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# Broken Symmetry in Ideal Chern Bands

Hui Liu  Department of Physics, Stockholm University, AlbaNova University
Center, 106 91 Stockholm, Sweden  Kang Yang  Dahlem Center for Complex Quantum
Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin,
Germany  Ahmed Abouelkomsan  Department of Physics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA  Zhao Liu  Zhejiang Institute
of Modern Physics, Zhejiang University, Hangzhou 310058, China  Emil J.
Bergholtz  Department of Physics, Stockholm University, AlbaNova University
Center, 106 91 Stockholm, Sweden

###### Abstract

Recent observations of the fractional anomalous quantum Hall effect in moiré
materials have reignited the interest in fractional Chern insulators (FCIs).
The chiral limit in which analytic Landau level-like single-particle states
form an “ideal” Chern band and local interactions lead to Laughlin-like FCIs
at $1/3$ filling, has been very useful for understanding these systems by
relating them to the lowest Landau level. We show, however, that, even in the
idealized chiral limit, a fluctuating quantum geometry is associated with
strongly broken symmetries and a phenomenology very different from that of
Landau levels. In particular, particle-hole symmetry is strongly violated and
e.g. at $2/3$ filling an emergent interaction driven Fermi liquid state with
no Landau level counterpart is energetically favoured. In fact, even the exact
Laughlin-like zero modes at $1/3$ filling have a non-uniform density tracking
the underlying quantum geometry. Moreover, by switching to a Coulomb
interaction, the ideal Chern band features charge density wave states with no
lowest Landau level counterpart.

_Introduction._ — The rapid development of techniques to fabricate van-der-
Waals heterostructures with moiré patterns is providing excitingly new
opportunities to realize and manipulate novel quantum phases of matter Geim
and Grigorieva (2013); Novoselov  _et al._ (2016). Fractional Chern insulators
(FCIs) Kol and Read (1993); Tang  _et al._ (2011); Sun  _et al._ (2011);
Neupert  _et al._ (2011); Sheng  _et al._ (2011); Regnault and Bernevig
(2011); Möller and Cooper (2009); Kapit and Mueller (2010); Bergholtz and Liu
(2013); Parameswaran  _et al._ (2013); Liu and Bergholtz (2024), which are
lattice analogues of the celebrated fractional quantum Hall (FQH) states in
continuum two-dimensional electron gases (2DEGs) Tsui  _et al._ (1982);
Laughlin (1983); Hansson  _et al._ (2017), have been theoretically predicted
and experimentally observed in graphene Abouelkomsan  _et al._ (2020);
Repellin and Senthil (2020); Ledwith  _et al._ (2020); Liu  _et al._ (2021);
Xie  _et al._ (2021); Lu  _et al._ (2023) and transition metal dichalcogenide
moiré materials Li  _et al._ (2021); Crépel and Fu (2023); Cai  _et al._
(2023); Zeng  _et al._ (2023); Park  _et al._ (2023); Xu  _et al._ (2023). The
exciting experimental evidence of FCIs has further attracted tremendous
theoretical attention to understand the phenomena emerging in experiments Dong
_et al._ (2023a); Zhou  _et al._ (2023); Wang  _et al._ (2024); Reddy  _et
al._ (2023); Morales-Durán  _et al._ (2023); Yu  _et al._ (2023); Abouelkomsan
_et al._ (2023a); Xu  _et al._ (2024); Dong  _et al._ (2023b).

Based on the analogy between FCIs and the FQH states, the emergence of FCI
states in moiré materials suggests the similarity of the partially filled
moiré band to a Landau level (LL) in 2DEGs Ledwith  _et al._ (2020); Wang  _et
al._ (2021); Morales-Durán  _et al._ (2023). This similarity imposes strong
conditions on the dispersion, topology, and quantum geometry Goerbig (2012);
Roy (2014); Jackson  _et al._ (2015); Claassen  _et al._ (2015); Ozawa and
Mera (2021); Mera and Ozawa (2021a); Varjas  _et al._ (2022); Mera and Ozawa
(2021b); Wang  _et al._ (2021); Ledwith  _et al._ (2020); Wang and Liu (2022);
Ledwith  _et al._ (2022); Mera  _et al._ (2022); Abouelkomsan  _et al._
(2023b); Estienne  _et al._ (2023); Ledwith  _et al._ (2023); Hetényi and
Lévay (2023) of the moiré band. In general, a moiré band is expected to mimic
a LL if (i) it is nearly flat in dispersion; (ii) it carries a nonzero Chern
number; (iii) its quantum geometry, characterized by the Berry curvature and
the Fubini-Study (FS) metric Provost and Vallee (1980); Cheng (2010), is
nearly uniform in the entire Brillouin zone (BZ); and (iv) its FS metric
varies approximately in sync with the Berry curvature. The last two conditions
originates from the special quantum geometry of a LL, i.e., the Berry
curvature $\Omega$ and the FS metric $g$ are constant and related to each
other by $g_{ij}=\left(n+\frac{1}{2}\right)|\Omega|\delta_{ij}$, where $n$ is
the LL index Ozawa and Mera (2021).

