Exact Many-Body Ground States from Decomposition of Ideal Higher Chern Bands Applications to Chirally Twisted Graphene Multilayers Junkai Dong1Patrick J. Ledwith1Eslam Khalaf2Jong Yeon Lee3and Ashvin Vishwanath1

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Exact Many-Body Ground States from Decomposition of Ideal Higher Chern Bands:

Applications to Chirally Twisted Graphene Multilayers

Junkai Dong,1, ∗Patrick J. Ledwith,1Eslam Khalaf,2Jong Yeon Lee,3and Ashvin Vishwanath1

1Department of Physics, Harvard University, Cambridge, MA 02138, USA

2Department of Physics, The University of Texas at Austin, TX 78712, USA

3Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

(Dated: October 26, 2022)

Motivated by the higher Chern bands of twisted graphene multilayers, we consider ﬂat bands

with arbitrary Chern number Cwith ideal quantum geometry. While C > 1 bands diﬀer from

Landau levels, we show that these bands host exact fractional Chern insulator (FCI) ground states

for short range interactions. We show how to decompose ideal higher Chern bands into separate

ideal bands with Chern number 1 that are intertwined through translation and rotation symmetry.

The decomposed bands admit an SU(C) action that combines real space and momentum space

translations. Remarkably, they also allow for analytic construction of exact many-body ground

states, such as generalized quantum Hall ferromagnets and FCIs, including ﬂavor-singlet Halperin

states and Laughlin ferromagnets in the limit of short-range interactions. In this limit, the SU(C)

action is promoted to a symmetry on the ground state subspace. While ﬂavor singlet states are

translation symmetric, the ﬂavor ferromagnets correspond to translation broken states and admit

charged skyrmion excitations corresponding to a spatially varying density wave pattern. We conﬁrm

our analytic predictions with numerical simulations of ideal bands of twisted chiral multilayers

of graphene, and discuss consequences for experimentally accessible systems such as monolayer

graphene twisted relative to a Bernal bilayer.

I. INTRODUCTION

The discovery of correlated states in twisted bilayer

graphene (TBG) has inspired interest in the study of

topological ﬂat bands [1–6]. The interplay of band

topology and strong interactions in TBG makes it an

ideal candidate to realize strongly correlated topological

physics. A simpliﬁed “chiral” model[7], where intrasub-

lattice moir´e tunneling is ignored, yields exactly ﬂat topo-

logical bands at the magic angle, thereby capturing the

most essential features of the system. The chiral model

hosts analytically-attainable wavefunctions [7] that are

equivalent to those of the LLL in an inhomogeneous mag-

netic ﬁeld [8]. Its mathematical properties and mysteries

have turned the chiral model into a sub-ﬁeld of study of

its own right [7–19].

At the same time, it was shown [8] that the chi-

ral model satisﬁes a quantum-geometric[20–29] iden-

tity known as the “trace condition”. It was recently

shown that the trace condition enables intra-band vor-

tex attachment[30, 31], which has strong implications

for interacting physics. For example, it directly enables

the construction of FQHE-like trial states which are ex-

act ground states of the system under short-range re-

pulsive interactions. For TBG, this led to the analytic

prediction[8] of fractional Chern insulators (FCI)[32–41],

states that host the fractional quantum Hall eﬀect at

fractional ﬁlling of a Chern band ﬁrst observed[42] in

Hofstadter bands [43–47]. FCIs in TBG were simultane-

ously predicted in numerical works [48–50] and were re-

cently experimentally observed [51] in a small magnetic

∗junkaidong@g.harvard.edu

ﬁeld that eﬀectively restored the ideal chiral limit [52].

We will refer to trace condition satisfying bands as ideal

bands; they have also been referred to as “vortexable”

[31]. The trace condition is also related to momentum

space holomorphicity [21, 23, 24, 29, 31, 53], a powerful

analytic tool.

