{\mathcal{B}}(\bm{k}^{\prime},\bm{k})=\int_{\bm{k}}^{\bm{k}^{\prime}}\bm{A}(\bm{q})\cdot
d\bm{q}$ with the Berry connection
$\bm{A}(\bm{k})=\frac{\mathcal{B}}{2}(-k_{y},k_{x})$ playing the role of the
vector potential in the symmetric gauge. We remark that Eq 14 is identical to
that obtained for the lowest Landau level with magnetic length
$\ell_{B}^{2}=\mathcal{B}$, but describes an unbounded spectrum of
parabolically dispersing electrons.

We will utilize this limit as the parent band in the Hamiltonian Eq 1. The
single-particle dispersion in $\mathcal{H}_{0}$ is given by
$\mathcal{E}(\bm{k})=\frac{|\bm{k}|^{2}}{2m}$ (a constant offset has been
neglected), and the form factors in $\hat{\mathcal{H}}_{\mathrm{pot,int}}$ are
given by Eq 14. We define the operators $c_{\bm{k}}^{\dagger[\mathcal{B}]}$ to
create a fermion in the microscopic state
$e^{i\bm{k}\cdot\bm{r}}\ket{u_{\bm{k}}^{\mathcal{B}}}$ in the ideal parent
band, which satisfy
$\innerproduct{u_{\bm{k}^{\prime}}^{\mathcal{B}}}{u_{\bm{k}}^{\mathcal{B}}}=\mathcal{F}_{\mathcal{B}}(\bm{k}^{\prime},\bm{k})$.
We will typically neglect the $\mathcal{B}$ superscript on
$c^{\dagger}_{\bm{k}}$ when it is unimportant.

##  III Periodic potential

Figure 2:  The ideal parent band in a periodic scalar potential. For the
$C_{4}$ symmetric potential, (a) illustrates the effective momentum space
square lattice Hofstadter model at each $\bm{k}$, (b) shows the gap between
the first and second minibands, and the Chern number as a function of
$\mathcal{B}$, and (c) shows the Berry curvature distribution. For the $C_{3}$
symmetric potential, (d) shows the effective triangular lattice Hofstadter
model, (e) shows the Chern number as a function of $\mathcal{B}$ and potential
shape parameter $\phi$, and (f) shows the Berry curvature. Parameters used for
(b,e) are $m=\frac{1}{2}$, $g=1$, and $U_{0}=1$ ($U_{0}=5$) and (c,f)
$\mathcal{B}\Omega_{\mathrm{BZ}}=2\pi$, $U_{0}=1$ ($U_{0}=0.5$,$\phi=\pi$),
for the $C_{4}$ ($C_{3}$) symmetric potential.

We now consider the effect of the periodic potential
$\hat{\mathcal{H}}_{\mathrm{pot}}$. Let us focus first on the $C_{4}$
symmetric potential with period $a$,

$\begin{split}U(\bm{r})=&-2U_{0}\left(\cos(\frac{2\pi x}{a})+\cos(\frac{2\pi
y}{a})\right)\\\ =&-U_{0}\sum_{j=1}^{4}e^{i\bm{b}_{j}\cdot\bm{r}}\end{split}$
(15)

where $\bm{b}_{j}=g(\cos(\frac{\pi(j-1)}{2}),\sin(\frac{\pi(j-1)}{2}))$ are
reciprocal lattice vectors, and $g=\frac{2\pi}{a}$.

The Hamiltonian takes the form
$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\mathrm{pot}}$,
with

$\begin{split}\hat{\mathcal{H}}_{0}=&\sum_{\bm{k},\bm{g}}\mathcal{E}(\bm{k}+\bm{g})c^{\dagger}_{\bm{k},\bm{g}}c_{\bm{k},\bm{g}}\\\
\hat{\mathcal{H}}_{\mathrm{pot}}=&-U_{0}e^{-\frac{\mathcal{B}}{4}g^{2}}\sum_{\bm{k},\bm{g},j}c^{\dagger}_{\bm{k},\bm{g}+\bm{b}_{j}}e^{i\frac{\mathcal{B}}{2}(\bm{k}+\bm{g})\crossproduct\bm{b}_{j}}c_{\bm{k},\bm{g}}\end{split}$
(16)

where $c_{\bm{k},\bm{g}}=c_{\bm{k}+\bm{g}}$ and, from this point onwards,
$\bm{k}$ are momenta within the first Brillouin zone (which remains a good
quantum number), and $\bm{g}$ are reciprocal lattice vectors. The periodic
potential causes the original dispersion $\mathcal{E}(\bm{k})$ to split into
multiple minibands. The prospect of engineering topological minibands from
parent bands with non-trivial topology has been recently explored [65, 66, 67,
68, 69, 70]. In general, the miniband structure and topology depend crucially
on the quantum geometry of the original band 111L. Fu, private
communications..

For each $\bm{k}$,
$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\mathrm{pot}}$ can
be mapped exactly to a momentum space version of the Hofstadter tight binding
model [72] on the square lattice of sites
$\bm{g}_{n_{1}n_{2}}=n_{1}\bm{b}_{1}+n_{2}\bm{b}_{2}$,
($n_{1},n_{2}\in\mathbb{Z}$), with hopping amplitude
$-U_{0}e^{-\frac{\mathcal{B}}{4}g^{2}}$, in the presence of a uniform a
magnetic field $\mathcal{B}$ and a parabolic confining potential
$\mathcal{E}(\bm{k}+\bm{g}_{n_{1}n_{2}})$, as illustrated in Fig 2a. This
Hofstadter model has an effective magnetic flux of
$\Phi=\mathcal{B}\Omega_{\mathrm{BZ}}$ through each plaquette, where
$\Omega_{\mathrm{BZ}}=g^{2}$. Crucially, the properties of the Hofstadter
model depends only on the magnetic flux modulo $2\pi$. This implies a relation
between the Hamiltonian with $\mathcal{B}$ to that with
$\mathcal{B}+2\pi/\Omega_{\mathrm{BZ}}$. We now establish this periodicity,
and use it to make exact statements about the the resulting minibands.

Consider the unitary transformation

$\hat{W}c_{\bm{k},\bm{g}}^{\dagger[\mathcal{B}]}\hat{W}^{\dagger}=W_{\bm{g}}(\bm{k})c_{\bm{k},\bm{g}}^{\dagger[\mathcal{B}+2\pi/\Omega_{\mathrm{BZ}}]}$
(17)

which shifts $\mathcal{B}$ in the creation/annihilation operators, but more
importantly applies a phase factor

$W_{\bm{g}}(\bm{k})=\exp{i\pi\left(\frac{\bm{k}\crossproduct\bm{g}}{\Omega_{\mathrm{BZ}}}+\omega(\bm{g})\right)}$
(18)

where $\omega(\bm{g})=g_{x}g_{y}/g^{2}=n_{1}n_{2}$. This transformation leaves
the kinetic term $\hat{\mathcal{H}}_{0}$ invariant (it affects the microscopic
wavefunctions created by $c^{\dagger}_{\bm{k}}$, but the form of the second
quantized Hamiltonian is unchanged)

The reason for the phase factors is so that $\hat{W}$ will transform
$\hat{\mathcal{H}}_{\mathrm{pot}}$, when viewed as a function of $\mathcal{B}$
and $U_{0}$, according to

$\hat{\mathcal{H}}_{\mathrm{pot}}[\mathcal{B}+2\pi/\Omega_{\mathrm{BZ}},U_{0}]=\hat{W}\hat{\mathcal{H}}_{\mathrm{pot}}[\mathcal{B},U_{0}e^{-\frac{\pi}{2}}]\hat{W}^{\dagger}$
(19)

Thus implying that the total spectrum of $\hat{\mathcal{H}}$ is periodic under
a shift of $\mathcal{B}$ by $2\pi/\Omega_{\mathrm{BZ}}$, provided we also
rescale the potential strength $U_{0}$. The phase factors $W_{\bm{g}}(\bm{k})$
can be interpreted as the “gauge transformation” which adds $2\pi$ flux to the
momentum space Hofstadter model, and the rescaling of $U_{0}$ corrects for the
quantum distance factor in $\mathcal{F}$.

Despite the spectrum being periodic, we now demonstrate that topology is not.
Specifically,

$C[\mathcal{B}+2\pi/\Omega_{\mathrm{BZ}},U_{0}]=C[\mathcal{B},U_{0}e^{-\frac{\pi}{2}}]-1$
(20)

where $C$ is the Chern number of (any) isolated miniband. Thus, counter-
intuitively, an _increase_ of the parent band Berry curvature by
$2\pi/\Omega_{\mathrm{BZ}}$ results in a _decrease_ of the miniband Chern
number by $1$.

To prove Eq 20, we consider the first quantized wavefunction of
$\hat{\mathcal{H}}[\mathcal{B}_{0},U_{0}e^{-\frac{\pi}{2}}]$ in a single
isolated band,
$|\psi_{\bm{k}}^{(0)}\rangle=\sum_{\bm{g}}\psi_{\bm{g}}^{(0)}(\bm{k})e^{i(\bm{k}+\bm{g})\cdot\bm{r}}|u^{\mathcal{B}_{0}}_{\bm{k}+\bm{g}}\rangle$.
As a result of Eq 19, the wavefunction

$|\psi^{(1)}_{\bm{k}}\rangle=\sum_{\bm{g}}W_{\bm{g}}(\bm{k})\psi^{(0)}_{\bm{g}}(\bm{k})e^{i(\bm{k}+\bm{g})\cdot\bm{r}}|u_{\bm{k}+\bm{g}}^{\mathcal{B}_{1}}\rangle$
(21)

is the corresponding eigenstate of $\mathcal{H}[\mathcal{B}_{1},U_{0}]$, where
$\mathcal{B}_{1}=\mathcal{B}_{0}+2\pi/\Omega_{\mathrm{BZ}}$. We use the
superscripts $(0)$ to denote things relating to the $\mathcal{B}_{0}$ model,
and $(1)$ to denote those relating to $\mathcal{B}_{1}$. The Berry curvature
is calculated from the cell-periodic part of the wavefunction,
$e^{-i\bm{k}\cdot\bm{r}}|\psi_{\bm{k}}^{(1)}\rangle$, which gives rise to

$B^{(1)}(\bm{k})=B^{\prime}(\bm{k})+\frac{\pi}{\Omega_{\mathrm{BZ}}}\sum_{\bm{g},\mu}g_{\mu}\partial_{k_{\mu}}|\psi^{(0)}_{\bm{g}}(\bm{k})|^{2}$
(22)

where $B^{\prime}$ is the Berry curvature computed from
$\sum_{\bm{g}}\psi_{\bm{g}}^{(0)}(\bm{k})e^{i\bm{g}\cdot\bm{r}}|u_{\bm{k}+\bm{g}}^{\mathcal{B}_{1}}\rangle$.
It can be shown that
$\int_{\mathrm{BZ}}B^{\prime}(\bm{k})d\bm{k}=\int_{\mathrm{BZ}}B^{(0)}(\bm{k})d\bm{k}=2\pi
C^{(0)}$ gives the original Chern number. The final Chern number is therefore

$C^{(1)}=\frac{1}{2\pi}\int_{\mathrm{BZ}}B^{(1)}(\bm{k})d\bm{k}=C^{(0)}-1$
(23)

where the $-1$ is due to second term in Eq 22, which comes from the
$W_{\bm{g}}(\bm{k})$ factor in $|\psi^{(1)}\rangle$. This result follows by
integrating, using
$|\psi^{(0)}_{\bm{g}}(\bm{k}+\bm{g}^{\prime})|=|\psi^{(0)}_{\bm{g}+\bm{g}^{\prime}}(\bm{k})|$,
and performing the $\bm{g}$ summation by parts, thus proving Eq 20.