Figure 1: Broken symmetry in an ideal Chern band. (a) Moiré band structure of
cTBG at the first magic angle $\theta\approx 1.13^{\circ}$ and $m_{z}=0.1\
\text{eV}$. The horizontal lines are the two flat bands with Chern number
$\mathcal{C}=\pm 1$. (b) Our main result: the occupation number of the many-
body ground state $n(\mathbf{k})$ as a function of the quantum geometry,
$\mathcal{G}(\mathbf{k})\equiv\text{log}[\text{tr}[g(\mathbf{k})]]=\text{log}[|\Omega(\mathbf{k})|]$,
for a FCI state at filling $\nu=1/3$ of the $\mathcal{C}=-1$ band and an
emergent Fermi liquid at $\nu=2/3$. This shows that ideal Chern bands strongly
breaks both the reciprocal-space translation symmetry and the particle-hole
symmetry that are present in Landau levels.

According to the criteria above, twisted bilayer graphene in the chiral limit
(cTBG) Tarnopolsky  _et al._ (2019) stands out as an extremely promising
platform to realize the LL physics in moiré systems. The valence and
conduction bands of cTBG are perfectly flat at magic twist angles Tarnopolsky
_et al._ (2019) and carry unit Chern numbers. Furthermore, their quantum
geometry are “ideal” Ledwith  _et al._ (2020); Wang  _et al._ (2021) in the
sense that the FS metric varies exactly in sync with the Berry curvature at
each ${\bf k}$ point of the BZ via $g_{ij}({\bf k})=\frac{1}{2}|\Omega({\bf
k})|\delta_{ij}$, although $g({\bf k})$ and $\Omega({\bf k})$ still fluctuate
with ${\bf k}$ Ledwith  _et al._ (2020). The cTBG bands are hence almost
identical to the $n=0$ lowest LL (LLL) except that their quantum geometry is
not uniform. These elegant properties lead to LLL-like single-body wave
functions, and exact zero modes for pseudopotential-like short-range
interactions in cTBG flat bands, just like the Laughlin states in the LLL Wang
_et al._ (2021). The ideal quantum geometry also emerges for flat bands with
higher Chern number in the chiral limit of multilayered graphene Wang and Liu
(2022); Ledwith  _et al._ (2022). While the aforementioned effective continuum
descriptions are most relevant for moiré systems, the Kapit-Mueller model
Kapit and Mueller (2010) and arbitrary Chern number generalizations thereof
Behrmann  _et al._ (2016) provide analogous ideal band geometry in tight-
binding models and exact parent Hamiltonians for bosonic states. In general,
ideal Chen bands have been described as LLLs in curved space Estienne  _et
al._ (2023).

Given the glaring similarity between the ideal flat bands in cTBG [cf. Fig.
1(a)] and the LLL, one may naturally expect that their only difference,
namely, the fluctuating quantum geometry of the former, would not cause
significantly distinct physics in cTBG flat bands from that in the LLL.
However, in this work we present strong evidence against this expectation. We
find that the particle-hole symmetry in a single cTBG flat band is still
strikingly broken by the non-uniform, albeit ideal quantum geometry [see Fig.
1(b)]. Remarkably, even a pseudopotential-like local interaction can induce an
unconventional Fermi liquid state at band filling $\nu=2/3$ with no LL
counterpart Läuchli  _et al._ (2013). The ground state at $\nu=1/3$ remains an
exact zero-energy Laughlin-like state — which, however, also exhibits a non-
uniform particle density caused by the fluctuating FS metric tensor [Fig.
1(b)]. Furthermore, the Coulomb interaction favors a charge density wave
instead of the composite Fermi liquid at $\nu=1/4$, which again distinguishes
the cTBG ideal bands from the LLL.

_Setup._ — The valley and spin polarized Hamiltonian of cTBG takes the form of
Tarnopolsky  _et al._ (2019)

$\displaystyle
H=\begin{pmatrix}-v_{\text{F}}\mathbf{\sigma}_{\theta/2}\cdot\nabla&wT(\mathbf{r})\\\
wT^{\dagger}(\mathbf{r})&-iv_{\text{F}}\mathbf{\sigma}_{-\theta/2}\cdot\nabla\end{pmatrix},$
(1)

where
$\mathbf{\sigma}_{\theta/2}=e^{-i\theta\sigma_{z}/4}(\sigma_{x},\sigma_{y})e^{i\theta\sigma_{z}/4}$
and $v_{\text{F}}$ is the Fermi velocity of monolayer graphene.
$T(\mathbf{r})=\sum_{n=1}^{3}T_{n}e^{-i\mathbf{q}_{n}\cdot\mathbf{r}}$
describes the coupling between the two layers, where
$T_{n+1}=\kappa\sigma_{0}+\cos(n\phi)\sigma_{x}+\sin(n\phi)\sigma_{y}$ with
$\phi=2\pi/3$, $\mathbf{q}_{1}=k_{\theta}(0,-1)$, and
$\mathbf{q}_{2,3}=k_{\theta}(\pm\sqrt{3}/2,1/2)$. We fix $w=0.11\ \text{eV}$.
The size of moiré Brillouin zone (MBZ) is controlled by
$k_{\theta}=2k_{\text{D}}\sin(\theta/2)$, where $k_{\text{D}}=4\pi/(3a_{0})$
with $a_{0}$ the graphene lattice constant. The lattice constant of the moiré
pattern is $a_{M}=a_{0}/(2\sin(\theta/2))$.