Multilayer twisted graphene systems consisting of

nchirally stacked[54–56] graphene layers, i.e. AB,

ABC, etc., stacked with a single twist on top of

mchirally stacked layers are a natural extension of

TBG. Experimentally, twisted double bilayer graphene,

(n, m) = (2,2) [57–64], and twisted mono-bilayer

graphene (n, m) = (1,2) [65–72], have been fabricated

and a nearly quantized anomalous Hall eﬀect has repeat-

edly been observed with C > 1 [57, 63, 65, 66]. This

has generated extensive theoretical interest in study-

ing topological physics in these systems at diﬀerent ﬁll-

ings [30, 73–85]. Furthermore, symmetry broken Chern

insulators have been found experimentally at half ﬁll-

ing [70]. A variety of these states was recently identi-

ﬁed numerically together with signs of an approximate

SU(2) symmetry [86], although its origin has not been

understood. FCIs have also been studied in these sys-

tems [29, 79], and other systems with higher Chern

bands [40, 44, 46, 87–92], though analytic progress has

largely been limited to toy models. These advances mo-

tivate the need for a systematic understanding of corre-

lation phenomena in higher Chern bands, including both

symmetry-broken and fractional Chern insulators.

Chiral models have been constructed for the twisted

chiral multilayers in Refs. [30, 80] through neglecting cer-

tain interlayer tunneling terms. Chiral twisted multilayer

graphene yields ﬂat and ideal higher Chern bands that

are analytically tractable when tuned to the same magic

arXiv:2210.13477v1 [cond-mat.mes-hall] 24 Oct 2022

angle as chiral TBG. By combining ideality, which allows

the construction of trial wavefunctions for topologically

ordered states, with higher Chern number in an exper-

imentally feasible system, these models provide a novel

and unexplored platform to study and realize interacting

topological phases. Ref. [80] numerically found model

FCI states with particle entanglement suggestive of par-

ticular “color-singlet” [40] Halperin states, though a gen-

eral analytic understanding of the range of possible states

in such a system remains lacking.

One approach to understanding correlated states in a

C > 1 band is to employ the hybrid Wannier functions

to decompose the band into CChern 1 bands by enlarg-

ing the unit cell [93]. However, this approach does not

respect the ideality condition: an ideal Chern C > 1

band generally decomposes into a set of C= 1 non-ideal

bands such that analytic techniques cannot be applied

to the decomposed basis. Furthermore, this decomposi-

tion assumes broken translation symmetry in a speciﬁc

direction from the outset, making it diﬃcult to under-

stand translation-unbroken states like the FCIs observed

in Ref. [80]. This naturally leads to the question: is it

possible to decompose an ideal C > 1 band into ideal

C= 1 bands and in a way that captures both transla-

tionally symmetric and translation-breaking states? A

partial answer to this question was provided in Ref. [30]

which showed that it is impossible to decompose a generic

ideal C > 1 bands in terms of orthogonal and ideal C= 1

bands.

Here, we show that by lifting the orthogonality con-

straint, it is possible to decompose a generic ideal C > 1

into Cideal Chern 1 bands. From this decomposition,

we reveal a hidden non-unitary SU(C) action among the

decomposed bands combining real space and momentum

space translations. We note that although this SU(C)

action is generally non-unitary and is not a symmetry of

the Hamiltonian, it is often a symmetry of the ground

state manifold for short-range repulsive interactions, en-

abling us to make sharp predictions regarding the many-

body ground state at partial ﬁlling. Furthermore, the

SU(C) structure is reminiscent of the structure of multi-

component Landau levels [40], allowing us to interpret

these ground states in terms of more familiar correlated

states in multi-component quantum Hall systems. It

should be emphasized that the states we obtain are phys-

ically distinct and have non-trivial translation symmetry

breaking patterns visible through real-space charge den-

sity, compared to multi-component Landau levels which

have uniform charge density.

By employing this decomposition, we analytically con-

struct ground states at a variety of ﬁllings of ideal Chern

C > 1 bands and make concrete predictions on their re-

alization. First, we identify a ground state manifold of

charge density waves (CDWs) at ﬁlling 1/C with emer-

gent SU(C) symmetry. Such states can be interpreted

as generalized quantum Hall ferromagnets in the decom-

posed basis. An immediate consequence of this iden-

tiﬁcation is the existence of charged skyrmion textures

which correspond to a characteristic winding pattern of

the CDW order parameter (cf. Fig. 1). Such a wind-

ing pattern can be readily observed with local charge

probes such as STM, which has already been employed

in these systems [71, 72, 94–98]. Second, we characterize

the structure of translation symmetric fractional Chern

insulator states at ﬁllings 1/(2Cs + 1), for each positive

integer s, by establishing a direct analogy with ﬂavor-

singlet Halperin states. Finally, we discuss a manifold of

translation-breaking Laughlin states that appear at low

ﬁllings 1/C(2s+ 1) where fractionalization and topolog-

ical order coexists with CDW order. All our results are

veriﬁed by numerical exact diagonalization (ED) on the

chiral model for twisted mono-bilayer graphene. These

results enable us to make experimental predictions for

graphene multilayers.