Fig 2b shows the gap and Chern number of the first band, calculated
numerically, as a function of the flux, confirming this result. Fig 2c shows
the Berry curvature of the first miniband, demonstrating this counter-
intuitive feature. Near the $\gamma$ point of the BZ, $B(\bm{k})$ is positive
(inherited from the positive parent Berry curvature), but due to superlattice
effects, a gap is opened up near the zone boundaries that give rise to
negative $B(\bm{k})$, precisely overcancelling the parent Berry curvature to
give a negative Chern number.

We can also consider a $C_{3}$ symmetric potential,

$U(\bm{r})=-\sum_{j=1,3,5}U_{0}e^{i\phi}e^{i\bm{b}_{j}\cdot\bm{r}}+c.c.$ (24)

where $\bm{b}_{j}=g(\cos(\frac{\pi(j-1)}{3}),\sin(\frac{\pi(j-1)}{3}))$ and
$\phi$ is a parameter that controls the shape of the potential. There are a
few minor differences from the $C_{4}$ symmetric potential. First, the
Hamiltonian maps to a momentum space Hofstadter model on the triangular
lattice $\bm{g}_{n_{1}n_{2}}=n_{1}\bm{b}_{1}+n_{2}\bm{b}_{2}$, instead of the
square lattice, with the same confining parabolic potential (Fig 2d). The
hopping amplitudes are $-U_{0}e^{-\frac{\mathcal{B}}{4}g^{2}\pm i\phi}$ which,
in addition to the uniform $\mathcal{B}$, give rise to an effective
(staggered) magnetic flux of
$\Phi_{\pm}=\frac{1}{2}\mathcal{B}\Omega_{\mathrm{BZ}}\pm 3\phi$ on each
plaquette, where $\Omega_{\mathrm{BZ}}=\sqrt{3}g^{2}/2$ (each triangular
plaquette has an area of $\Omega_{\mathrm{BZ}}/2$). The triangular lattice
Hofstadter model comes back to itself with the addition of $\pi$ flux per
plaquette, but with an opposite sign of hopping amplitudes. This manifests as
the periodicity relation

$\hat{\mathcal{H}}_{\mathrm{pot}}[\mathcal{B}_{0}+2\pi/\Omega_{\mathrm{BZ}},U_{0}]=\hat{W}\hat{\mathcal{H}}_{\mathrm{pot}}[\mathcal{B}_{0},-U_{0}e^{-\frac{\pi}{\sqrt{3}}}]\hat{W}^{\dagger}$
(25)

with $W$ given as in Eq 18, except with $\omega(\bm{g})=(n_{1}-1)(n_{2}-1)$.
Provided that the spectrum is gapped so Chern number is well defined,

$C[\mathcal{B}_{0}+2\pi/\Omega_{\mathrm{BZ}},U_{0}]=C[\mathcal{B}_{0},-U_{0}e^{-\frac{\pi}{\sqrt{3}}}]-1$
(26)

obeys a similar periodicity relation. Fig 2(e,f) shows the phase diagrams and
Berry curvature for the $C_{3}$ symmetric potential.

A consequence of these results is that, for $\mathcal{B}=2\pi N/\Omega_{BZ}$,
where $N\in\mathbb{Z}$, each isolated miniband must carry Chern number $C=-N$.
This follows from both Eq 20 or Eq 26 and the fact that the Chern number must
be zero for $\mathcal{B}=0$ where there is no source of time-reversal symmetry
breaking. This Chern number is equal in magnitude, but opposite in sign, to
the integrated parent Berry curvature over the BZ. This is true even in the
limit of an infinitely strong potential.

##  IV Strong interactions

We now consider the effect of interactions $\hat{\mathcal{H}}_{\mathrm{int}}$,
in the absence of a periodic potential so continuous translation symmetry is
present. In a topologically trivial parent band ($\mathcal{B}=0$), strong
interactions can lead to spontaneous breaking of continuous translation
symmetry in the form of a Wigner crystal. As we shall demonstrate, strong
interactions will also lead to spontaneously broken continuous translation
symmetry in the presence of strong parent Berry curvature, but with non-zero
Chern number, thus realizing an AHC.

In general, the periodicity in $\mathcal{B}$ is no longer an exact feature of
the interacting problem. This is because the interactions involve quartic
terms from different momenta $\bm{k}$, effectively coupling the momentum space
Hofstadter models from different $\bm{k}$. Nevertheless, we show that there
still remains a remnant of this periodicity, and that it results in
topological states with the _same_ Chern number as the parent Berry curvature
$\mathcal{B}$ (opposite to our conclusions for periodic potential).

We consider the Hamiltonian
$\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\mathrm{int}}$, where
$\hat{\mathcal{H}}_{\mathrm{int}}$ is given in Eq 4. We allow for continuous
translation symmetry breaking down to a discrete unit cell with reciprocal
lattice vectors $\bm{g}$ with $C_{3}$ symmetry. We assume the period is
determined by the density of electrons such that there is one electron per
unit cell, $\Omega_{\mathrm{BZ}}=4\pi^{2}n$ where $n$ is the electron density,
as expected for a Wigner crystal. We consider decomposing the interaction into
two terms, the Hartree term in which $\bm{k}_{1}=\bm{k}_{4}$,
$\bm{k}_{2}=\bm{k}_{3}$, and the Fock term in which $\bm{k}_{1}=\bm{k}_{3}$,
$\bm{k}_{2}=\bm{k}_{4}$ (within the BZ). That is,
$\hat{\mathcal{H}}_{\mathrm{int}}\approx\hat{\mathcal{H}}_{\mathrm{H}}+\hat{\mathcal{H}}_{\mathrm{F}}$,

$\begin{split}\hat{\mathcal{H}}_{\mathrm{H}}=\frac{1}{2A}\sum_{\begin{subarray}{c}\bm{k}_{1}\bm{k}_{2}\\\
\bm{g}_{1}\bm{g}_{2}\bm{g}_{3}\bm{g}_{4}\end{subarray}}\tilde{V}^{\bm{k}_{1}\bm{k}_{2}\bm{k}_{2}\bm{k}_{1}}_{\bm{g}_{1}\bm{g}_{2}\bm{g}_{3}\bm{g}_{4}}({c}_{\bm{k}_{1}\bm{g}_{1}}^{\dagger}c_{\bm{k}_{1}\bm{g}_{4}})({c}_{\bm{k}_{2}\bm{g}_{2}}^{\dagger}{c}_{\bm{k}_{2}\bm{g}_{3}})\\\
\hat{\mathcal{H}}_{\mathrm{F}}=-\frac{1}{2A}\sum_{\begin{subarray}{c}\bm{k}_{1}\bm{k}_{2}\\\
\bm{g}_{1}\bm{g}_{2}\bm{g}_{3}\bm{g}_{4}\end{subarray}}\tilde{V}^{\bm{k}_{1}\bm{k}_{2}\bm{k}_{1}\bm{k}_{2}}_{\bm{g}_{1}\bm{g}_{2}\bm{g}_{3}\bm{g}_{4}}({c}_{\bm{k}_{1}\bm{g}_{1}}^{\dagger}c_{\bm{k}_{1}\bm{g}_{3}})({c}_{\bm{k}_{2}\bm{g}_{2}}^{\dagger}{c}_{\bm{k}_{2}\bm{g}_{4}})\end{split}$
(27)

where

$\begin{split}\tilde{V}^{\bm{k}_{1}\bm{k}_{2}\bm{k}_{3}\bm{k}_{4}}_{\bm{g}_{1}\bm{g}_{2}\bm{g}_{3}\bm{g}_{4}}=&V(\bm{k}_{1}+\bm{g}_{1}-\bm{k}_{4}-\bm{g}_{4})\mathcal{F}(\bm{k}_{1}+\bm{g}_{1},\bm{k}_{4}+\bm{g}_{4})\\\
&\times\mathcal{F}(\bm{k}_{2}+\bm{g}_{2},\bm{k}_{3}+\bm{g}_{3})\times\delta(\dots)\end{split}$
(28)

and the $\delta(\dots)$ ensures total momentum conservation. Both
$\hat{\mathcal{H}}_{\mathrm{H}}$ and $\hat{\mathcal{H}}_{\mathrm{F}}$ are, at
this point, still fully interacting. The HF approximation follows from a mean-
field treatment of the interactions,
$(c^{\dagger}c)(c^{\dagger}c)\approx\langle c^{\dagger}c\rangle
c^{\dagger}c+c^{\dagger}c\langle c^{\dagger}c\rangle-\langle
c^{\dagger}c\rangle\langle c^{\dagger}c\rangle$, in which
$\hat{\mathcal{H}}_{\mathrm{H,F}}$ give rise to the Hartree and Fock terms
respectively, and then solving the set of equations self-consistently.

Let us again consider the unitary transformation $\hat{W}$ from Eq 25. Indeed,
somewhat remarkably, $\hat{\mathcal{H}}_{\mathrm{H}}$, viewed as a functional
of $\mathcal{B}$ and the Fourier transform of the density-density interaction
$V(q)$, satisfies the exact periodicity

$\hat{\mathcal{H}}_{\mathrm{H}}[\mathcal{B}+2\pi/\Omega_{BZ},V(q)]=\hat{W}\hat{\mathcal{H}}_{\mathrm{H}}[\mathcal{B},V(q)e^{-\frac{\pi
q^{2}}{\Omega_{BZ}}}]\hat{W}^{\dagger}$ (29)

However, the same does not apply to the $\mathcal{H}_{\mathrm{F}}$. Instead,

$\hat{\mathcal{H}}_{\mathrm{F}}[\mathcal{B}+2\pi/\Omega_{BZ},V(q)]=\hat{Q}\hat{\mathcal{H}}_{\mathrm{F}}[\mathcal{B},V(q)e^{-\frac{\pi
q^{2}}{\Omega_{\mathrm{BZ}}}}]\hat{Q}^{\dagger}$ (30)

where $\hat{Q}$ is defined

$\hat{Q}c_{\bm{k},\bm{g}}^{\dagger[\mathcal{B}]}\hat{Q}^{\dagger}=W_{\bm{g}}^{*}(\bm{k})c_{\bm{k},\bm{g}}^{\dagger[\mathcal{B}+2\pi/\Omega_{\mathrm{BZ}}]}$
(31)

Thus, there is not a single unitary transformation that accomplishes the
mapping for both terms simultaneously.

