Origin of Model Fractional Chern Insulators in All Topological Ideal Flatbands Explicit Color-entangled Wavefunction and Exact Density Algebra Jie Wang1 2 3Semyon Klevtsov4and Zhao Liu5y

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Origin of Model Fractional Chern Insulators in All Topological Ideal Flatbands:

Explicit Color-entangled Wavefunction and Exact Density Algebra

Jie Wang,1, 2, 3, ∗Semyon Klevtsov,4and Zhao Liu5, †

1Center for Mathematical Sciences and Applications,

Harvard University, Cambridge, MA 02138, USA

2Department of Physics, Harvard University, Cambridge, MA 02138, USA

3Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA

4IRMA, Universit´e de Strasbourg, UMR 7501, 7 rue Ren´e Descartes, 67084 Strasbourg, France

5Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China

It is commonly believed that nonuniform Berry curvature destroys the Girvin-MacDonald-

Platzman algebra and as a consequence destabilizes fractional Chern insulators. In this work we

disprove this common lore by presenting a theory for all topological ideal ﬂatbands with nonzero

Chern number C. The smooth single-particle Bloch wavefunction is proved to admit an exact color-

entangled form as a superposition of Clowest Landau level type wavefunctions distinguished by

boundary conditions. Including repulsive interactions, Abelian and non-Abelian model fractional

Chern insulators of Halperin type are stabilized as exact zero-energy ground states no matter how

nonuniform Berry curvature is, as long as the quantum geometry is ideal and the repulsion is

short-ranged. The key reason behind is the existence of an emergent Hilbert space in which Berry

curvature can be exactly ﬂattened by adjusting wavefunction’s normalization. In such space, the

ﬂatband-projected density operator obeys a closed Girvin-MacDonald-Platzman type algebra, mak-

ing exact mapping to C−layered Landau levels possible. In the end we discuss applications of the

theory to moir´e ﬂatband systems with a particular focus on the fractionalized phase and spontaneous

symmetry breaking phase recently observed in graphene based twisted materials.

I. INTRODUCTION

Flatbands are ideal platforms for realizing exotic quan-

tum phases of matter. Of particular interests are topo-

logical ﬂatbands where fractionalized topological phases

are possible [1–7]. During recent years, there has been

numerous progress in creating and engineering ﬂatband

systems such as by stacking and twisting [8–11]. Under-

standing the stability of fractionalized phases with re-

spect to tuning parameters such as energy dispersion,

wavefunction geometry, interaction range is crucial for

material engineering, experimental realization and the-

oretical characterization of these exotic phases. Along

this path, exact statements are particularly important.

In this work, we present an exact condition for the stabil-

ity of fractional Chern insulators (FCIs), i.e. the lattice

version of fractional quantum Hall (FQH) states [3–7].

Our results also provide a framework to systematically

explore the interplay between wavefunction’s geometry

and interactions, and are useful in guiding material engi-

neering toward realizing exotic quantum phases of mat-

ter.

Landau levels (LLs) are the simplest topological Chern

ﬂatbands, with unit Chern number C= 1 and exactly

zero bandwidth. They are well known for their extreme

uniformness that is crucial for the stability of FQH states:

LLs are covariant under any smooth area preserving de-

formations [12,13] which is encoded in the LL projected

∗jiewang.phy@gmail.com

†zhaol@zju.edu.cn

density algebra initially noticed by Girvin, MacDonald,

and Platzman (GMP) for lowest Landau level (LLL)

states [14,15]:

[ˆρq1,ˆρq2] = eq∗

1q2l2

B−eq1q∗

2l2

Bˆρq1+q2,(1)

where ˆρqis the LLL projected density operator, qis

the complex coordinate of momentum which is q=

(qx+iqy)/√2 in an isotropic LL, and lBis the mag-

netic length. The density operator deforms the shape of

LLs while preserving their LL index [12–15]. The GMP

algebra is important for FQH physics in many aspects

including the conformal ﬁeld theory mapping [16,17],

constructing exact pseudopotential projectors that pro-

tect the stability of FQH states [18,19] and others. The

GMP algebra was initially derived based on the holomor-

phicity of LLL wavefunctions [20] which arises from the

non-commutativity of guiding centers and in fact applies

to any LL [21].