The paper is organized as follows. We begin with

an overview of the central theoretical results as well as

some consequences for experiment in Sec. II. Next, we re-

view the physical model for chiral graphene multilayers

in Sec. III and quantum geometry techniques to under-

stand the single particle physics in Sec. IV. We present

the central technical result, the decomposition of an ideal

higher Chern band, in Sec. V for C= 2. We study quan-

tum ferromagnets, which manifest as topological CDWs,

and their associated skyrmion description at half ﬁlling

for a spinless C= 2 band in Sec. VI and a spinful C= 2

band in Sec. VII. We discuss the possible fractional Chern

insulators, both translation symmetric and translation

broken, in Sec. VIII. We generalize the physical implica-

tions to ideal bands with C > 1 in Sec. IX. We conclude

with some future questions in Sec. X.

II. SUMMARY OF RESULTS

We begin by brieﬂy summarizing our results and dis-

cussing their main implications. By now, a full under-

standing of ideal C= 1 bands has been achieved: they

have been related to the LLL of a Dirac particle in a mag-

netic ﬁeld [8, 29], which enables the construction of exact

many body Laughlin-like ground states for short range

interactions [31]. Our main technical achievement in this

work is the generalization of such understanding for any

ideal higher Chern band. In particular, we provide a gen-

eral explicit construction to decompose any ideal higher

Chern band into ideal but non-orthogonal Chern 1 bands

with a non-unitary SU(C) action. Compared to the hy-

brid Wannier decomposition [93], our procedure has the

distinct advantages of preserving the ideal band geome-

try and allowing access to translation broken states in all

possible directions in the same basis. The preservation

of ideal band geometry means we leverage the analytic

knowledge of ideal C= 1 bands to analytically study in-

teracting phases – in particular, FCIs – in C > 1 bands.

The decomposition provides analytic many-body

states that are the exact ground states at certain frac-

FIG. 1. Main results of our paper. (a) Plots of the density proﬁle of a CDW-skyrmion that arises out of a manifold of topological

charge density waves from the ideal C= 2 band in chiral twisted monolayer-bilayer graphene. Windings of the order parameter

corresponds to windings of the translation breaking pattern. We zoom into diﬀerent regions to highlight diﬀerent regions that

correspond to particular topological charge density waves that repeat in a 2 ×2 unit cell. (b) shows many-body spectrum from

ED at diﬀerent ﬁllings for the ideal C= 2 band in chiral twisted monolayer-bilayer graphene: boxed energy levels correspond to

the ground state manifold of model FCIs. The ﬁrst panel is the translation invariant (332) Halperin state at ν= 1/5 with the

interaction V2δ00 (ˆ

ri−ˆ

rj) and the expected ground state degeneracy of 5. The second panel is the translation broken Laughlin

state at ν= 1/6 under a screened Coulomb interaction with a ground state degeneracy of 21 that combines the Laughlin

degeneracy of 3 with the generalized-ferromagnetic degeneracy of Ne+ 1 = 7. Both of these collections of states originate from

the depicted analytic vortex attachment construction summarized in Sec. IV.

tional ﬁllings for short range interactions. For example,

for a spinless band with Chern number Cthese ground

states include topological CDWs at 1/C ﬁlling. At lower

ﬁllings, we construct Halperin-like translationally sym-

metric FCIs at ν= 1/(2Cs + 1) and Laughlin-like trans-

lation broken FCIs at ν= 1/C(2s+ 1) for any positive

integer s.