The presence of this exact mapping of the quartic Hamiltonians
$\hat{\mathcal{H}}_{\mathrm{H}}$ and $\hat{\mathcal{H}}_{\mathrm{F}}$
individually is quite remarkable. It is especially interesting considering
that, in the periodic potential case, the momentum space Hofstadter model
fails to exhibit the exact $2\pi$ flux-per-plaquette periodicity when generic
longer-range hoppings are included (that is, the
$\hat{\mathcal{H}}_{\mathrm{pot}}$ we consider only satisfies the exact
$2\pi/\Omega_{\mathrm{BZ}}$ periodicity because the potential $U(\bm{r})$ is
taken to consist of first harmonic terms $\bm{b}_{j}$). Nevertheless, both
$\hat{\mathcal{H}}_{\mathrm{H}}$ and $\hat{\mathcal{H}}_{\mathrm{F}}$, which
in principle contains correlated hopping terms of all ranges, enjoys this
exact mapping without restriction.

We now argue that the Fock term will dominate when $\mathcal{B}$ is
significant. Suppose $\mathcal{B}=2\pi N/\Omega_{\mathrm{BZ}}$. By repeated
applications of $\hat{Q}^{\dagger}$,
$\hat{\mathcal{H}}_{\mathrm{H}}+\hat{\mathcal{H}}_{\mathrm{F}}$ can be related
to a zero-$\mathcal{B}$ Hamiltonian

$\begin{split}\hat{Q}^{\dagger
N}(\hat{\mathcal{H}}_{\mathrm{H}}&[\mathcal{B},V(q)]+\hat{\mathcal{H}}_{\mathrm{F}}[\mathcal{B},V(q)])\hat{Q}^{N}=\\\
&\hat{Q}^{\dagger
N}\hat{\mathcal{H}}_{\mathrm{H}}[0,\tilde{V}(q)]\hat{Q}^{N}+\hat{\mathcal{H}}_{\mathrm{F}}[0,\tilde{V}(q)]\end{split}$
(32)

where $\tilde{V}(q)=V(q)e^{-\frac{\pi N}{\Omega_{\mathrm{BZ}}}q^{2}}$ is an
effective interaction that is suppressed at large $q$ by a Gaussian factor.
This corresponds to a real space interaction $\tilde{V}(r)\sim
e^{-\frac{\Omega_{\mathrm{BZ}}r^{2}}{8\pi
N}}I_{0}(\frac{\Omega_{\mathrm{BZ}}r^{2}}{8\pi N})$ that is smoothed off at
short distances. This Hamiltonian differs from a true trivial parent band
model only in the presence of various phase factors in
$\hat{\mathcal{H}}_{\mathrm{H}}$ introduced by the $\hat{Q}$. However, since
$\hat{\mathcal{H}}_{\mathrm{H}}$ only involves momentum transfer $q$ with a
minimum of $q=g$ (the $\tilde{V}(q=0)$ term is just the total charging
energy), its magnitude is more strongly suppressed by the Gaussian factor than
$\hat{\mathcal{H}}_{\mathrm{F}}$, which also involves $\tilde{V}(q)$ with
$0<q<g$. As a result, $\hat{\mathcal{H}}_{\mathrm{H}}$ can be neglected for
large enough $\mathcal{B}$. Neglecting $\hat{\mathcal{H}}_{\mathrm{H}}$, the
mapping in Eq 30 applies to the fully interacting Hamiltonian
$\hat{\mathcal{H}}_{0}+\mathcal{H}_{\mathrm{F}}$ exactly.

Now, suppose we have a self-consistent solution to the HF mean field
Hamiltonian at $\mathcal{B}=2\pi N/\Omega_{\mathrm{BZ}}$, with sufficiently
strong interactions to create a charge gap, where the first quasiparticle band
is filled. In the Fock dominated regime, the Hamiltonian is unitarily related
to a zero-$\mathcal{B}$ problem with an effective interaction $\tilde{V}(q)$.
The corresponding zero-$\mathcal{B}$ state describes a Wigner crystal with a
filled quasiparticle band of trivial topology. By the same argument for the
change in topology in Eq 20, except with the phase factors
${W}_{\bm{g}}^{*}(\bm{k})$, we conclude that the filled band of the
$\mathcal{B}=2\pi N/\Omega_{\mathrm{BZ}}$ model must have a Chern number of
$C=+N$, i.e. it describes an AHC.

Note that the Chern number is positive, which contrasts to the negative Chern
band in the presence of a strong periodic potential. This indicates that the
two limits, strong potential versus strong interactions, are distinct and,
when both present, will generically compete with one another. This result is
unexpected, as the two effects are typically complementary in a topologically
trivial band.

Figure 3:  Energetic and quantum geometric properties of the model. (a) The
charge gap $\Delta$ and bandwidth $W$ of the filled band obtained from solving
the mean-field HF Hamiltonian self-consistently, as a function of Coulomb
interaction strength $V_{C}$. (b) The Berry curvature fluctuation ($\delta B$)
and trace condition violation (TCV) of the filled band are shown. (c,d) show
corresponding quantities for a non-interacting model with a periodic potential
$U_{0}$ instead. All energies are measured in units of $\frac{g^{2}}{2m}$.

To verify our analysis, we numerically solve the full self-consistent HF
equations in Fig 3. We take the Coulomb interaction $V(q)=V_{C}/q$, with no
periodic potential, and focus on $\mathcal{B}=2\pi/\Omega_{\mathrm{BZ}}$. The
HF mean-field Hamiltonian is solved self-consistently at filling $\nu=1$
electron per unit cell. The calculation is performed in the
$\ket{u_{\bm{k}+\bm{g}}}$ basis by keeping states up to a cutoff
$|\bm{g}|<\Lambda$. Fig 3a shows the bandwidth of the filled quasiparticle
band $W$, and charge gap $\Delta$, as a function of interaction strength
$V_{C}$. Above a critical $V_{C}$, $\Delta$ becomes non-zero signaling a
transition into a crystalline state with spontaneously broken continuous
translation symmetry, while the bandwidth $W$ becomes small. Numerical
integration of the Berry curvature, properly taking into account the
wavefunction overlaps of the parent band, reveals a Chern number $C=1$, in
agreement with the above analysis in the Fock-dominated regime.

To contrast, we show in Fig 3c the analogous results for a non-interacting
Hamiltonian ($V_{C}=0$), in the $C_{3}$ symmetric potential $U(\bm{r})$ with
$\phi=\pi$, at the same $\mathcal{B}=2\pi/\Omega_{\mathrm{BZ}}$. For any
$U_{0}>0$, the gap $\Delta>0$ and bandwidth $W$ becomes very small with
increasing $U_{0}$. Integration of the Berry curvature reveals that the first
band is topological with $C=-1$, as expected.

We now turn to the quantum geometric properties of the resulting bands. We
consider two quantum geometric indicators that have been proposed to probe the
suitability of a band to realizing fractionalized phases at partial filling.
We consider the Berry curvature fluctuation

$(\delta
B)^{2}=\frac{\Omega_{\mathrm{BZ}}}{4\pi^{2}}\int_{\mathrm{BZ}}(B(\bm{k})-\bar{B})^{2}d\bm{k}$
(33)

where $\bar{B}=\Omega_{\mathrm{BZ}}^{-1}\int_{\mathrm{BZ}}B(\bm{k})d\bm{k}$,
and the average trace condition violation

$\mathrm{TCV}=\frac{1}{2\pi}\int_{\mathrm{BZ}}\left(\Tr[g^{FS}_{\mu\nu}(\bm{k})]-|B(\bm{k})|\right)d\bm{k}$
(34)

which measures the amount by which the trace inequality,
$\Tr[g^{\mathrm{FS}}_{\mu\nu}(\bm{k})]\geq|B(\bm{k})|$, is exceeded. A band
with $\mathrm{TCV}=0$ is said to be “ideal” for the realization of fractional
Chern insulators [9, 53, 54, 55, 56, 57, 58, 59, 60].

We show $\delta B$ and TCV for the self-consistent HF band of the interacting
model in Fig 3b, and the corresponding plot for the case of the periodic
potential without interaction in Fig 3d. The Berry curvature distribution for
both cases becomes increasingly uniform with $V_{C}$ or $U_{0}$. However, they
differ strikingly in terms of their TCV. While the interacting $C=1$ AHC
becomes increasingly ideal with strong interactions, the non-interacting
$C=-1$ CI with a periodic potential is the opposite. This striking difference
will be explained in the next section, based on an analytic ansatz for the
wavefunctions of these two states.

##  V Wavefunction ansatz

Figure 4:  Wavefunction ansatz and ideal AHC. (a) Illustration of the
wavefunction for the trivial $C=0$ state in the $\mathcal{B}=0$ parent band,
where $\xi$ is a tuning parameter; $\xi\rightarrow\infty$ describes the limit
of strong interaction or potential. By utilizing the transformation $\hat{Q}$
($\hat{W}$), we arrive at a $C=1$ AHC ($C=-1$ CI) state for the
$\mathcal{B}=2\pi/\Omega_{\mathrm{BZ}}$ model in the Fock (potential)
dominated regime, respectively. (b) The trace condition violation (TCV) is
shown as a function of $\xi$, demonstrating that the AHC becomes ideal as
$\xi\rightarrow\infty$.

The mappings established in Sec III and IV are incredibly powerful. They
manage to relate the ground states of topological quantum systems, the $C=N$
AHC or the $C=-N$ Chern insulator, to those of a topologically trivial Wigner
crystal or band insulator, which can essentially be described classically. In
this section, we leverage this relation to write down analytic wavefunctions
based on a simple Gaussian ansatz for the trivial band insulator. Remarkably,
this analytic expression fully captures and explains the emergence of the
ideal HF band of the strongly interacting AHC.

First, let us consider the ground state wavefunction of the zero-$\mathcal{B}$
Hamiltonian in an infinitely strong periodic potential (with a single
potential minimum per unit cell). This limit is equivalent to an array of
harmonic oscillators localized at each potential minimum $\bm{R}$, each of
which are described by the Gaussian wavefunction
$\psi_{\bm{R}}(\bm{r})\sim\exp{-\alpha(\bm{r}-\bm{R})^{2}}$. The equivalent
momentum space picture is a single isolated flat band, with the wavefunctions
described by
$\psi_{\bm{g}}^{(0)}(\bm{k})=\mathcal{N}_{\bm{k}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{4\xi^{2}}}$
where $\xi$ is a momentum space localization length, and
$\mathcal{N}_{\bm{k}}$ is a normalization constant, as illustrated in Fig 4a.
Although we have considered a strong periodic potential, the ground state in
the presence of an infinitely strong repulsive interaction is a classical
Wigner crystal, which can also be well described by a Slater determinant of
such states. We take this wavefunction as an ansatz and treat $\xi$, the
momentum space localization length, as an effective parameter reflecting the
overall strength of the interaction or periodic potential. This ansatz becomes
exact in the strong interaction or potential limit in which
$\xi\rightarrow\infty$.