We focus on the chiral limit by setting $\kappa=0$ and work at the first magic
angle Tarnopolsky  _et al._ (2019) throughout the paper. In this case, the
valence and conduction bands are perfectly flat with zero energy and possess
ideal quantum geometry. Introducing an extra sublattice potential
$m_{z}\tau_{0}\sigma_{z}$, where $\tau_{0}$ acting on the layer index, can
lift the two-fold degeneracy, leading to two isolated flat bands with Chern
number $-1$ and $+1$, respectively [see Fig. 1(a)]. The precise value of
$m_{z}$ does not affect our results because it only shifts the band energy by
a constant without changing the band eigenvector.

Without losing generality, we consider the interacting physics in the lower
flat band. We project the electron-electron interaction to this single valence
band and reach

$\displaystyle
H^{\text{proj}}=\sum_{\mathbf{k}}V_{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}c^{\dagger}_{\mathbf{k}_{1}}c^{\dagger}_{\mathbf{k}_{2}}c_{\mathbf{k}_{3}}c_{\mathbf{k}_{4}},$
(2)

with $c^{\dagger}_{\mathbf{k}}$ ($c_{\mathbf{k}}$) being the fermionic
creation (annihilation) operator with momentum $\mathbf{k}$ in the valence
band. The normal ordering convention here is consistent with obtaining exact
zero-modes at $\nu=1/3$ Ledwith  _et al._ (2020). The band eigenvectors have
been encoded into the matrix element
$V_{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}$ (see the
Supplemental Material Sup  for more details). The single-electron dispersion
is dropped because it only contributes a constant. We use exact
diagonalization to extract the low-lying energy spectrum and the associated
eigenstates for finite systems including $N$ electrons in $N_{s}$ moiré unit
cells at band filling $\nu=N/N_{s}$, which can provide direct evidence to
distinguish FCIs from competing phases, such as the Fermi liquid and the
charge density wave, etc.

_Broken symmetry in ideal Chern bands._ — Our main result centers around the
striking breaking of reciprocal-space translation symmetry and particle-hole
symmetry in the ideal flat band of cTBG, even when we employ a
pseudopotential-like short-range repulsive interaction whose Fourier transform
is $V(\mathbf{q})=V_{0}-V_{1}a_{M}^{2}|\mathbf{q}|^{2}$ Haldane (1983);
Ledwith  _et al._ (2020); Sup . While this interaction gives rise to zero-
energy Laughlin-like model FCIs at band filling $\nu=1/3$, the particle
occupation number $n(\mathbf{k})$ is not a constant at $1/3$ as for the
conventional Laughlin state in the LLL. By contrast, $n(\mathbf{k})$ exhibits
a clear linear dependence on the quantum geometry characterized by
$\mathcal{G}(\mathbf{k})\equiv\log[\text{tr}[g(\mathbf{k})]]=\log[|\Omega(\mathbf{k})|]$
[Fig. 1(b)]. This variation of $n(\mathbf{k})$ is a signature of the breaking
of translation symmetry in the reciprocal space, which further suggests a
significant center-of-mass dependence of the model Laughlin-like wavefunction
within the ideal flat of cTBG Ledwith  _et al._ (2020); Wang  _et al._ (2021).
Notably, as shown in Fig. 2 the quantum geometry in terms of $\mathcal{G}$
[Fig. 2(a)], has a similar structure in the moiré BZ as the energy of a single
hole [Fig. 2(b)], thus providing a physical rationale for the inhomogeneous
(reciprocal space) occupation density $n(\mathbf{k})$ Abouelkomsan  _et al._
(2023b).

Figure 2: Quantum geometry and single hole dispersion. (a) The quantum
geometry $\mathcal{G}(\mathbf{k})$ and (b) the single hole energy
$-E_{\text{h}}(\mathbf{k})$ as a function of momentum
$\mathbf{k}=(q_{x},q_{y})$.

Moving to banding filling $\nu=2/3$, we find that the occupation number
$n(\mathbf{k})$ completely deviates from the value $2/3$ of the $\nu=2/3$ FQH
state in the LLL. Instead, as displayed in Fig. 1(b), a sharp jump from
$n(\mathbf{k})\approx 1$ to $n(\mathbf{k})\approx 0$ signifies the emergence
of a Fermi liquid, with a stable Fermi surface around $\mathcal{G}\approx
7.6$. The very distinct behavior of $n(\mathbf{k})$ at $\nu=2/3$ from that at
$\nu=1/3$ indicates the breaking of particle-hole symmetry in the ideal cTBG
bands. Such particle-hole asymmetry in Chern bands has been identified in
various non-ideal models: first in a checkerboard lattice Läuchli  _et al._
(2013) and, recently, in moiré systems Abouelkomsan  _et al._ (2020); Reddy
and Fu (2023) where it is related to the fluctuating quantum geometry
Abouelkomsan  _et al._ (2023b). That these effects persist even for electrons
with pseudopotential-like interaction in the ideal cTBG bands further
emphasizes the fundamental difference between Chern bands and the LLL, and
disapproves of the overoptimistic expectation that the LLL physics can be
perfectly reproduced in the cTBG band given the ideal quantum geometry and the
Laughlin-like wave functions in the latter.