The topological CDWs are best understood as a man-

ifold of generalized quantum Hall ferromagnets (FM),

where translations act as pseudospin rotations on the fer-

romagnets. The pseudospin ferromagnets are generically

translation symmetric only in a C×Cunit cell. Crys-

talline symmetries, both translations and discrete rota-

tions, emerge as rotations in the order parameter mani-

fold. For C= 2, this manifold is a sphere and diﬀerent

topological CDW states may be understood as diﬀerent

pseudospin directions, pictorially represented in Fig. 2(a)

and summarized in Tab. 2(b). The ±X, ±Z, ±Yaxes

of the pseudospin sphere correspond to states that pre-

serve the translations along a1,2,3, where a1,2are prim-

itive lattice vectors and a3=−a1−a2. For rotational

symmetry, we focus on the three-fold C3zcase which

is relevant for graphene moir`e systems, and other rota-

tional symmetries may be understood similarly. We ﬁnd

that C3zacts as a 120◦rotation around a particular n0

axis of the pseudospin sphere. There are also rotations

C(i)

3z=TaiC3zT−aiaround lattice vectors aiand corre-

sponding invariant axes ni; these are distinct operations

because we have broken translations.

We construct exact model FCIs with the vortex at-

tachment procedure [30, 31] reviewed in Sec. IV. By at-

taching vortices to the fully ﬁlled Chern Cband, we ar-

rive at an explicit ﬁrst quantized wavefunction for the

translationally-symmetric Halperin states. If we instead

choose a C= 1 topological charge density wave as a

parent state we obtain the translation broken Laughlin

states. We numerically verify these predictions in ex-

act diagonalization for the C= 2 band of chiral twisted

monolayer-bilayer graphene, see Fig. 1(b).

Finally, based on our analysis in the ideal limit, we

make concrete predictions for the ground states and ex-

citations of twisted mono-bilayer graphene at several ﬁll-

ings as follows. Note, here we allow for valley and spin

degeneracy, hence the ﬁlling νranges from 0 ≤ν≤4.

We will focus on states doped on top of a ν= 3 spin and

valley polarized state such that there is a single empty

C= 2 band left over.

1. At ν= 3+1/2, we expect that the ground state will

be a continuous manifold of spin polarized topo-

logical CDWs, or generalized quantum Hall fer-

romagnets. The observed topological CDWs in

Ref. [70] are special points in a continuous manifold

of generalized quantum Hall ferromagnets. Our

main prediction is that there will be associated

charged skyrmion textures with a particular wind-

ing of translation breaking patterns illustrated in

Fig. 1(a). These textures can be directly probed

by STM.

2. At ν= 3+1/5, we expect that the ground state will

be a translationally invariant Halperin (332) state:

see ﬁrst panel of Fig. 1(b) for numerical spectrum.

FIG. 2. Sphere of ﬂavor ferromagnets (FMs) at half ﬁlling

for a C3zsymmetric ideal band with C= 2. Each point on

the sphere correspond to a topological CDW with unit cell

2a1×2a2. (a) We show the order parameter sphere of fer-

romagnets: red arrows correspond to C3zsymmetric CDWs

(second and third insets of Fig. 1(a)) and yellow dots corre-

spond to CDWs that preserve certain translation symmetries

Tai(ﬁrst inset of Fig. 1(a)). C3zrotations in real space cor-

respond to rotations around red arrows in order parameter

space. Table (b) explains the representation of special CDWs

as special ferromagnets on the sphere and crystalline symme-

tries as rotations. (c) We show the real space unit cell and

the lattice vectors.

In particular, the Hall conductivity (or the slope

of the gapped feature on the Landau fan diagram)

will be 2/5.

3. At ν= 3 + 1/6, we expect that the ground state

will be translation breaking ferromagnetic Laughlin

states: see second panel of Fig. 1(b) for numerical

spectrum. These states will in general be transla-

tion breaking striped phases with 1/3 Hall conduc-

tivity.