Based on this, we can write down the corresponding wavefunctions for the
$\mathcal{B}=2\pi/\Omega_{BZ}$ Hamiltonian. These follow from applications of
either $\hat{W}$ or $\hat{Q}$. The wavefunctions are

$|\psi_{\bm{k}}^{\pm}\rangle=\mathcal{N}_{\bm{k}}\sum_{\bm{g}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{4\xi^{2}}\pm
i\pi\left[\frac{\bm{k}\crossproduct\bm{g}}{\Omega_{\mathrm{BZ}}}+\omega(\bm{g})\right]}e^{i(\bm{k}+\bm{g})\cdot\bm{r}}|u_{\bm{k}+\bm{g}}^{2\pi/\Omega_{\mathrm{BZ}}}\rangle$
(35)

where the $+$ ($-$) corresponds to the case of strong potential (interaction).
The full many-body state will be a Slater determinant of
$|\psi_{\bm{k}}^{\pm}\rangle$.

We directly compute the TCV of the bands described by
$|\psi_{\bm{k}}^{\pm}\rangle$ in Fig 4b, as a function of $\xi$. The behavior
is similar to that observed numerically from HF in Fig 3(b,d). Namely, in the
strong potential limit $|\psi_{\bm{k}}^{+}\rangle$, TCV increases with
increasing $\xi$, while in the strong interacting limit
$|\psi_{\bm{k}}^{-}\rangle$, TCV decreases with increasing $\xi$ becoming more
ideal. For large $\xi$, the trace of the Fubini-Study metric (and the Berry
curvature) is $\bm{k}$-independent, and an analytic expression can be computed
directly from $|\psi^{\pm}_{\bm{k}}\rangle$,

$\Tr[g_{\mu\nu}^{\mathrm{FS}(\pm)}]=\frac{2\pi}{\Omega_{\mathrm{BZ}}}+(1\pm
1)\frac{4\pi^{2}\xi^{2}}{\Omega_{\mathrm{BZ}}^{2}}+\frac{1}{2\xi^{2}}+\mathcal{O}(e^{-\frac{2\pi^{2}\xi^{2}}{\Omega_{BZ}}})$
(36)

thus, in the Fock-dominated AHC, the $\xi^{2}$ term is zero and
$\Tr[g]-|B|\rightarrow 0$ becomes perfectly ideal at large $\xi$, in good
agreement with Fig 4b. Thus, interactions separate out an ideal AHC Chern band
from the parent band (reminiscent of how ideal higher Chern bands can be
decomposed into ideal Chern 1 bands [73]).

##  VI Discussion

###  VI.1 Rhombohedral graphene

We first discuss our findings in the context of recent experimental and
theoretical studies of rhombohedral pentalayer graphene on hBN [19, 41, 42,
43, 44, 45]. The system is described by the parent band structure of R5G,
which features an electron band with a high concentration of Berry curvature
at the band edge. Although the microscopic model features many detail-
dependent terms, the origin of the Berry curvature can be understood from the
effective low-energy two-band model describing states on the top/bottom-most
layers [64],

$H_{\mathrm{RLG}}(\bm{k})=\begin{pmatrix}\Delta&-t(\frac{\hbar
v}{t})^{L}(k_{x}-ik_{y})^{L}\\\ -t(\frac{\hbar
v}{t})^{L}(k_{x}+ik_{y})^{L}&-\Delta\\\ \end{pmatrix}$ (37)

where $L=5$ is the number of layers, and $\Delta$ is the effective potential
difference between the top/bottom layers which can be tuned by an external
displacement field, and $(v,t)\approx(0.9\times
10^{6}\mathrm{m/s},0.3\mathrm{eV})$. $H_{\mathrm{RLG}}$ describes one of a
pair of spin-degenerate electron states near the $K$ valley, which also exists
with their time reversed partner at the $K^{\prime}$ valley. We propose that
our theory captures the essence of a single spin-valley component of the full
model.

Experimentally, the integer and fractional QAH states are only observed for
one sign of displacement field [19]. Theoretical studies [41, 42, 43, 44, 45]
find that the integer and fractional QAH states appear only when the relevant
parent band is localized, by the displacement field, to be on the layer
located furthest from the interface with the moiré superlattice. In this
regime, self-consistent HF at filling factor $\nu=1$ electron per moiré unit
cell robustly find a spontaneously spin-valley-polarized state with an
extremely flat and nearly ideal quasiparticle band. Exact diagonalization at
fractional fillings of this HF band reveals fractional Chern insulators, in
agreement with the experiment. The ideal HF band persists even in the absence
of the moiré superlattice with hBN [43, 42, 45]. It should be remarked that
these theoretical studies use models for R5G that differ in various minor
details, yet all arrive at the robust conclusion of the ideal HF band.

Our theory naturally explains the emergence of this ideal HF band. Namely, it
is adiabatically connected to the ideal AHC in our model at
$\mathcal{B}=2\pi/\Omega_{\mathrm{BZ}}$ with strong interactions. Let us first
verify that the experimentally relevant Berry flux per BZ is indeed close to
$2\pi$. To estimate this, we note that the Berry curvature distribution from
$H_{\mathrm{RLG}}$ is concentrated at the band edge and contains a total of
$5\pi$ Berry flux. This Berry flux is localized within an effective momentum
space region of radius $\bar{k}\approx t/(\hbar v)$, beyond which the kinetic
energy sharply increases as $E\sim(k/\bar{k})^{5}$ making those states
irrelevant for the low-energy physics. This gives an effective average Berry
curvature seen by the low-energy electrons of $\bar{\mathcal{B}}\approx
5/\bar{k}^{2}$. Meanwhile, the relevant density at which QAH is observed is
set by the moiré superlattice at the hBN/graphene interface, with a period
$a\approx 11.5$nm. Thus, using
$\Omega_{\mathrm{BZ}}=8\pi^{2}/(\sqrt{3}a^{2})$, the average Berry flux per BZ
is $\bar{\mathcal{B}}\Omega_{\mathrm{BZ}}\approx 1.2\times 2\pi$, which is
indeed close to $2\pi$.

At $\mathcal{B}\Omega_{\mathrm{BZ}}\approx 2\pi$, our model predicts at least
two competing phases, a $C=1$ ideal AHC that is favored by strong
interactions, and a $C=-1$ non-ideal CI that is favored by a strong periodic
potential. The HF integer QAH state in R5G is spin-valley polarized and: has
same Chern sign as the parent Berry curvature 222We thank Junkai Dong and Yves
Kwan for pointing this out., is very ideal, and appears even when the moiré
superlattice is absent. These are all in agreement with our expectations based
on the ideal AHC in the Fock-dominated regime of our theory. Furthermore, its
absence for the opposite sign of displacement field, i.e. when the electrons
are localized on the graphene layer proximate to the moiré, can be explained
by the fact that the moiré essentially acts as a periodic potential, which we
have shown competes with interactions. Thus, while it is likely that a weak
periodic potential may help to stabilize and pin a commensurate AHC, a strong
periodic potential would eventually lead to a weakening of the state and
eventually a topological phase transition out of it.

###  VI.2 Future directions

The exact $\mathcal{B}\rightarrow\mathcal{B}+2\pi/\Omega_{\mathrm{BZ}}$
periodicity of the many-body interacting Hamiltonian
$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\mathrm{F}}$
implies that all properties of the AHC in this model can be obtained from the
corresponding properties of the corresponding WC. This includes the properties
of all excited states as well. It would be interesting to analyze the
resulting dynamics [75] of the AHC through the lens of this mapping. To this
end, it is therefore important to determine the effects of the interaction
terms that were neglected in making the Fock-dominated
$\hat{\mathcal{H}}_{\mathrm{int}}\approx\mathcal{H}_{\mathrm{F}}$
approximation. While our numerical HF analysis of the ground state with
Coulomb potential is in good agreement with the physics being Fock-dominated,
the neglected terms are likely to be important for other physics, such as
those of excitations above the ground state.

There are many interesting direction for further study of our model, as well
as possible extensions. Although we have focused in this paper on the case
where either the potential or interactions dominate, there may be intermediate
phases between the two that would not have an analog in the trivial parent
band model. Another interesting question is the role of Berry curvature
distribution: while we have focused only on the uniform case, it would be
interesting to explore finite-$S$ models and the effect of the Berry curvature
distribution by tuning $M$ in Eq 12. It would also be interesting to consider
a “valley-ful” extension of our model, in which we allow for a time reversed
partner of the parent band, which may allow for interesting valley-coherent
states or excitations.

Finally, we may ask what the implications of an “ideal” AHC is, in the absence
of a periodic potential. While the HF bands may be ideal for fractionalized
states, the concept of fractional filling becomes somewhat dubious since the
unit cell is itself _defined_ by the electron density. However, an interesting
observation is that the Berry curvature density $\mathcal{B}$ also sets a
natural unit cell, $\Omega_{\mathrm{BZ}}=2\pi/\mathcal{B}$, in which there is
$2\pi$ Berry flux per BZ. Fractional filling $p/q<1$ of this unit cell is
equivalent to full filling of a system with a smaller Brillouin zone (enlarged
unit cell) $\Omega_{\mathrm{BZ}}^{\prime}=(p/q)\Omega_{\mathrm{BZ}}$, and an
effective Berry flux of $\mathcal{B}\Omega_{\mathrm{BZ}}^{\prime}=(p/q)2\pi$
per BZ. Depending on the details of the energetics, we speculate that the
system may prefer fractionally filling the $\Omega_{\mathrm{BZ}}$ unit cell,
rather than full filling of the $\Omega_{\mathrm{BZ}}^{\prime}$ unit cell. In
this scenario, the ideal AHC band may lead to the exotic possibility of a
“fractional anomalous Hall crystal” that spontaneously breaks continuous
translation symmetry to form a fractional Chern insulator.

_Note added —_ While finalizing our manuscript, a preprint appeared [76] also
studying the formation of the AHC; while there is little direct overlap, our
findings are all consistent.

###### Acknowledgements.

We are indebt to Liang Fu for valuable discussions at the inception of this
project. We acknowledge helpful discussions with Aidan P. Reddy, Yves Kwan,
Sid Parameswaran, Steve Kivelson, Sri Raghu, Senthil Todadri, Mike Zaletel,
Dan Parker, Patrick Ledwith, Junkai Dong, and Zi-ang Hu. TD acknowledges
support from a startup fund at Stanford University. TT is supported by the
Stanford Graduate Fellowship. The computations are performed using resources
provided by the Stanford Research Computing Center.