_Evidence for the FCIs and the emergent Fermi liquid._ — We now provide more
numerical evidence on the FCIs at $\nu=1/3$ and the Fermi liquid at $\nu=2/3$
in the cTBG ideal band. The interaction is still chosen as the
pseudopotential-like local repulsion
$V(\mathbf{q})=V_{0}-V_{1}a_{M}^{2}|\mathbf{q}|^{2}$.

The most straightforward signature of the $\nu=1/3$ Laughlin-like model FCIs
is the three exactly zero-energy ground states with momenta predicted by the
Haldane statistics Regnault and Bernevig (2011); Bernevig and Regnault (2012).
This can be clearly seen in the low-lying energy spectrum for any finite
system [Fig. 3(a)]. The ground-state topological order can be further
diagnosed by the particle-cut entanglement spectrum Li and Haldane (2008);
Sterdyniak  _et al._ (2011). Here, we divide the whole system into $N_{A}$ and
$N_{B}=N-N_{A}$ electrons. By tracing out the subsystem $B$, we effectively
create holes in the system, so the reduced density matrix of the subsystem
$A$,
$\rho_{A}=\text{tr}_{B}[\frac{1}{3}\sum_{i=1}^{3}|\Psi_{i}\rangle\langle\Psi_{i}|]$,
contains the information of the quasihole excitations above the ground states
$|\Psi_{i}\rangle$. In Fig. 3(b), we display the particle-cut entanglement
spectrum, which is the spectrum of $-\log\rho_{A}$, for the three zero modes.
There is a narrow band at the bottom of the entanglement spectrum, in which
the number of levels matches the quasihole counting of the $\nu=1/3$ Laughlin
state. More entanglement levels appear at very high entanglement energies
above $\xi\approx 50$. However, as these levels correspond to exponentially
small eigenvalues of $\rho_{A}$ beyond the machine precision limit, we
attribute them to numerical noise. Therefore, the entanglement spectrum of the
zero modes at $\nu=1/3$ in fact has an infinite entanglement gap, which is a
remarkable feature of model wavefunctions Li and Haldane (2008); Behrmann  _et
al._ (2016). Both the energy spectrum and the ground-state entanglement
spectrum strongly indicate the existence of Laughlin-like model FCIs at
$\nu=1/3$ and rule out the possibility of a trivial charge density wave.

Figure 3: FCIs and Fermi Liquids. (a) The many-body energy spectrum at band
filling $\nu=1/3$ on the $4\times 6$ (blue $\cdot$) and $5\times 6$ clusters
(black +), respectively. We calculate five lowest energies in each momentum
sector. (b) The associated particle-cut entanglement spectrum of the three
zero modes on the $5\times 6$ cluster for $N_{A}=5$. The number of levels in
the narrow band at the bottom is $23256$, which matches the quasihole counting
of the $\nu=1/3$ Laughlin state. The very high-energy grey levels are
numerical noises. (c) The many-body energy spectrum at band filling $\nu=2/3$.
(d) The ground-state occupation number of electrons at $\nu=1/3$ and $\nu=2/3$
as a function of single-hole energy. (e)-(f) The occupation number of
electrons for the lowest $9$ excited states at $\nu=2/3$ on the $5\times 6$
cluster as a function of the quantum geometry and the single-hole energy,
respectively. Here, the black hexagon and the bold line represent the averaged
occupation number over the lowest $20$ energy states.

At band filling $\nu=2/3$, the absence of topological degeneracy in the energy
spectrum, as illustrated in Fig. 3(c), already rules out the presence of FCIs.
Given the dispersionless nature of the ideal cTBG band, the Fermi liquid
feature in Fig. 1(b) must originate solely from the electron-electron
correlation, whose role can be exposed more clearly in the hole picture. It
has been pointed out in early works that the electron-electron interaction
leads to an extra single-hole dispersion
$E_{\text{h}}(\mathbf{k})=\sum_{\mathbf{k}^{\prime}\in\text{MBZ}}\big{(}V_{\mathbf{k}^{\prime}\mathbf{k}\mathbf{k}^{\prime}\mathbf{k}}+V_{\mathbf{k}\mathbf{k}^{\prime}\mathbf{k}\mathbf{k}^{\prime}}-V_{\mathbf{k}\mathbf{k}^{\prime}\mathbf{k}^{\prime}\mathbf{k}}-V_{\mathbf{k}^{\prime}\mathbf{k}\mathbf{k}\mathbf{k}^{\prime}}\big{)}$
after the particle-hole transformation Läuchli  _et al._ (2013); Abouelkomsan
_et al._ (2020). This hole dispersion in general dominates over the hole-hole
interaction at relatively large fillings and gives rise to the correlation-
induced Fermi liquid. We confirm the correlated nature of the Fermi liquid in
cTBG ideal bands by drawing the ground-state occupation number of electrons as
a function of $E_{\text{h}}(\mathbf{k})$. As shown in Fig. 3(d),
$n(\mathbf{k})$ at $\nu=2/3$ has an overall trend to drop from $1$ to $0$
following the hole dispersion $E_{\text{h}}(\mathbf{k})$. The zigzags in
$n(\mathbf{k})$ for some system sizes result from the hole-hole interaction
which distorts the Fermi surface of electrons. In contrast, $n(\mathbf{k})$
for the $\nu=1/3$ FCI only weakly varies with the hole dispersion. The Fermi
liquid behavior at $\nu=2/3$ even extends beyond the ground state, where
excited states possess features of particle-hole excitations near the emergent
Fermi surface [see Figs. 3(e) and (f)].