Topological ﬂatbands realized in lattice systems are

more complicated than LLs. They can carry arbitrary

integer Chern number, and typically do not exhibit uni-

form Berry curvature. Due to this complication, nei-

ther the holomorphicity of wavefunction nor the exact

projected density algebra exist, thereby Eq. (1) becomes

approximate. As a consequence FCIs are no longer ex-

pected to be exact and their stability is supposed to be

reduced [22]. There have been important attempts to-

wards restoring the GMP algebra in ﬂatband systems.

For instance, Ref. [23] highlighted the importance of the

Fubini-Study metric gab

kwhich is the intrinsic wavefunc-

tion distance measure. Given constant Berry curvature

Ωk= Ω and the so-called trace condition Trgk= Ωk,

arXiv:2210.13487v1 [cond-mat.mes-hall] 24 Oct 2022

Ref. [23] derived a GMP algebra for all topological ﬂat-

bands of C 6= 0. However, the assumption of constant

Berry curvature oversimpliﬁed the ﬂatband problem.

In this work, we relax the above assumption by fo-

cusing on ideal ﬂatbands satisfying the following condi-

tions [24]:

gab

k=1

2ωabΩk,Ωk>0 for ∀k.(2)

The ωab is a constant uni-modular matrix. The Berry

curvature is assumed to be positive deﬁnite but not nec-

essarily uniform. Recently, it is known that Eq. (2) is

necessarily [25] and suﬃciently [26–28] equivalent to the

momentum-space holomorphicity, which is a key geo-

metric property of the momentum-space (= boundary-

condition space). For all C= 1 ideal ﬂatbands such

as those realized in the chiral model of twisted bilayer

graphene [29,30] and others [31–33], it has been shown

that momentum-space holomorphicity highly constrains

the wavefunction to admit a universal form descend-

ing from the LLL wavefunction [24]. Such simplicity

enables exact construction of many-body model wave-

functions [30] as well as exact projective pseudopotential

Hamiltonians [24], hence the ideal condition plays an im-

portance role in experimental realization and identiﬁca-

tion of FCIs [34,35]. Recently the ideal condition Eq. (2)

has been generalized to a larger family called “vortexabil-

ity” [36] by allowing nonlinear real-space embedding [37].

While much progress are made for C= 1, less is known

for generic cases of C>1. Recently, solvable models

based on twisted multilayer graphene sheets were pro-

posed which realize ideal ﬂatbands of arbitrary Chern

number [38,39]. Remarkably, exact Halperin type FCIs

were found by numerical diagonalization [38]. Moti-

vated by the C= 1 case, it was conjectured that the

momentum-space holomorphicity is also the fundamen-

tal reason for their emergence [38], however, the nature

of the Bloch wavefunction and why FCIs are stable are

still far from thorough understanding.

In this work, we prove that the ideal quantum geo-

metric condition, without assuming the ﬂatness of Ωk

or gkthemselves, is suﬃcient to guarantee exact GMP

algebra and model FCI states occurring as exact zero-

energy ground states of proper short-ranged interactions

in all C ≥ 1 ideal ﬂatbands. We achieve these results

by pointing out the importance of an emergent Hilbert

space in which the wavefunctions’ normalization factors

are adjusted to ﬂatten Berry curvature in an exact man-

ner. Importantly, this leads to a simple algebra for

the projected coordinates identical to the guiding cen-

ter algebra in LLs, and enables an exact derivation for

the single-particle wavefunction in ideal ﬂatbands. The

general form Bloch wavefunction is found to be a non-

linear superposition of LLL wavefunctions of Cdistinct

boundary conditions, generalizing the previously pro-

posed color-entangled wavefunctions with uniform Berry

curvature [40–42] to the case of ﬂuctuating Berry cur-

vature. The density algebra is proved to be closed, ex-

tending the GMP algebra initially derived for C= 1 LLs

to generic C ≥ 1 ideal ﬂatbands. The closeness of the

density algebra directly leads to the exact FCIs stabi-

lized by generic M−body repulsive interactions which

are Halperin type when C>1 and typically non-Abelian

when M > 2. Thus the ideal condition Eq. (2) provides a

general and exact statement for stabilizing FCIs regard-

less of the nonuniformness of Berry curvature.

The paper is structured as follows. In Sec. II, we pose

the problem by showing unusual numerical observations.