We end our summary of results with some cautionary

comments on application to experiments. First, realis-

tic systems will not be in the chiral limit, which means

that the higher Chern band will not have exactly ideal

quantum geometry, and may be dispersive. However, the

band geometry and ﬂatness could be improved by exter-

nal perturbations such as out of plane electric and mag-

netic ﬁelds as believed to occur in magic angle graphene

[52], bringing the system closer to the ideal limit. Sec-

ond, in most of our discussions, we have assumed a spin-

less band for simplicity; exotic magnetic orders at half

ﬁlling such as the tetrahedral antiferromagnetic (TAF)

order exist in spinful bands [86, 99–104]. Remarkably,

after adding spin we ﬁnd that the TAF states lie in the

same CP3manifold as the previously mentioned charge

density waves. Furthermore, a magnetic ﬁeld may be

used to favor the spin-polarized submanifold that hosts

charge-density skyrmions. A ﬁnal caveat is that typi-

cal twisted graphene systems contain twist angle disorder

which leads to large inhomogeneous heterostrain. Strain

greatly increases single particle dispersion and pushes the

band away from ideality. We leave a detailed quantita-

tive study of the realistic system that addresses the above

issues to future work.

III. CHIRAL TWISTED GRAPHENE

MULTILAYERS

In this section, we review the chiral twisted graphene

multilayers: they give rise to ideal higher Chern bands

and closely describe experimentally relevant systems. We

use them as our primary numerical example. Experimen-

tally, these graphene multilayers have been fabricated

both as twisted monolayer-bilayer (mono-bi) [65–68] and

twisted bilayer-bilayer (bi-bi) [57–61] graphene systems.

The following discussion will closely follow recent works

which introduced these models [30, 80].

We consider n+mgraphene layers. The ﬁrst nlayers

are untwisted and chirally stacked on each other: chiral

stacking means that the successive layers have Bernal

stacking AB or BA. These nchirally stacked layers are

then twisted by the magic angle θof chiral TBG, and put

on top of another mlayers of chirally stacked graphene.

The mono-bilayer system corresponds to n= 1, m = 2 and

the bilayer-bilayer system corresponds to n= 2, m = 2.

For concision, consider electrons of a speciﬁc valley and

spin ﬂavor. We make the approximation that the electron

can only tunnel between neighboring layers, and that the

interlayer tunneling is on-site (this neglects e.g. trigonal

warping terms [105]). The Hamiltonian can be written

H=hm,σ TM

T†

Mhn,σ0(1)

where hn,σ describes the nuntwisted chirally stacked

graphene layers with chirality σ=±depending on

whether the layers have AB (+1) or BA (−1) stack-

ing. TMonly tunnels between layers nand n+ 1 and

is identical to the moir´e tunneling of chiral TBG, where

intra-sublattice tunneling is switched oﬀ. Its explicit di-

mensionless form is given by

TM=α0U(r)

U∗(−r) 0 (2)

where U1(r) = P2

n=0 e2πin

3−iqn·r,qn=Cn

3(0,−1)T, and

C3is a 120◦rotation. The parameter α=w1/vkθis

a dimensionless tunneling strength, where w1≈110

meV and the angle-dependent kinetic energy scale is

vkθ=v|Ktop −Kbot|, where vis the Fermi velocity of

the Dirac cone and Ktop,bot are the graphene K-points

for the top and bottom stacks respectively. The chiral

FIG. 3. Band structure of twisted monolayer-bilayer

graphene. The higher Chern band is highlighted in red. We

use the chiral model (1) with α= 0.586 and β= 2; the two

Chern bands are degenerate and both of them have zero en-

ergy.

TBG magic angle corresponds to α≈0.586 [7].

The form of hn,σ is tri-diagonal:

hn,±=





−iσ · ∇ T±0. . .

T†

±−iσ · ∇ T±. . .

0T†

±−iσ · ∇ . . .

. . . . . . . . . ...







(3)

The diagonal terms describe the Dirac cones of each layer,

and the oﬀ diagonal terms describe the tunneling due to

two kinds of Bernal stacking: T±=βσx±iσy

Most importantly, the model hosts zero energy ﬂat

bands at particular values of αidentical to the magic an-

gles for chiral TBG. These zero energy bands are closely

related to the chiral symmetry of the model: {H, σz}= 0.