We emphasize that the symmetry breaking in cTBG ideal bands with
pseudopotential-like interactions is a phenomenon observed across various
fillings. For the filling $\nu\in[1/3,1/2)$, such as $\nu=2/5$ and $3/7$, and
their complements, our results show a consistent behavior, where an FCI
emerges at band filling $\nu$ but disappears at band filling $1-\nu$ (see the
SM for more details Sup ).

_Charge density wave at $\nu=1/4$._ — In previous sections, the striking
differences between the ideal cTBG band and the LLL have been revealed at
fillings where FQH states are expected in the LLL. Now we consider other
fillings where there are no incompressible FQH states in the LLL. As a
concrete example, we consider $\nu=1/4$, where the Coulomb interaction
$V_{\text{C}}(\mathbf{q})\sim 1/|\mathbf{q}|$ in the LLL were found to
stabilize the Fermi sea of composite fermions Hossain  _et al._ (2019); Wang
(2019) (except in the wide quantum well which is beyond the scope of our
work). We suppose the electrons in the cTBG interact via the bare Coulomb
potential also. Surprisingly, at $\nu=1/4$ of the ideal cTBG band, we observe
a charge density wave (CDW) rather than the Fermi sea of composite fermions.

We verify the CDW phase from two aspects. For finite samples whose single-
electron momenta allowed by the periodic boundary conditions include the three
inequivalent ${\bf M}$ points of the MBZ, we find four approximately
degenerate ground states in the low-energy spectrum separated by the momenta
${\bf M}_{i=1,2,3}$ [Fig. 4(a)]. This is a strong signal of the charge
distribution with order momenta ${\bf M}_{i}$. The charge order can be
diagnosed more reliably by the structure factor

$\displaystyle
S(\mathbf{q})=\frac{1}{N_{s}}(\langle\rho(\mathbf{q})\rho(-\mathbf{q})\rangle-N^{2}\delta_{{\bf
q},{\bf 0}}),$ (3)

where $\rho_{\bf q}$ is the density operator projected to the ideal cTBG band.
Remarkably, we find pronounced peaks only at the three ${\bf M}$ points [Fig.
4(b)], confirming the ground states as ${\bf M}$-CDWs. The unit cell of the
charge distribution in these CDWs is quadrupled moiré unit cell. Similar CDWs
were reported in realistic TBG beyond the chiral limit Wilhelm  _et al._
(2021). Our results demonstrate that they can survive even in the ideal cTBG
band. The fluctuation of the Berry curvature and the quantum metric tensor of
the ideal cTBG band, which distinguishes it from the LLL, could be a major
mechanism for the formation of these CDWs that are absent in the LLL.

Figure 4: CDWs from the Coulomb interaction in the ideal cTBG band. (a) The
many-body energy spectrum at band filling $\nu=1/4$ for the bare Coulomb
interaction. We calculate three lowest levels in each momentum sector. (b) The
ground-state structure factor in the MBZ. Here, we use tilted samples to
include the three inequivalent ${\bf M}$ points. The $+$ (black in (a)) and
$\times$ (blue in (a)) symbols correspond to clusters with $32$ and $36$ moiré
unit cells, respectively Sup .

_Discussion._ — By employing both a pseudopotential-like short-range
interaction and a Coulomb interaction, we have discovered that, even the
topological flat band in cTBG exhibits an ideal quantum geometry, yet
significant dissimilarities persist when compared to the LLL. In the former
scenario, the breaking of particle-hole symmetry at band filling $\nu=2/3$
leads to a correlation-induced Fermi liquid, while at $\nu=1/3$ a FCI is
supported although also this state has a non-uniform occupation density
adapting to the underlying geometry. The breakdown of correspondence between
the flat band in cTBG and the LLL also provides hints to other possible
competing orders in this ideal setting. In particular, we unveiled a charge
density wave at band filling $\nu=1/4$ in the latter case, characterized by
three clear peaks in the structure factor. This contrasts with the anticipated
gapless composite Fermi liquid of the LLL physics, highlighting compelling
deviations in cTBG. In conclusion, while the ideal cTBG band does indeed
elegantly preserve the presence of zero modes of certain short range
interactions, the phase diagram is qualitatively altered compared to that of
the LLL.

It should be noted that the quantum geometric quantities are of limited
fundamental importance in lattice models due to their dependence on the
embedding of the orbitals into real space. A more orbital-independent approach
should go back to continuum models obtained from first principle calculations.
In moiré systems, the underlying (e.g., graphene) orbitals are highly
localized compared to the moiré length scale. Thus the system has a natural
continuum description that for practical purposes alleviates the fundamental
ambiguity of lattice embeddings.