## References

  * Andrei  _et al._ [2021] E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moiré materials, Nature Reviews Materials 6, 201 (2021).
  * Mak and Shan [2022] K. F. Mak and J. Shan, Semiconductor moiré materials, Nature Nanotechnology 17, 686 (2022).
  * Bistritzer and MacDonald [2011] R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proceedings of the National Academy of Sciences 108, 12233 (2011).
  * Cao  _et al._ [2018a] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature 556, 43 (2018a).
  * Cao  _et al._ [2018b] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, _et al._ , Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature 556, 80 (2018b).
  * Wu  _et al._ [2019] F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. MacDonald, Topological insulators in twisted transition metal dichalcogenide homobilayers, Physical review letters 122, 086402 (2019).
  * Devakul  _et al._ [2021] T. Devakul, V. Crépel, Y. Zhang, and L. Fu, Magic in twisted transition metal dichalcogenide bilayers, Nature communications 12, 6730 (2021).
  * Liu and Bergholtz [2022] Z. Liu and E. J. Bergholtz, Recent developments in fractional chern insulators, arXiv preprint arXiv:2208.08449 (2022).
  * Parameswaran  _et al._ [2013] S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum hall physics in topological flat bands, Comptes Rendus Physique 14, 816 (2013).
  * Bergholtz and Liu [2013] E. J. Bergholtz and Z. Liu, Topological flat band models and fractional chern insulators, International Journal of Modern Physics B 27, 1330017 (2013).
  * Tang  _et al._ [2011] E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum hall states, Physical review letters 106, 236802 (2011).
  * Neupert  _et al._ [2011] T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum hall states at zero magnetic field, Phys. Rev. Lett. 106, 236804 (2011).
  * Regnault and Bernevig [2011] N. Regnault and B. A. Bernevig, Fractional chern insulator, Phys. Rev. X 1, 021014 (2011).
  * Sheng  _et al._ [2011] D. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum hall effect in the absence of landau levels, Nature communications 2, 389 (2011).
  * Park  _et al._ [2023a] H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, _et al._ , Observation of fractionally quantized anomalous hall effect, Nature 622, 74 (2023a).
  * Xu  _et al._ [2023] F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y. Gu, K. Watanabe, T. Taniguchi, B. Tong, J. Jia, Z. Shi, S. Jiang, Y. Zhang, X. Liu, and T. Li, Observation of integer and fractional quantum anomalous hall effects in twisted bilayer ${\mathrm{mote}}_{2}$, Phys. Rev. X 13, 031037 (2023).
  * Cai  _et al._ [2023] J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, _et al._ , Signatures of fractional quantum anomalous hall states in twisted mote2, Nature 622, 63 (2023).
  * Zeng  _et al._ [2023] Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Knüppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional chern insulator in moiré mote2, Nature 622, 69 (2023).
  * Lu  _et al._ [2024] Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Fractional quantum anomalous hall effect in multilayer graphene, Nature 626, 759 (2024).
  * Kang  _et al._ [2024] K. Kang, B. Shen, Y. Qiu, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Observation of the fractional quantum spin hall effect in moiré mote2, arXiv preprint arXiv:2402.03294 (2024).
  * Wigner [1934] E. Wigner, On the interaction of electrons in metals, Phys. Rev. 46, 1002 (1934).
  * Kivelson  _et al._ [1986] S. Kivelson, C. Kallin, D. P. Arovas, and J. R. Schrieffer, Cooperative ring exchange theory of the fractional quantized hall effect, Physical review letters 56, 873 (1986).
  * Halperin  _et al._ [1986] B. I. Halperin, Z. Tešanović, and F. Axel, Compatibility of crystalline order and the quantized hall effect, Phys. Rev. Lett. 57, 922 (1986).
  * Kivelson  _et al._ [1987] S. Kivelson, C. Kallin, D. P. Arovas, and J. R. Schrieffer, Cooperative ring exchange and the fractional quantum hall effect, Physical Review B 36, 1620 (1987).
  * Tešanović  _et al._ [1989] Z. Tešanović, F. Axel, and B. Halperin, “hall crystal”versus wigner crystal, Physical Review B 39, 8525 (1989).
  * Ceperley and Bernu [2004] D. M. Ceperley and B. Bernu, Ring exchanges and the supersolid phase of ${}^{4}\mathrm{H}\mathrm{e}$, Phys. Rev. Lett. 93, 155303 (2004).
  * Feynman [1953] R. P. Feynman, Atomic theory of the $\lambda$ transition in helium, Phys. Rev. 91, 1291 (1953).
  * Leggett [1970] A. J. Leggett, Can a solid be ”superfluid”?, Phys. Rev. Lett. 25, 1543 (1970).
  * Chen  _et al._ [2020] G. Chen, A. L. Sharpe, E. J. Fox, Y.-H. Zhang, S. Wang, L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi, _et al._ , Tunable correlated chern insulator and ferromagnetism in a moiré superlattice, Nature 579, 56 (2020).
  * Zhang  _et al._ [2019] Y.-H. Zhang, D. Mao, Y. Cao, P. Jarillo-Herrero, and T. Senthil, Nearly flat chern bands in moiré superlattices, Phys. Rev. B 99, 075127 (2019).
  * Han  _et al._ [2023a] T. Han, Z. Lu, G. Scuri, J. Sung, J. Wang, T. Han, K. Watanabe, T. Taniguchi, H. Park, and L. Ju, Correlated insulator and chern insulators in pentalayer rhombohedral stacked graphene, arXiv preprint arXiv:2305.03151 (2023a).
  * Lu  _et al._ [2023] Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Fractional quantum anomalous hall effect in a graphene moire superlattice, arXiv preprint arXiv:2309.17436 (2023).
  * Han  _et al._ [2023b] T. Han, Z. Lu, Y. Yao, J. Yang, J. Seo, C. Yoon, K. Watanabe, T. Taniguchi, L. Fu, F. Zhang, and L. Ju, Large quantum anomalous hall effect in spin-orbit proximitized rhombohedral graphene (2023b), arXiv:2310.17483 [cond-mat.mes-hall] .
  * Zhou  _et al._ [2021a] H. Zhou, T. Xie, T. Taniguchi, K. Watanabe, and A. F. Young, Superconductivity in rhombohedral trilayer graphene, Nature 598, 434–438 (2021a).
  * Zhou  _et al._ [2021b] H. Zhou, T. Xie, A. Ghazaryan, T. Holder, J. R. Ehrets, E. M. Spanton, T. Taniguchi, K. Watanabe, E. Berg, M. Serbyn, and A. F. Young, Half- and quarter-metals in rhombohedral trilayer graphene, Nature 598, 429–433 (2021b).
  * de la Barrera  _et al._ [2022] S. C. de la Barrera, S. Aronson, Z. Zheng, K. Watanabe, T. Taniguchi, Q. Ma, P. Jarillo-Herrero, and R. Ashoori, Cascade of isospin phase transitions in bernal-stacked bilayer graphene at zero magnetic field, Nature Physics 18, 771–775 (2022).
  * Seiler  _et al._ [2022] A. M. Seiler, F. R. Geisenhof, F. Winterer, K. Watanabe, T. Taniguchi, T. Xu, F. Zhang, and R. T. Weitz, Quantum cascade of correlated phases in trigonally warped bilayer graphene, Nature 608, 298–302 (2022).
  * Zhou  _et al._ [2022] H. Zhou, L. Holleis, Y. Saito, L. Cohen, W. Huynh, C. L. Patterson, F. Yang, T. Taniguchi, K. Watanabe, and A. F. Young, Isospin magnetism and spin-polarized superconductivity in bernal bilayer graphene, Science 375, 774–778 (2022).
  * Liu  _et al._ [2023] K. Liu, J. Zheng, Y. Sha, B. Lyu, F. Li, Y. Park, Y. Ren, K. Watanabe, T. Taniguchi, J. Jia, _et al._ , Spontaneous broken-symmetry insulator and metals in tetralayer rhombohedral graphene, Nature nanotechnology , 1 (2023).
  * Han  _et al._ [2023c] T. Han, Z. Lu, G. Scuri, J. Sung, J. Wang, T. Han, K. Watanabe, T. Taniguchi, L. Fu, H. Park, _et al._ , Orbital multiferroicity in pentalayer rhombohedral graphene, Nature 623, 41 (2023c).
  * Dong  _et al._ [2023a] Z. Dong, A. S. Patri, and T. Senthil, Theory of fractional quantum anomalous hall phases in pentalayer rhombohedral graphene moiré structures, arXiv preprint arXiv:2311.03445 (2023a).
  * Zhou  _et al._ [2023] B. Zhou, H. Yang, and Y.-H. Zhang, Fractional quantum anomalous hall effects in rhombohedral multilayer graphene in the moiréless limit and in coulomb imprinted superlattice, arXiv preprint arXiv:2311.04217 (2023).
  * Dong  _et al._ [2023b] J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Anomalous hall crystals in rhombohedral multilayer graphene i: Interaction-driven chern bands and fractional quantum hall states at zero magnetic field, arXiv preprint arXiv:2311.05568 (2023b).
  * Guo  _et al._ [2023] Z. Guo, X. Lu, B. Xie, and J. Liu, Theory of fractional chern insulator states in pentalayer graphene moiré superlattice, arXiv preprint arXiv:2311.14368 (2023).
  * Kwan  _et al._ [2023] Y. H. Kwan, J. Yu, J. Herzog-Arbeitman, D. K. Efetov, N. Regnault, and B. A. Bernevig, Moiré fractional chern insulators iii: Hartree-fock phase diagram, magic angle regime for chern insulator states, the role of the moiré potential and goldstone gaps in rhombohedral graphene superlattices, arXiv preprint arXiv:2312.11617 (2023).
  * Parameswaran [2024] S. Parameswaran, Anomalous hall crystals or moiré chern insulators? spontaneous versus explicit translational symmetry breaking in graphene pentalayers, Journal Club for Condensed Matter Physics 10.36471/JCCM_January_2024_02 (2024).
  * Vishwanath [2023] A. Vishwanath, Zero field fractional hall states, now in graphene heterostructures, Journal Club for Condensed Matter Physics 10.36471/JCCM_December_2023_03 (2023).
  * Park  _et al._ [2023b] Y. Park, Y. Kim, B. L. Chittari, and J. Jung, Topological flat bands in rhombohedral tetralayer and multilayer graphene on hexagonal boron nitride moiré superlattices, Phys. Rev. B 108, 155406 (2023b).
  * Jung  _et al._ [2015] J. Jung, A. M. DaSilva, A. H. MacDonald, and S. Adam, Origin of band gaps in graphene on hexagonal boron nitride, Nature communications 6, 6308 (2015).
  * Jung  _et al._ [2017] J. Jung, E. Laksono, A. M. DaSilva, A. H. MacDonald, M. Mucha-Kruczyński, and S. Adam, Moiré band model and band gaps of graphene on hexagonal boron nitride, Phys. Rev. B 96, 085442 (2017).
  * Wallbank  _et al._ [2013] J. R. Wallbank, A. A. Patel, M. Mucha-Kruczyński, A. K. Geim, and V. I. Fal’ko, Generic miniband structure of graphene on a hexagonal substrate, Phys. Rev. B 87, 245408 (2013).
  * Resta [2011] R. Resta, The insulating state of matter: a geometrical theory, The European Physical Journal B 79, 121 (2011).
  * Roy [2014] R. Roy, Band geometry of fractional topological insulators, Phys. Rev. B 90, 165139 (2014).
  * Wang  _et al._ [2021] J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact landau level description of geometry and interaction in a flatband, Phys. Rev. Lett. 127, 246403 (2021).
  * Ledwith  _et al._ [2022] P. J. Ledwith, A. Vishwanath, and D. E. Parker, Vortexability: A unifying criterion for ideal fractional chern insulators (2022), arXiv:2209.15023 [cond-mat.str-el] .
  * Claassen  _et al._ [2015] M. Claassen, C. H. Lee, R. Thomale, X.-L. Qi, and T. P. Devereaux, Position-momentum duality and fractional quantum hall effect in chern insulators, Phys. Rev. Lett. 114, 236802 (2015).
  * Ledwith  _et al._ [2020] P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Fractional chern insulator states in twisted bilayer graphene: An analytical approach, Phys. Rev. Res. 2, 023237 (2020).
  * Lee  _et al._ [2017] C. H. Lee, M. Claassen, and R. Thomale, Band structure engineering of ideal fractional chern insulators, Phys. Rev. B 96, 165150 (2017).
  * Jackson  _et al._ [2015] T. S. Jackson, G. Möller, and R. Roy, Geometric stability of topological lattice phases, Nature communications 6, 8629 (2015).
  * Mera and Ozawa [2021] B. Mera and T. Ozawa, Kähler geometry and chern insulators: Relations between topology and the quantum metric, Phys. Rev. B 104, 045104 (2021).
  * Tarnopolsky  _et al._ [2019] G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene, Phys. Rev. Lett. 122, 106405 (2019).
  * Song and Bernevig [2022] Z.-D. Song and B. A. Bernevig, Magic-angle twisted bilayer graphene as a topological heavy fermion problem, Phys. Rev. Lett. 129, 047601 (2022).
  * Bernevig  _et al._ [2006] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin hall effect and topological phase transition in hgte quantum wells, Science 314, 1757 (2006), https://www.science.org/doi/pdf/10.1126/science.1133734 .
  * Min and MacDonald [2008] H. Min and A. H. MacDonald, Electronic structure of multilayer graphene, Progress of Theoretical Physics Supplement 176, 227–252 (2008).
  * Zeng  _et al._ [2024a] Y. Zeng, T. M. R. Wolf, C. Huang, N. Wei, S. A. A. Ghorashi, A. H. MacDonald, and J. Cano, Gate-tunable topological phases in superlattice modulated bilayer graphene (2024a), arXiv:2401.04321 [cond-mat.mes-hall] .
  * Tan  _et al._ [2024] T. Tan, A. P. Reddy, L. Fu, and T. Devakul, Designing topology and fractionalization in narrow gap semiconductor films via electrostatic engineering (2024), arXiv:2402.03085 [cond-mat.str-el] .
  * Su  _et al._ [2022] Y. Su, H. Li, C. Zhang, K. Sun, and S.-Z. Lin, Massive dirac fermions in moiré superlattices: A route towards topological flat minibands and correlated topological insulators, Phys. Rev. Res. 4, L032024 (2022).
  * Ghorashi  _et al._ [2023] S. A. A. Ghorashi, A. Dunbrack, A. Abouelkomsan, J. Sun, X. Du, and J. Cano, Topological and stacked flat bands in bilayer graphene with a superlattice potential, Phys. Rev. Lett. 130, 196201 (2023).
  * Ghorashi and Cano [2023] S. A. A. Ghorashi and J. Cano, Multilayer graphene with a superlattice potential, Phys. Rev. B 107, 195423 (2023).
  * Suri  _et al._ [2023] N. Suri, C. Wang, B. M. Hunt, and D. Xiao, Superlattice engineering of topology in massive dirac fermions, Phys. Rev. B 108, 155409 (2023).
  * Note [1] L. Fu, private communications.
  * Hofstadter [1976] D. R. Hofstadter, Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239 (1976).
  * Dong  _et al._ [2023c] J. Dong, P. J. Ledwith, E. Khalaf, J. Y. Lee, and A. Vishwanath, Many-body ground states from decomposition of ideal higher chern bands: Applications to chirally twisted graphene multilayers, Phys. Rev. Res. 5, 023166 (2023c).
  * Note [2] We thank Junkai Dong and Yves Kwan for pointing this out.
  * Grüner [1988] G. Grüner, The dynamics of charge-density waves, Rev. Mod. Phys. 60, 1129 (1988).
  * Zeng  _et al._ [2024b] Y. Zeng, D. Guerci, V. Crépel, A. J. Millis, and J. Cano, Sublattice structure and topology in spontaneously crystallized electronic states (2024b), arXiv:2402.17867 [cond-mat.str-el] .