The chiral symmetry arises because the model is purely

oﬀ diagonal in the sublattice basis:

H=0D†

D0AB

,(4)

where the subscript AB denotes that the matrix is block

diagonalized in its sublattice. Thus, we may choose zero

energy eigenstates of Hto be eigenstates of σz. At the

magic angle α= 0.586, two zero energy bands labeled by

the sublattice index Aand Bemerge independent of the

value of β[30, 80]. These bands carry Chern numbers,

whose signs depend on the stacking chiralities (σ, σ0) and

whose values depend on (n,m).

For example, when both upper nlayers and lower m

layers are AB-stacked with σ=σ0= 1, the Chern num-

ber for the A-sublattice polarized ﬂatband (zero-energy)

is CA=n, while for the B-sublattice polarized ﬂatband

CB=−m. In particular, both mono-bilayer and bilayer-

bilayer systems give rise to Chern bands with |C|= 2

for this stacking orientation. For an analytic derivation

of the ﬂat band wavefunctions ψk(r) for any stacking

(σ, σ0), in terms of the n=m= 1 chiral TBG wave-

functions, see Ref. [30]. We numerically plot the band

structure of mono-bi with the chiral model in Fig. 3; the

higher Chern band is highlighted in red.

In the next section we will review ideal band geometry

and show that the zero modes (ﬂat bands) of magic-angle

chiral graphene multilayers in Eq. (4) attain the ideal ge-

ometry. Crucially, the zero mode operator Donly depend

on coordinates x, y and the antiholomorphic derivative

∂=1

2(∂x+i∂y): D(r, ∂x, ∂y) = D(r, ∂). This means

that

[D, z]=0,where z=x+iy. (5)

This property of Dwill be equivalent to the “vortex-

attachment” description of ideal quantum geometry.

IV. QUANTUM GEOMETRY OF IDEAL

CHERN BANDS

In this section we review our characterization of ideal

band geometry via the equivalent perspectives of the

“trace condition,” momentum-space holomorphicity, and

real-space “vortexability.” We will review the utility of

each of these perspectives. Finally, we will conclude by

developing certain technical tools for working with ideal

bands that we will use in the subsequent decomposition

section.

Consider a set of bands with total Chern number C≥

0 and periodic wavefunctions uka(r) = e−ik·rψka(r),

where ais a band index. The set of bands is ideal if

they satisfy the trace condition deﬁned as one of the fol-

lowing three equivalent conditions [31]:

(i) The inequality tr g(k)≥Ω(k) is saturated, where

gµν (k) = Re ηµν (k) is the Fubini Study metric and

Ω(k) = −µν Im ηµν is the Berry curvature. [20, 23,

24, 33]. Here, ηis the quantum metric deﬁned by

ηµν (k) = X

ah∂kνuka|Q(k)∂kµuka(6)

and Q(k) = I−Pka|ukaihuka|.

(ii) The periodic wavefunctions |ukai=e−ik·r|ψkai

may be chosen to be holomorphic functions of k=

kx+ikyin a suitable choice of gauge. Non-unitary

gauge transformations are generically required to

reach this gauge, and the wavefunctions |ukaiare

generically not orthonormal.

(iii) The Bloch wavefunctions ψsatisfy zψ =Pzψ where

z=x+iy and P=Pka|ψkaihψka|is the projector

onto the bands of interest, written here using an

orthonormal basis |ψkai.

See Ref. [24] for the equivalence of (i) and (ii) and de-

tailed description of the associated K¨ahler geometry on

the Brillouin Zone (BZ). The condition (iii) and its gen-

eralizations, together with its relationship to conditions

(i) and (ii), is discussed in Ref. [31]. Pictorially, these

relations are shown in Fig. 4. See also Ref. [106] which

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ExactMany-BodyGroundStatesfromDecompositionofIdealHigherChernBands:ApplicationstoChirallyTwistedGrapheneMultilayersJunkaiDong,1,PatrickJ.Ledwith,1EslamKhalaf,2JongYeonLee,3andAshvinVishwanath11DepartmentofPhysics,HarvardUniversity,Cambridge,MA02138,USA2DepartmentofPhysics,TheUniversityofTexasatAust...

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Exact Many-Body Ground States from Decomposition of Ideal Higher Chern Bands Applications to Chirally Twisted Graphene Multilayers Junkai Dong1Patrick J. Ledwith1Eslam Khalaf2Jong Yeon Lee3and Ashvin Vishwanath1

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