We also note that normal ordering prescription matters. We have used a
convention used in the literature, normal ordering with respect to an empty
vaccum, which leads to Laughlin-like zero modes of electrons at $\nu=1/3$
Ledwith  _et al._ (2020). If instead normal ordered with respect to charge
neutrality, the interaction Hamiltonian differs by extra single-particle terms
which can be thought as Hartree Fock corrections from the remote bands
Bernevig  _et al._ (2021) but the particle-hole asymmetry within the single
band still exists hence our results still hold with the difference that FCIs
are no longer exact zero modes of the pseudopotential. We stress that the
particle-hole asymmetry considered here is the one emerging within a given
(ideal) band which is different from the question of symmetry of the model as
a whole.

The glaring differences between ideal Chern bands and Landau levels identified
in this work prompts a natural question that, what criteria can accurately
determine to what extent an ideal moiré Chern band aligns with the physics of
the Landau levels more generally. Understanding this question becomes crucial
in realization of non-Abelian topological phases and their associated quantum
computing applications in moiré systems, which has been experimentally
demonstrated that FCIs can be realized without the need of high magnetic field
and exhibit robustness at relatively high temperatures.

While FCIs have been experimentally observed in twisted bilayer graphene Xie
_et al._ (2021) at fillings $\nu=4-1/3$ and $\nu=4-2/3$ of the conduction
band, we would like to stress that our emphasis here is different as we are
studying the spin and valley polarized limit of chiral TBG. It is believed
that the state at $\nu=4-1/3$ is spin and valley polarized Parker  _et al._
(2021) while the state at $\nu=4-2/3$ (in the chiral limit) is spin-singlet
Repellin and Senthil (2020). Our results are concerned with particle-hole
asymmetry in a band with a single flavor. In the presence of multi-component
degrees of freedom, the single-hole dispersion takes a more complicated form
than the form in sup  therefore the conclusions can differ. It is also
possible that external magnetic fields could diminish the existing particle-
hole asymmetry at zero magnetic field approaching a more LLL-like limit. We
leave this to future study.

On the other hand, robust spin-valley polarization has been observed in moiré
TMD systems Cai  _et al._ (2023); Zeng  _et al._ (2023); Park  _et al._
(2023); Xu  _et al._ (2023) but FCIs were found only at hole doping
$\nu_{h}=2/3$ of the valence band while absent at the particle-hole dual
filling $\nu_{h}=1/3$ reflecting strong particle-hole asymmetry similar to our
study. While chiral TBG is significantly different from moiré TMDs, we believe
that our results provide general and relevant insights to this problem. We
have shown that even in an idealized model, translation and particle-hole
symmetries is strongly violated even in absence of band mixing, hence such
symmetries should not be expected in any model derived starting from an
underlying lattice structure.

###### Acknowledgements.

Acknowledgements.— We acknowledge discussions with Jie Wang. H. Liu and E. J.
Bergholtz were supported by the Swedish Research Council (VR, grant
2018-00313), the Wallenberg Academy Fellows program of the Knut and Alice
Wallenberg Foundation (2018.0460) and the Göran Gustafsson Foundation for
Research in Natural Sciences and Medicine. Z. Liu was supported by the
National Natural Science Foundation of China (Grant No. 12350403). K.Yang is
supported by the ANR-DFG project (TWISTGRAPH). A. Abouelkomsan is supported by
the Knut and Alice Wallenberg Foundation (2022.0348).

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##  I Supplemental Materials to ‘Broken Symmetry in Ideal Chern Bands’

In this supplemental material, we provide the details of the interaction
Hamiltonian, the evidence of particle-hole symmetry breaking at other
fillings, the spectral flow with respect to magnetic flux insertion, the
choice of samples, and the relation between the hole energy and the quantum
geometry.

##  II The interaction Hamiltonian and the single-hole dispersion

The two-body interaction projected to a single band with specific valley and
spin takes the form of

$\displaystyle
H^{\text{proj}}=\sum_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}}V_{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}c^{\dagger}_{\mathbf{k}_{1}}c^{\dagger}_{\mathbf{k}_{2}}c_{\mathbf{k}_{3}}c_{\mathbf{k}_{4}},$
(4)

with the interaction matrix element

$\displaystyle V_{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\mathbf{\mathbf{q}}}V(\mathbf{q})\sum_{\tau,\tau^{\prime}}\sum_{\\{m,n\\}=-d}^{d}\delta_{\mathbf{k}_{1}+\mathbf{k}_{2}+(m_{1}+m_{2})\mathbf{G}_{1}+(n_{1}+n_{2})\mathbf{G}_{2},\mathbf{k}_{3}+\mathbf{k}_{4}+(m_{3}+m_{4})\mathbf{G}_{1}+(n_{3}+n_{4})\mathbf{G}_{2}}\cdot$
(5)
$\displaystyle\delta_{\mathbf{k}_{1}-\mathbf{k}_{4}+(m_{1}-m_{4})\mathbf{G}_{1}+(n_{1}-n_{4})\mathbf{G}_{2},\mathbf{q}}\mu^{*}_{m_{1},n_{1},\tau}(\mathbf{k}_{1})\mu^{*}_{m_{2},n_{2},\tau^{\prime}}(\mathbf{k}_{2})\mu_{m_{3},n_{3},\tau^{\prime}}(\mathbf{k}_{3})\mu_{m_{4},n_{4},\tau}(\mathbf{k}_{4}).$