##  Appendix A analytical results on the trace condition violation of ansatz
wave functions

We showed numerically in the main text Fig.4 that with the gauge
transformation, $\hat{Q}$ (Fock) and $\hat{W}$ (Potential), the TCV of the
$C=\pm 1$ band decrease/increase as a function of $\xi$ at large $\xi$. We now
give a proof for this behaviour.

Notice first that at large $\chi$ limit, the berry curvature $B(k)$ becomes
uniform for both $\hat{Q}$ and $\hat{W}$ ansatz wave functions. We will not
prove this statement, but satisfy ourselves to say that this is the behaviour
that can be easily observed numerically. Thus the term
$\int|B(\bm{k})|d\bm{k}$ in TCV is just $2\pi$ for both $\hat{Q}$ and
$\hat{W}$ ansatz wave functions. We study the behaviour $\int
Tr[g_{\mu\nu}^{FS}(\bm{k})]d\bm{k}$ for two ansatz wave functions at large
$\xi$ limit. We will focus on the square lattice case to avoid cluttered
notation, and the proof can be easily generalized to triangular lattice.

For $\hat{Q}$ ansatz (Assume that
$\mathcal{B}=\frac{2\pi}{\Omega_{BZ}}=\frac{2\pi}{g^{2}}$).

$\begin{split}&\ket{\psi^{-}(\bm{k})}\\\
&=\sum_{\bm{g}}\mathcal{N}(\bm{k})e^{-\frac{(\bm{k}+\bm{g})^{2}}{4\xi^{2}}}e^{-i\frac{\pi}{g^{2}}[g_{x}g_{y}+k_{x}g_{y}-k_{y}g_{x}]}\ket{u_{\bm{k}+\bm{g}}}e^{i\bm{g}\cdot\bm{r}}\end{split}$
(38)

If we replace the summation by an integral (we will discuss this approximation
afterwards)

$\begin{split}&\ket{\psi^{-}(\bm{k})}\\\ &=\int
d\bm{g}\frac{1}{\sqrt{2\pi\xi^{2}}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{4\xi^{2}}}e^{-i\frac{\pi}{g^{2}}[g_{x}g_{y}+k_{x}g_{y}-k_{y}g_{x}]}\ket{u_{\bm{k}+\bm{g}}}e^{i\bm{g}\cdot\bm{r}}\end{split}$
(39)

The $\mathcal{N}(\bm{k})\approx\frac{1}{\sqrt{2\pi\xi^{2}}}$ is obtained by
requiring $\ket{\psi^{-}}$ to be normalized under this approximation. Notice
that
$|\innerproduct{\psi^{-}(\bm{k})}{\psi^{-}(\bm{k}+d\bm{k})}|^{2}=1-g_{\mu\nu}dk_{\mu}dk_{\nu}$.
We then obtain the following

$\begin{split}&|\innerproduct{\psi^{-}(\bm{k})}{\psi^{-}(\bm{k}+dk_{x})}|^{2}\\\
&=|\int
d\bm{g}\frac{1}{2\pi\xi^{2}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{2\xi^{2}}}e^{\frac{{-dk_{x}^{2}}}{4\xi^{2}}}e^{-\frac{(k_{x}+g_{x})dk_{x}}{2\xi^{2}}}e^{-i\frac{\pi}{g^{2}}dk_{x}g_{y}}e^{\frac{-\pi
dk_{x}^{2}}{2g^{2}}}\\\ &e^{\frac{i\pi(k_{y}+g_{y})dk_{x}}{2g^{2}}}|^{2}\\\
&=e^{\frac{-\pi^{2}(dk^{2}_{x})}{g^{2}}}e^{-\frac{dk^{2}_{x}}{2\xi^{2}}}|\int
d\bm{g}\frac{1}{2\pi\xi^{2}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{2\xi^{2}}}e^{-\frac{(k_{x}+g_{x})dk_{x}}{2\xi^{2}}}|^{2}\end{split}$
(40)

By symmetry, it is obvious that the term inside $|..|$ is of order
$1+L_{1}dk^{2}_{x}$. Thus we have an expression of $g_{xx}$ and similarly
$g_{yy}$

$\begin{split}g_{xx}=\frac{\pi}{g^{2}}+\frac{1}{2\xi^{2}}-2L_{1}\\\
g_{yy}=\frac{\pi}{g^{2}}+\frac{1}{2\xi^{2}}-2L_{2}\end{split}$ (41)

Where we have

$L_{1}+L_{2}=\int
d\bm{g}\frac{1}{2\pi\xi^{2}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{2\xi^{2}}}\frac{(\bm{k}+\bm{g})^{2}}{8\xi^{4}}=\frac{1}{4\xi^{2}}$
(42)

Thus at large $\xi$, we have

$Tr[g_{\mu\nu}^{FS}(\hat{Q})]=\frac{2\pi}{g^{2}}+\frac{1}{2\xi^{2}}$ (43)

We now turn to the $\hat{W}$ ansatz wave function. The analysis is exactly the
same, except there will be a sign change in the gauge transformation