In Sec. III, we review key analytical results for ideal

ﬂatbands, emphasizing their real and momentum-space

boundary conditions and deﬁne the emergent Hilbert

space. We show in Sec. IV a well deﬁned guiding center in

the ideal ﬂatband problems which leads to the derivation

of single-particle wavefunctions in Sec. V. In Sec. VI, the

density algebra is derived and the origin of model FCIs is

explained. We discuss application of the theory to FCIs

and symmetry breaking phases [34,43] in moir´e ﬂatband

systems in Sec. VII.

II. MOTIVATION FROM UNUSUAL

NUMERICAL OBSERVATIONS

The ideal ﬂatbands are not abstract concepts, rather

they can be realized in concrete models including Dirac

fermion based models [29,33,38,39] and Kapit Mueller

models [31,32,44,45]. In particular, in the chiral twisted

multilayer graphene models [38,39] ideal ﬂatbands of ar-

bitrary C 6= 0 are realized exactly. In this section, we use

this model as a concrete platform to numerically study

the interacting spectra, which point out a couple of im-

portant and unusual aspects of the projected density op-

erator that motivates the theory to be discussed in the

following sections.

For self-consistency, we ﬁrst brieﬂy review the chiral

twisted multilayer graphene models [38,39], although the

model details are not crucial for the current discussion.

Details about the chiral twisted graphene model and be-

yond can be found in the literature [29,30,46–51]. The

chiral twisted multilayer model consists of two sheets of

graphene multilayers each of which has n−layers and is

A/B stacked without twist [52–54]. The top sheets are

twisted relative to the bottom sheets as a whole by a twist

angle θ. See Fig. 1(a) for illustration of the setup. At

magic twist angles, the model exhibits two exactly dis-

persionless degenerate bands showing in Fig. 1(b), which

have Chern number C=±n. Such degeneracy can be

easily lifted by sublattice contrasting potential, and we

focus on the C=nband. The ﬂatband wavefunction is

derived in Refs. [38,39], and its components on the out-

most layers are found to be responsible for the high Chern

number and zero-mode FCIs. We hence project the wave-

function to the outmost layers to get an eﬀective single-

component wavefunction, denoted as φBloch

k(r). It can

be numerically veriﬁed that φBloch

k(r) is smooth in both

kand r. The Berry curvature associated to φBloch

k(r),

deﬁned through the standard deﬁnition,

Ωk=ab∂a

kAb

k,Aa

k≡ −ihuBloch

k|∂a

kuBloch

ki,(3)

is plotted in Fig. 1(c), which integrates to the Chern

number C=n. Here uBloch

k(r) = e−ik·rφBloch

k(r) is the

cell-periodic part of the Bloch wavefunction. Its ideal

geometry condition can be either analytically proved or

numerically veriﬁed to be satisﬁed [38,39]. See Fig. 1(d)

for the plot of the trace condition for the n= 2 model

where errors are ﬁnite-size artifact that will vanish in the

inﬁnite-size limit.

We then consider interacting phenomena when the ﬂat-

band is partially ﬁlled. We focus on ﬁnite systems on the

torus geometry at band ﬁlling ν=N/(N1N2), where N

is the number of particles and N1,2is the number of unit

cells in each primitive direction of the lattice. The two-

body density-density interaction takes the form of

H=X

v(q) : ˆρqˆρ−q:,(4)

where v(q) is the Fourier transform of the interaction po-

tential, ˆρqis the band-projected density operator, and : :

is the normal ordering. Because of the translation invari-

ance by lattice vectors, the energy spectrum of Hcan

be resolved by the center of mass momentum (K1, K2).

We are particularly interested in the repulsive interac-

tion vm(r1−r2) = Pqvm(q)eiq·(r1−r2), whose Fourier

transform vm(q) has series expansion of orders no higher

than m:

vm(q) = c0+c1|q|2+c2|q|4+... +cm|q|2m.(5)

The above series expansion is equivalent as interaction

range expansion, as the real space form of the |q|2nterm

is given by the 2nth derivative of the contact interaction

δ(r1−r2). In particular, the v0and v1are the shortest

interactions for bosons and fermions, respectively [55].