Here, $V(\mathbf{q})$ indicates either the pseudopotential-like short-range
interaction or the bare Coulomb interaction, $\mathbf{G}_{1}$ and
$\mathbf{G}_{2}$ denote the two moiré reciprocal operators, respectively, and
$\tau$ ($\tau^{\prime}$) represents the orbital index. $\mu_{m,n}(\mathbf{k})$
is the eigenvector of the valence and topological flat band at
$m\mathbf{G}_{1}+n\mathbf{G}_{2}$ sector for momentum
$\mathbf{k}\in\text{MBZ}$. In numerical simulations, we choose a cutoff $d=8$,
which is enough to make both the single-particle bandstructure and the many-
body energy spectrum converge.

The breaking of particle-hole symmetry is originated from the inhomogeneous
single-hole dispersion $E_{\text{h}}(\mathbf{k})$ (in other words, not a
constant). Here, we derive the concrete from of $E_{\text{h}}(\mathbf{k})$.
The original interaction Hamiltonian in momentum space reads,

By doing the particle-hole transformation $c^{\dagger}_{\mathbf{k}}\rightarrow
c_{\mathbf{k}}$, we have

$\displaystyle
H^{\text{proj}}\rightarrow\sum_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}}V_{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}c^{\dagger}_{\mathbf{k}_{1}}c^{\dagger}_{\mathbf{k}_{2}}c_{\mathbf{k}_{3}}c_{\mathbf{k}_{4}}+\sum_{\mathbf{k}}E_{\text{h}}(\mathbf{k})c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}},$
(6)

with
$E_{\text{h}}(\mathbf{k})=\sum_{\mathbf{k}^{\prime}}(V_{\mathbf{k}^{\prime}\mathbf{k}\mathbf{k}^{\prime}\mathbf{k}}+V_{\mathbf{k}\mathbf{k}^{\prime}\mathbf{k}\mathbf{k}^{\prime}}-V_{\mathbf{k}\mathbf{k}^{\prime}\mathbf{k}^{\prime}\mathbf{k}}-V_{\mathbf{k}^{\prime}\mathbf{k}\mathbf{k}\mathbf{k}^{\prime}})$.

##  III Spectral flow for $\nu=1/3$ and $3/7$

Here, we examine the topological nature of the system at band filling
$\nu=1/3$ and $3/7$ by introducing an external magnetic flux. As shown in Fig.
5, for both fillings, the quasi-degenerate FCI ground states remain isolated
from the ground states, and evolve into each other with the magnetic flux,
which indicates their corresponding fractional transmission.

##  IV Results of other fillings

Figure 5: The low-lying energy spectrum as a function of magnetic flux $\Phi$
at band filling $v=1/3$ (a) and $3/7$ (b) with a system size $4\times 6$ and
$4\times 7$ clusters, respectively. Here, we use a pseudopotential-like short-
range interaction.  Figure 6: (a1-c1) are the low-lying energy spectrum for
band filling $\nu=2/5$, $3/7$, and $4/9$, at system size $5\times 6$, $4\times
7$, and $3\times 9$ clusters, respectively. (a2-c2) are the energy spectrum
for their complementary band fillings $\nu=3/5$, $3/7$, and $5/9$,
respectively. (d1) and (d2) are the occupation number of ground states at
different fillings as a function of quantum geometry and the single-hole
dispersion, respectively. Here, we use a pseudopotential-like short-range
interaction and only selective five lowest energies in ED simulations.

In this section, we provide a detailed study for other fillings. Like the FQHE
in the LLL, varying the partial filling of the flat band in cTBG $\nu$, from
$1/3$ to $1/2$ (here, we choose $\nu=2/5$, $3/7$, and $4/9$), indeed leads to
a energy gap closing between FCI ground states and excitation states, shown in
Fig. 6(a1-c1).

At $\nu=2/5$, five quasi-degenerate ground states appear at momentum sectors
satisfying the Haldane statistics and are separated from excitation states
with a clear energy gap.

For $\nu=3/7$, although the energy gap is less conspicuous, seven quasi-
degenerate ground states persist at the right momentum sectors, maintaining
isolation from other states. This is further validated by introducing a
magnetic flux, where the seven ground states never mix with other higher
energy states, making the energy gap visible. Alternatively, employing a
tilted sample configuration reveals the clear separation of the seven ground
states from excitation states (see Fig. 7)(a).

Approaching $\nu=4/9$, near the phase transition, both clear FCI ground states
and energy gap are absent, as evident from both the rectangular sample, shown
in Fig. 6(c1) and the tilted sample shown in Fig. 7(c)

Figure 7: (a-b) the low-lying energy spectrum of the tilted sample for banding
filling $\nu=3/7$ and $4/9$, at system size $4\times 7$ and $3\times 9$
clusters, respectively. (c-d) are the energy spectrum of their particle-hole
partner $\nu=4/7$ and $5/9$, respectively. Here, we use a pseudopotential-like
short-range interaction and only selective three lowest energies in ED
simulations.