$\begin{split}&|\innerproduct{\psi^{+}(\bm{k})}{\psi^{+}(\bm{k}+dk_{x})}|^{2}\\\
&=|\int
d\bm{g}\frac{1}{2\pi\xi^{2}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{2\xi^{2}}}e^{\frac{{-dk_{x}^{2}}}{4\xi^{2}}}e^{-\frac{(k_{x}+g_{x})dk_{x}}{2\xi^{2}}}e^{i\frac{\pi}{g^{2}}dk_{x}g_{y}}e^{\frac{-\pi
dk_{x}^{2}}{2g^{2}}}\\\ &e^{\frac{i\pi(k_{y}+g_{y})dk_{x}}{2g^{2}}}|^{2}\\\
&=e^{\frac{-\pi^{2}(dk^{2}_{x})}{g^{2}}}e^{-\frac{dk^{2}_{x}}{2\xi^{2}}}|\int
d\bm{g}\frac{1}{2\pi\xi^{2}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{2\xi^{2}}}e^{-\frac{(k_{x}+g_{x})dk_{x}}{2\xi^{2}}}e^{i\frac{2\pi}{g^{2}}dk_{x}g_{y}}|^{2}\end{split}$
(44)

In that case, we will have

$\begin{split}&|\int
d\bm{g}\frac{1}{2\pi\xi^{2}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{2\xi^{2}}}e^{-\frac{(k_{x}+g_{x})dk_{x}}{2\xi^{2}}}e^{i\frac{2\pi}{g^{2}}dk_{x}g_{y}}|^{2}\\\
&=|\int
d\bm{\eta}\frac{1}{2\pi\xi^{2}}e^{-\frac{\bm{\eta}^{2}}{2\xi^{2}}}e^{-\frac{\eta_{x}dk_{x}}{2\xi^{2}}}e^{i\frac{2\pi}{g^{2}}dk_{x}\eta_{y}}|^{2}\\\
&\approx|1+\int
d\bm{\eta}\frac{e^{-\frac{\bm{\eta}}{2\xi^{2}}}}{2\pi\xi^{2}}\frac{dk^{2}_{x}}{2}(\frac{2i\pi}{g^{2}}{\eta_{y}}-\frac{\eta_{x}}{2\xi^{2}})^{2}|^{2}\end{split}$
(45)

Thus, we will have

$\begin{split}&L_{1}+L_{2}\\\ &=\int
d\bm{g}\frac{1}{2\pi\xi^{2}}e^{-\frac{(\bm{k}+\bm{g})^{2}}{2\xi^{2}}}(\bm{k}+\bm{g})^{2}(\frac{1}{8\xi^{4}}-\frac{2\pi^{2}}{g^{4}})\\\
&=\frac{1}{4\xi^{2}}-\frac{4\pi^{2}\xi^{2}}{g^{4}}\end{split}$ (46)

Then at large $\xi$

$Tr[g_{\mu\nu}^{FS}(\hat{W})]=\frac{2\pi}{g^{2}}+\frac{1}{2\xi^{2}}+\frac{8\pi^{2}\xi^{2}}{g^{4}}$
(47)

Now, we notice that when we replaced the $\sum_{\bm{g}}$ by $\int d\bm{g}$, we
are effectively ignoring the $\bm{k}$ dependence of the trace. We could have
been more careful with the momentum dependence. But it turns out that those
terms are not super important in understanding the physics. Let me now show
this point: The properly normalized $\hat{Q}$ ansatz wavefunction
$\ket{\psi^{-}(\bm{k})}$ is

$\begin{split}&\ket{\psi^{-}(\bm{k})}\\\
&=\sum_{\bm{g}}\mathcal{N}(\bm{k})e^{-\frac{(\bm{k}+\bm{g})^{2}}{4\xi^{2}}}e^{-i\frac{\pi}{g^{2}}[g_{x}g_{y}+k_{x}g_{y}-k_{y}g_{x}]}\ket{u_{\bm{k}+\bm{g}}}e^{i\bm{g}\cdot\bm{r}}\end{split}$
(48)

Where the normalization constant is the following, with
$z_{x/y}=\frac{k_{x/y}\pi}{g}$, $q=\exp(-\frac{2\pi^{2}\xi^{2}}{g^{2}})$, and
$\theta$ is the Jacobi theta function
$\theta(z,q)=\sum_{n=-\infty}^{\infty}q^{n^{2}}\exp(2inz)$

$\mathcal{N}(\bm{k})=\frac{g}{\sqrt{2\pi}\xi}[\mathcal{\theta}(z_{x},q)\mathcal{\theta}(z_{y},q)]^{-\frac{1}{2}}$
(49)

In that case, we find that

$\begin{split}&Tr[g^{FS}_{\mu\nu}(\hat{Q})]=\frac{2\pi}{g^{2}}+\frac{1}{2\xi^{2}}\\\
&-\frac{2\pi^{2}q}{g^{2}}cos(\frac{2k_{x}\pi}{g})-\frac{2\pi^{2}q}{g^{2}}cos(\frac{2k_{y}\pi}{g})+O(q^{2})\end{split}$
(50)

So we see that at large $\xi$, all the momentum dependence are suppressed by
$q=\exp(-\frac{2\pi^{2}\xi^{2}}{g^{2}})$, which is negligible when compared
with $\frac{1}{\xi^{2}}$

For the $\hat{W}$ ansatz, we can do similar things, and arrive at the results
of

$\begin{split}&Tr[g^{FS}_{\mu\nu}(\hat{W})]=\frac{2\pi}{g^{2}}+\frac{1}{2\xi^{2}}+\frac{8\pi^{2}\xi^{2}}{g^{4}}\\\
&-\frac{2\pi^{2}q}{g^{2}}cos(\frac{2k_{x}\pi}{g})-\frac{2\pi^{2}q}{g^{2}}cos(\frac{2k_{y}\pi}{g})\\\
&-\frac{32\pi^{4}\xi^{4}q}{g^{6}}cos(\frac{2k_{x}\pi}{g})-\frac{32\pi^{4}\xi^{4}q}{g^{6}}cos(\frac{2k_{y}\pi}{g})+O(q^{2})\end{split}$
(51)

Again, we realized that all the momentum dependence is suppressed by powers of
$q$.

##  Appendix B more on Chern number mapping

In the main text, we sketched how the transformation $\hat{W}$ and $\hat{Q}$
will change the Chern number. In this appendix section, we work out that in
more details. We will focus on the strong-potential, square lattice case,
though it is obvious how the calculation can be translated to other cases.

The Hamiltonian at $\mathcal{B}_{0}$ with potential strength $U_{0}$ takes the
form,
$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\mathrm{pot0}}$.

$\begin{split}\hat{\mathcal{H}}_{0}=&\sum_{\bm{k},\bm{g}}\mathcal{E}(\bm{k}+\bm{g})c^{\dagger[\mathcal{B}_{0}]}_{\bm{k},\bm{g}}c^{[\mathcal{B}_{0}]}_{\bm{k},\bm{g}}\\\
\hat{\mathcal{H}}_{\mathrm{pot0}}=&-U_{0}e^{-\frac{\mathcal{B}_{0}}{4}g^{2}}\sum_{\bm{k},\bm{g},j}c^{\dagger[\mathcal{B}_{0}]}_{\bm{k},\bm{g}+\bm{b}_{j}}e^{i\frac{\mathcal{B}_{0}}{2}(\bm{k}+\bm{g})\crossproduct\bm{b}_{j}}c^{[\mathcal{B}_{0}]}_{\bm{k},\bm{g}}\end{split}$
(52)

The Hamiltonian at $\mathcal{B}_{1}=\mathcal{B}_{0}+\frac{2\pi}{g^{2}}$ with
potential strength $U_{0}\exp(\frac{\pi}{2})$ takes the form,
$\hat{\mathcal{H}_{1}}=\hat{\mathcal{H}}_{01}+\hat{\mathcal{H}}_{\mathrm{pot1}}$,
with

$\begin{split}\hat{\mathcal{H}}_{01}=&\sum_{\bm{k},\bm{g}}\mathcal{E}(\bm{k}+\bm{g})c^{\dagger[\mathcal{B}_{1}]}_{\bm{k},\bm{g}}c^{[\mathcal{B}_{1}]}_{\bm{k},\bm{g}}\\\
\hat{\mathcal{H}}_{\mathrm{pot1}}=&-U_{0}e^{-\frac{\mathcal{B}_{0}}{4}g^{2}}\sum_{\bm{k},\bm{g},j}c^{\dagger[\mathcal{B}_{1}]}_{\bm{k},\bm{g}+\bm{b}_{j}}e^{i\frac{\mathcal{B}_{1}}{2}(\bm{k}+\bm{g})\crossproduct\bm{b}_{j}}c^{[\mathcal{B}_{1}]}_{\bm{k},\bm{g}}\end{split}$
(53)

Where again we should stress that $c^{\dagger[\mathcal{B}]}_{\bm{k},\bm{g}}$
creates $e^{i\bm{k}\cdot\bm{r}}\ket{u^{\mathcal{B}}_{\bm{k+g}}}$. If we define

$d^{[\mathcal{B}_{1}]}_{\bm{k},\bm{g}}\equiv
c^{[\mathcal{B}_{1}]}_{\bm{k},\bm{g}}W_{\bm{g}}(\bm{k})$ (54)

Then $\hat{\mathcal{H}_{1}}$, written in terms of operators $d^{\dagger}$/$d$
operators, is formally identical to $\hat{\mathcal{H}}$, written in terms of
operators $c^{\dagger[\mathcal{B}]}$/$c^{[\mathcal{B}]}$. By formally
equivalent, we mean that the matrix we need to diagonalize when solving this
non-interacting Hamiltonian is exactly the same in these two cases, and hence
the numerical eigenvalues and numerical eigenvectors are identical in
$\mathcal{B}_{0}$ and $\mathcal{B}_{1}$ case. To see this point, just consider
one example term

$\begin{split}c^{\dagger[\mathcal{B}_{1}]}_{\bm{k},\bm{g}+\bm{b}_{j}}e^{i\frac{\mathcal{B}_{1}}{2}(\bm{k}+\bm{g})\crossproduct\bm{b}_{2}}c^{[\mathcal{B}_{1}]}_{\bm{k},\bm{g}}\\\
=d^{\dagger[\mathcal{B}_{1}]}_{\bm{k},\bm{g}+\bm{b}_{j}}e^{i\frac{\mathcal{B}_{0}}{2}(\bm{k}+\bm{g})\crossproduct\bm{b}_{2}}d^{[\mathcal{B}_{1}]}_{\bm{k},\bm{g}}\end{split}$
(55)

Notice,crucially, this equality depends on the fact that we take only the the
first harmonics in the periodic potential.