We ﬁrst consider two particles interacting with vm,

where mis even for bosons and odd for fermions. While

it is unrealistic for bosons to occupy the ﬂatband near

charge neutrality of twisted double bilayer graphene, we

can still use that setup to examine the band property. Re-

markably, for any choice of mand Chern number C, the

two-particle spectrum always shows a dispersive “band”

with a ﬁxed number of nonzero eigenvalues at each mo-

mentum k, no matter what the lattice size is. The dimen-

sion of this ﬁnite-energy “band” is precisely (m+ 1)Cfor

fermions and (m+1)C+1 for bosons [56]. The “band” dis-

persion reﬂects the fact that Berry curvature is nonuni-

form. Apart from these ﬁnite-energy levels, there are also

massive zero modes whose dimension increases with the

system size. The typical example with m= 1,C= 2 is

demonstrated in Fig. 1(e) for two fermions. Such two-

particle spectra are diﬀerent from those in either LLs or

generic ﬂatbands. For generic ﬂatbands without ideal

geometry, the zero modes would not appear and the di-

mension of ﬁnite-energy levels is hence lattice-size de-

pendent [5,57]. In a LL, as the vminteraction only picks

FIG. 1. (a) Illustration of the chiral twisted multilayered

graphene model, where solid and empty dots represent the A

and B sublattice, respectively, and the interlayer tunneling is

assumed to be chiral and represented by the dashed line. The

φ1,...,n and φ0

1,...,n label the layer component. (b) The band

structure where a tiny sublattice bias potential is used to split

the two-fold degeneracy of the ﬂatbands. The resulting two

bands are entirely sublattice polarized. (c) and (d) The Berry

curvature and the trace condition of uBloch

kdeﬁned in the

main text. The Berry curvature is positive deﬁnite. (e) The

energy spectrum of the two-body vm=1 interaction for two

fermions on a N1×N2= 5 ×5 lattice. At each momentum

k, there are precisely (m+ 1)C= 4 non-zero eigenvalues.

(f) The energy spectrum of the three-body onsite interaction

for N= 10 bosons on a N1×N2= 5 ×3 lattice. The exact

six-fold degenerate zero-energy ground states are non-Abelian

model FCIs which are analogous to the FQH non-Abelian spin

singlet state (NASS). In (a)-(f) we use n= 2, namely the

twisted double bilayer graphene model, as an example.

out two-particle coherent states of relative angular mo-

mentum no greater than m, both a ﬁnite-energy band

whose dimension is independent of the system size and

massive zero modes appear, however, the “band” is non-

dispersive due to the continuous translation symmetry.

Therefore, Fig. 1(e) clearly shows there is an emergent

projector property associated to the density operator of

ideal ﬂatbands which however, unlike in LLs, is inﬂu-

enced by the nonuniform Berry curvature.

We can further generalize Eq. (4) to multibody interac-

tions. In Fig. 1(f), the spectrum of the three-body onsite

interaction is plotted for bosons at ν= 2/3 in C= 2

band. In this case, a six-fold exact ground-state degen-

eracy at zero energy is obtained. These zero modes are

expected to be lattice analogs of the non-Abelian spin

singlet states, which are generalizations of the Abelian

Halperin topological order in bilayer LLL [58]. This re-

sult indicates that suitable short-range M-body repul-

sions can stabilize model FCIs of both Abelian and non-

Abelian types in high Chern number ideal ﬂatbands.

Moreover it points out that two-body interaction is not

special: there should be a general property that the pro-

jected density operator obey in order to give rise to FCIs

stabilized by generic M-body interactions [59].

To summarize, numerical studies on interacting two-

particle spectra and three-body interactions point out

the importance of the ﬂatband projected density oper-

ator which exhibits emergent projector properties and

internal Cdegrees of freedom playing the role of layers,

although the underlying Berry curvature can be highly

nonuniform. It motivates the following two questions:

•What internal degrees of freedom in the scalar-

valued wavefunction uk(r) play the role of layers?

•Why the nonuniform Berry curvature does not

destabilize the Abelian and non-Abelian FCIs?

Thorough this work, we will show the answers to these

two questions by deriving the general form of C 6= 0 ideal

ﬂatband wavefunctions, illustrating their physical mean-

ings and studying their density algebra which we ﬁnd

generalizes the GMP algebra Eq. (1) in a nontrivial way.