We then shift the focus to band filling within $(1/2,2/3)$, specifically,
$\nu=3/5$, $4/7$, and $5/9$, the complimentary partners. Surprisingly, the
energy spectrum of each of these fillings, both from the rectangular sample
and the tilted shows an absence of FCI ground states or an associated energy
gap. This is again opposite to the conventional understanding that the flat
band in cTBG has an identical physics with the LLL, where the presence of a
FCI phase is expected.

We further apply an analysis of the occupation number, like what we have
discussed in the main text, there is an emergent Fermi liquid formed for
$\nu\in(1/2,2/3)$, while for $\nu\in(1/3,1/2)$, the system shows a FCI
behavior, even for $\nu=4/9$, which is quite close the phase transition at
$\nu=1/2$. This thus aligns with our main conclusion.

##  V Details on the numerical simulations

In this section, we provide the geometry we use for the rectangular and tilted
sample. Note that we run all ED simulations in momentum space. Here, the
rectangular sample means we use the moiré reciprocal vectors $\mathbf{G}_{1}$
and $\mathbf{G}_{2}$. For the tilted sample, the spanning vector is a linear
combination of these two moiré reciprocal vectors Repellin  _et al._ (2014).

All calculations on rectangular samples (Fig. 1, Fig. 2 in the main text, and
Fig. 6 in this supplemental materials), are with system size $(N_{x},N_{y})$
and spanning vectors $\mathbf{T}_{1}=(N_{x},0)$ and
$\mathbf{T}_{2}=(0,N_{y})$.

In the calculation of the charge density wave (Fig. 3 in this supplemental
materials), we use two system sizes. For the case of $32$ sites, we use the
spanning vectors $\mathbf{T}_{1}=(2,-4)$ and $\mathbf{T}_{2}=(6,4)$. For the
case of $36$ sites, we use the spanning vectors $\mathbf{T}_{1}=(6,0)$ and
$\mathbf{T}_{2}=(0,6)$.

In the calculation of the energy spectrum of tilted samples (Fig. 7 in this
supplemental materials), for band filling $\nu=3/7$ and $4/7$, we use a system
size $28$ clusters, and the spanning vectors $\mathbf{T}_{1}=(4,-2)$ and
$\mathbf{T}_{2}=(2,6)$. At band filling $\nu=4/9$ and $5/9$, we use a system
size $27$ clusters, and the spanning vectors $\mathbf{T}_{1}=(6,3)$ and
$\mathbf{T}_{2}=(3,6)$.

We note that, the energy scale difference between the rectangular sample and
the tilted sample, shown in Fig. 6 and Fig. 7, is up to a constant in
numerical simulations.

##  VI Hole dispersion and band geometry

The Bloch wavefunction of an ideal Chern band takes the form
$\psi_{\mathbf{k}}(\mathbf{r})=N_{\mathbf{k}}B(\mathbf{r})\phi^{\rm
LLL}_{\mathbf{k}}(\mathbf{r})$ where $N_{\mathbf{k}}$ is a $k$-dependent
normalization, and $B(\mathbf{r})$ is a $k$-independent quasi-periodic
function with $\phi^{\rm LLL}_{\mathbf{k}}(\mathbf{r})$ the LLL wavefunction.
Therefore the matrix elements
$V_{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}$ are related to
the LLL form factors Wang  _et al._ (2021) as

$V_{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}=(\prod_{i=1}^{4}N_{\mathbf{k}_{i}})F_{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}$
(7)

where $F_{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}$ is a
function of the LLL form factors and the interactions. The normalization
factor $N_{\mathbf{k}}$ is the major difference from a LL and controls the
band geometry fluctuations. The Berry curvature $\Omega_{\mathbf{k}}$ and the
FS metric is given by $|\Omega_{\mathbf{k}}|=|-1+\Delta_{\mathbf{k}}\log
N_{\mathbf{k}}|={\rm tr}\>g_{\mathbf{k}}$, where $\Delta_{\mathbf{k}}$ is the
Laplacian operator. The hole energy is related to the normalization factor as

$\displaystyle
E_{h}(\mathbf{k})=N_{\mathbf{k}}^{2}\sum_{\mathbf{k^{\prime}}}N_{\mathbf{k}^{\prime}}^{2}\big{(}F_{\mathbf{k}\mathbf{k}^{\prime}\mathbf{k}\mathbf{k}^{\prime}}+F_{\mathbf{k}^{\prime}\mathbf{k}\mathbf{k}^{\prime}\mathbf{k}}-F_{\mathbf{k}\mathbf{k}^{\prime}\mathbf{k}^{\prime}\mathbf{k}}-F_{\mathbf{k}^{\prime}\mathbf{k}\mathbf{k}\mathbf{k}^{\prime}}\big{)}$
(8)

So in ideal bands, there is an even closer connection between hole dispersion
and the non-uniformity of the quantum geometry, directly from the
normalization $N_{\mathbf{k}}$ which controls the fluctuations of Berry
curvature and FS metric.