The numerical eigenvalues being identical means that the physical spectrum is
the same. However, the numerical eigenvectors in these cases are written in
two different basis, hence leading to different Chern number. Assume that the
Bloch wavefunctions at $\mathcal{B}_{0}$ for a single isolated band is

$|\psi_{\bm{k}}^{(0)}\rangle=\sum_{\bm{g}}\psi_{\bm{g}}^{(0)}(\bm{k})e^{i(\bm{k}+\bm{g})\cdot\bm{r}}|u^{\mathcal{B}_{0}}_{\bm{k}+\bm{g}}\rangle$
(56)

Where again, we stress that the $\psi^{(0)}_{\bm{g}}(\bm{k})$ is the numerical
eigenvectors we obtained when diagonalizing the matrix, while the physical
wavefunction is written in the basis of
$\ket{u_{\bm{k+g}}^{\mathcal{B}_{0}}}$. If we were doing ordinary continuum
model for moiré system, $\ket{u}$ is simply 1. But here, the internal
structure of $\ket{u}$ is absolutely important when calculating quantum
geometric quantities involving inner produces of $\ket{\psi_{\bm{k}}}$ at
different $\bm{k}$. By the mapping we discussed above, the corresponding
wavefunction at $\mathcal{B}_{1}$ is

$|\psi_{\bm{k}}^{(1)}\rangle=\sum_{\bm{g}}\psi_{\bm{g}}^{(0)}(\bm{k})e^{i(\bm{k}+\bm{g})\cdot\bm{r}}W_{\bm{g}}(\bm{k})|u^{\mathcal{B}_{1}}_{\bm{k}+\bm{g}}\rangle$
(57)

The definition of pahse $W_{\bm{g}}(\bm{k})$ is given in the main text. Then
the Berry curvature of $|\psi_{\bm{k}}^{(0)}\rangle$ is ($\mu=x,y$, or more
formally $\mu=k_{x},k_{y}$)

$\begin{split}B^{(0)}(\bm{k})&=i\epsilon_{\mu\nu}[\partial_{\mu}(e^{i\bm{k}\cdot\bm{r}}\bra{\psi^{(0)}_{\bm{k}}})][\partial_{\nu}(e^{-i\bm{k}\cdot\bm{r}}\ket{\psi^{(0)}_{\bm{k}}})]\\\
&=b_{01}(\bm{k})+b_{02}(\bm{k})+b_{03}(\bm{k})\end{split}$ (58)

The following definitions are introduced

$\begin{split}&b_{01}(\bm{k})=i\epsilon_{\mu\nu}[\partial_{\mu}\psi^{(0)*}_{\bm{g}}(\bm{k})][\partial_{\nu}\psi^{(0)}_{g}(\bm{k})]\\\
&b_{02}(\bm{k})=\sum_{\bm{g}}\epsilon_{\mu\nu}[\partial_{\mu}|\psi^{(0)}_{\bm{g}}(\bm{k})|^{2}]A_{\nu}(\bm{k}+\bm{g})\\\
&b_{03}(\bm{k})=\sum_{\bm{g}}|\psi^{(0)}_{\bm{g}}(\bm{k})|^{2}F(\bm{k}+\bm{g})\\\
\end{split}$ (59)

Where we have also defined the following two parent band quantities (since
it’s parent band quantity, then $\bm{k}$ is not restricted to BZ.)

$\begin{split}&A_{\mu}(\bm{k})=i\innerproduct{u^{\mathcal{B}_{0}}_{\bm{k}}}{\partial_{\mu}u^{\mathcal{B}_{0}}_{\bm{k}}}\\\
&F(\bm{k})=\epsilon_{\mu\nu}\partial_{\mu}A_{\nu}(\bm{k})\end{split}$ (60)

Notice that these three sets of equations are completely general, even if the
parent band spinor $\ket{u^{\mathcal{B}_{0}}_{k+g}}$ are replaced by some
other spinors different from what is given in the main text. But $b_{02}$ and
$b_{03}$ will not contribute to the total Chern number, as their sum is a
total derivative of a periodic function (using
$|\psi^{(0)}_{\bm{g}+\bm{g_{i}}}(\bm{k})|^{2}=|\psi^{(0)}_{\bm{g}}(\bm{k}+\bm{g_{i}})|^{2}$)

$b_{02}(\bm{k})+b_{03}(\bm{k})=\sum_{\bm{g}}\epsilon_{\mu\nu}\partial_{\mu}[|\psi^{(0)}_{\bm{g}}(\bm{k})|^{2}A_{\nu}(\bm{k}+\bm{g})]\\\
$ (61)

Thus, we have the following conclusion

$2\pi C^{(0)}=\int_{BZ}B^{(0)}(\bm{k})d\bm{k}=\int_{BZ}b_{01}(\bm{k})d\bm{k}$
(62)

And correspondingly for $\ket{\psi^{(1)}_{\bm{k}}}$

$\begin{split}&2\pi C^{(1)}\\\
&=\int_{BZ}\sum_{\bm{g}}i\epsilon_{\mu\nu}[\partial_{\mu}(W^{*}_{\bm{g}}(\bm{k})\psi^{(0)*}_{\bm{g}}(\bm{k}))][\partial_{\nu}(W_{\bm{g}}(\bm{k})\psi^{(0)}_{g}(\bm{k}))]d\bm{k}\\\
&=\int_{BZ}\sum_{\bm{g}}i\epsilon_{\mu\nu}\partial_{\mu}\Biggl{\\{}W^{*}_{\bm{g}}(\bm{k})\psi^{(0)*}_{\bm{g}}(\bm{k})[\partial_{\nu}(W_{\bm{g}}(\bm{k})\psi^{(0)}_{g}(\bm{k}))]\Biggl{\\}}d\bm{k}\\\
&=\int_{BZ}\sum_{\bm{g}}i\epsilon_{\mu\nu}\partial_{\mu}(\psi^{(0)*}_{\bm{g}}(\bm{k})\partial_{\nu}\psi^{(0)}_{\bm{g}}(\bm{k}))d\bm{k}\\\
&+\sum_{\bm{g}}i\epsilon_{\mu\nu}\partial_{\mu}[|\psi^{(0)}_{\bm{g}}(\bm{k})|^{2}(W^{*}_{\bm{g}}(\bm{k})\partial_{\nu}W_{\bm{g}}(\bm{k}))]d\bm{k}\\\
&=2\pi
C^{(0)}+\sum_{\bm{g}}i\epsilon_{\mu\nu}\partial_{\mu}[|\psi^{(0)}_{\bm{g}}(\bm{k})|^{2}(W^{*}_{\bm{g}}(\bm{k})\partial_{\nu}W_{\bm{g}}(\bm{k}))]d\bm{k}\end{split}$
(63)

We will now focus on the second term on the last line. We give the following
definition

$\alpha_{\nu}(\bm{k})\equiv\sum_{\bm{g}}|\psi^{(0)}_{\bm{g}}(\bm{k})|^{2}(W^{*}_{\bm{g}}(\bm{k})\partial_{\nu}W_{\bm{g}}(\bm{k}))$
(64)

It is easy to verify that with of form of $W_{\bm{g}}(\bm{k})$ given in the
main text, $\alpha_{\mu}(\bm{k})$ has the following behaviour under the
translation of reciprocal lattice vectors

$\begin{split}\alpha_{x}(\bm{k}+g(0,1))=\alpha_{x}(\bm{k})-i\frac{\pi}{g}\\\
\alpha_{y}(\bm{k}+g(1,0))=\alpha_{y}(\bm{k})+i\frac{\pi}{g}\end{split}$ (65)

Which proves that $C^{(1)}=C^{(0)}-1$. Finally, we note that all these
calculations (square lattice and triangular lattice), can all be done in a
gauge-independent manner.

##  Appendix C Hartree-Fock Calculation

We will describe the operational perspective of the Hartree-Fock calculation
presented in the main text. Hartree-Fock approximation boils down to replacing
the interaction
$\hat{\mathcal{H}}_{int}\approx\hat{\mathcal{H}}_{H}+\hat{\mathcal{H}}_{F}$.
Where the definitions of these three Hamiltonian is given in the main text.
After doing the mean-field approximation in $\hat{\mathcal{H}}_{H}$ and
$\hat{\mathcal{H}}_{F}$, we obtain the following

$\begin{split}\hat{\mathcal{H}}_{\mathrm{H}}=\frac{1}{A}\sum_{\begin{subarray}{c}\bm{k}_{1}\bm{k}_{2}\\\
\bm{g}_{1}\bm{g}_{2}\bm{g}_{3}\bm{g}_{4}\end{subarray}}\tilde{V}^{\bm{k}_{1}\bm{k}_{2}\bm{k}_{2}\bm{k}_{1}}_{\bm{g}_{1}\bm{g}_{2}\bm{g}_{3}\bm{g}_{4}}\rho_{\bm{g_{4}},\bm{g_{1}}}(\bm{k_{1}})({c}_{\bm{k}_{2}\bm{g}_{2}}^{\dagger}{c}_{\bm{k}_{2}\bm{g}_{3}})\\\
\hat{\mathcal{H}}_{\mathrm{F}}=-\frac{1}{A}\sum_{\begin{subarray}{c}\bm{k}_{1}\bm{k}_{2}\\\
\bm{g}_{1}\bm{g}_{2}\bm{g}_{3}\bm{g}_{4}\end{subarray}}\tilde{V}^{\bm{k}_{1}\bm{k}_{2}\bm{k}_{1}\bm{k}_{2}}_{\bm{g}_{1}\bm{g}_{2}\bm{g}_{3}\bm{g}_{4}}\rho_{\bm{g_{3}},\bm{g_{1}}}(\bm{k_{1}})({c}_{\bm{k}_{2}\bm{g}_{2}}^{\dagger}{c}_{\bm{k}_{2}\bm{g}_{4}})\end{split}$
(66)

The total Hamiltonian will then be the sum of
$\hat{\mathcal{H}}_{HF}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{pot}+\hat{\mathcal{H}}_{H}+\hat{\mathcal{H}}_{F}$.
We have also defined the density matrix $\rho$ as the following

$\begin{split}&\rho_{\bm{g}_{4},\bm{g}_{1}}(\bm{k}_{1})=\langle
c^{\dagger}_{\bm{k_{1}g_{1}}}c_{\bm{k_{1}g_{4}}}\rangle\\\ &=\sum_{n\in
occ}\psi^{n}_{\bm{g_{4}}}(\bm{k}_{1})\psi^{*n}_{\bm{g_{1}}}(\bm{k}_{1})\mathcal{F}(\bm{k_{1}}+\bm{g_{1}},\bm{k_{1}}+\bm{g_{4}})\end{split}$
(67)

Where $\psi^{n}_{g}(\bm{k})$ is the numerical eigenvectors obtained by
diagonalizing $\hat{\mathcal{H}}_{HF}$.$n$ is the band index, $\bm{g}$ is the
component index, $\bm{k}$ labels the momentum in the BZ. The braket means
averaging over the Slater determinant ground state. The equations is solved
until convergence (of the density matrix/Hartree-Fock spectrum).

Our system is solved on a $9\times 9$ sites lattice. That is, the allowed
$\bm{k}$ are of the form ($N_{1},N_{3}=0,1...8$)

$\bm{k}=\frac{N_{1}}{9}\bm{b}_{1}+\frac{N_{3}}{9}\bm{b}_{3}$ (68)