III. ANALYTICAL PROPERTIES OF IDEAL

FLATBANDS

As we have shown, numerical experiments in ideal ﬂat-

bands point out the unusual eﬀects of ideal quantum

geometry on interacting problems. To address the two

questions at the end of the last section, we will focus on

analytical derivations in the remaining parts of this arti-

cle. We start with reviewing the ideal quantum geomet-

ric condition and emphasizing its relation to momentum-

space K¨ahler condition in this section. The momentum-

space K¨ahler condition gives novel properties of the Bloch

wavefunction in many aspects, such as the unique bound-

ary condition and the relation between Berry curvature

and normalization factor. These general results were de-

rived for ideal ﬂatbands in Ref. [24]. In the next section

Sec. IV, we will discuss new results about their impor-

tant implication for the well deﬁned guiding center which

establishes connection to LL physics in the presence of

nonuniform Berry curvature.

A. Ideal quantum geometry and K¨ahler geometry

We ﬁrst of all set up conventions in more detail. Our

exact results applies to arbitrary system size N1,2. The

system is spanned by primitive lattice vectors a1,2con-

taining an area |a1×a2|= 2πS and throughout this work

we set S= 1 as the unit area scale. The reciprocal/ mo-

mentum space is spanned by reciprocal lattice vectors bi

such that ai·bj= 2πδij is satisﬁed.

We consider a single component complex valued Bloch

wavefunction deﬁned on this torus. Its cell periodic part

uBloch

k(r) is lattice translational invariant following the

Bloch theorem uBloch

k(r) = uBloch

k(r+a) for arbitrary

lattice vectors a=m1a1+m2a2and m1,2∈Z. The

uBloch

k(r) is assumed to be smooth in both kand r. Mo-

mentum quantization restricts k= (n1/N1+φ1/2π)b1+

(n2/N2+φ2/2π)b2where n1,2∈Zand φ1,2are ﬁcti-

tious boundary condition ﬂuxes. Our result applies to

arbitrary φ1,2too so without loss of generality we set

φ1,2= 0. Because the ﬂux space (torus formed by con-

tinuous variable φ1,2) is equivalent to the momentum

space in the presence of translational symmetry, we use

these two spaces interchangeably. The discussions in this

work can be generalized to ﬂux space for systems without

translation symmetry where the notion of momentum is

not deﬁned.

The uBloch

k(r) satisﬁes the ideal quantum geometric

condition while not necessarily carrying ﬂat Berry curva-

ture. We assume uBloch

khas positive Chern number; neg-

ative Chern number is a simple generalization. As being

a positive integer, the Chern number can be factorized

into two positive integers C=C1C2and the factorization,

as we will discuss, is a gauge choice.

Systematic studies of ﬂatbands with momentum-space

holomorphicity start from Refs. [25,60] where the au-

thors found that the ideal condition is automatically

satisﬁed if the cell-periodic part of Bloch wavefunction

uBloch

k(r) is given by Eq. (6). Very recently, Refs. [26–28]

pointed out the inverse is also true: the ideal condition

generally implies uBloch

kcan be written as Eq. (6) under a

gauge choice up to a positive-deﬁnite normalization fac-

tor Nk:

uBloch

k(r) = Nkuholo

k(r),¯

∂kuholo

k= 0,(6)

where the holomorphic coordinate k(unbold letter) is de-

ﬁned as k≡ωakaand the associated anti-holomorphic

derivative is deﬁned as ¯

∂k≡ωa∂a

k. Technically, the no-

tion of holomorphicity ωais deﬁned from factorizing the

constant uni-modular matrix ωab and the anti-symmetric

tensor ab as follows [21]:

ωab =ωaωb∗+ωa∗ωb,

iab =ωa∗ωb−ωaωb∗.(7)

The vectors satisfy ωa∗ωa= 1, ωaωa= 0 where and

throughout the paper Einstein summation is implicitly

assumed, and their indices are raised or lowered by ωab.

Note that in the isotropic case (ωx, ωy) = (1, i)/√2 the

complex coordinate reduces to the conventional form k=

(kx+iky)/√2. We will work in the general case without

assuming isotropy.

It is worth to mention that ωahas an equivalent canon-

ical deﬁnition as being the constant null vector of the

quantum geometric tensor [24]:

Qab

kωb= 0 for ∀k,(8)

where the quantum geometric tensor is deﬁned by us-

ing the covariant derivative of wavefunctions Da

k=∂a

k−

iAa

k, whose real symmetric and imaginary anti-symmetric

parts are the Fubini-Study metric and the Berry curva-

ture, respectively:

Qab

k≡ hDa

kuk|Db

kuki=gab

k+iab

2Ωk.(9)

B. Implications from K¨ahler geometry

We have reviewed the relation between ideal quantum

geometric condition and momentum-space holomorphic-

ity [25–28]. In this section we discuss two implications

from K¨ahler geometry: the uniqueness of the boundary

condition and the relation between normalization factor

and Berry curvature. They are important in deriving

ideal ﬂatband wavefunction and their density algebra.

1. Uniqueness of the boundary condition

Following Ref. [24], we can generally denote the

momentum-space boundary boundary condition of uholo

as:

uholo

k+bi(r) = uholo

k(r)·exp (−ibi·r+iφk,bi),(10)

where the complex phase φk,b is not only required to be

holomorphic in kbut also is constrained to satisfy a co-

cycle relation [24],

−2πC=φb1,b2−φ0,b2+φ0,b1−φb2,b1.(11)

Such a constraint Eq. (11) is required from the simpliﬁed

Chern number formula derived for ideal ﬂatbands [24]:

C=1

2πi Idk ∂kln uholo

k(r),(12)

which means that the winding of the wavefunction in the

momentum space around the Brillouin zone at any ﬁxed

rmust reﬂect the topology of the wavefunction. The co-

cycle relation Eq. (11) implies the boundary condition

factor φk,b must be a linear function of holomorphic co-

ordinate k[24]. The phase φk,b is the so-called factor

of automorphy which appears in classifying holomorphic

line bundles [61]. Without loss of generality, we choose

the symmetric gauge,

φk,bj=Cb∗

j(−ik −ibj/2) + Cjπ, j = 1,2,(13)

where C1,2are the two positive integers dividing the

Chern number C=C1C2, and the ambiguity in their

choice will prove to be unimportant.

FIG. 2. The normalized ﬂatband wavefunction deﬁnes a

Hilbert space Hwhere the Berry curvature is nonuniform.

For ideal ﬂatbands, the wavefunction’s normalization can be

tuned such that the resulting wavefunction uk(r) deﬁne a

modiﬁed Hilbert space ˜

Hin which Berry curvature can be

exactly ﬂattened. As a consequence uk∈˜

Hperceives iden-

tical k-dependence as LLL wavefunctions. Such property

greatly simpliﬁes the problem, enabling a thorough charac-

terization of ﬂatband wavefunction from a momentum-space

formulation. As an illustration, we used kto represent the

2D momentum-space. The normalization and Berry curvature

are represented by the dashed and solid lines, respectively.

2. K¨ahler potential

The anti-holomorphic component of the Berry connec-

tion ¯

Ak≡ωaAa

kcan be computed from the standard

deﬁnition:

Ak≡ −ihuBloch

k|¯

∂kuBloch

ki,

=−iZd2rNkuholo∗

k(r)¯

∂kNkuholo

k(r),

=−iN−1

k¯

∂kNkZd2rN2

kuholo∗

k(r)uholo

k(r),

=−i¯

∂klog Nk.(14)

Then from Ωk=−i∂k¯

Ak−¯

∂kAkwhere Akis the

complex-conjugate of ¯

Ak, we obtain the expression for

the Berry curvature [24,62,63]:

Ωk=−2∂k¯

∂kln Nk.(15)

Eq. (15) has an important implication: for ideal ﬂat-

bands, the Berry curvature can be eﬀectively ﬂattened by

adjusted by the normalization of the wavefunction. More

precisely, by factorizing Nkinto,

Nk=e−C

4|k|2˜

Nk,(16)

the Berry curvature is accordingly split into Ωk=C+˜

Ωk,

i.e. the uniform and ﬂuctuating part. The ﬂuctuating

part ˜

Ωk=−2∂k¯

∂kln ˜

Nkaverages to zero when being

integrated over the Brillouin zone which does not con-

tributes to the quantization.

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OriginofModelFractionalChernInsulatorsinAllTopologicalIdealFlatbands:ExplicitColor-entangledWavefunctionandExactDensityAlgebraJieWang,1,2,3,SemyonKlevtsov,4andZhaoLiu5,y1CenterforMathematicalSciencesandApplications,HarvardUniversity,Cambridge,MA02138,USA2DepartmentofPhysics,HarvardUniversity,Cambri...

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Origin of Model Fractional Chern Insulators in All Topological Ideal Flatbands Explicit Color-entangled Wavefunction and Exact Density Algebra Jie Wang1 2 3Semyon Klevtsov4and Zhao Liu5y

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