14 is the new 12 when topology is intertwined with Mottness Peizhi Mai1 Jinchao Zhao1 Benjamin E. Feldman234 and Philip W. Phillips1

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1/4 is the new 1/2 when topology is intertwined with

Mottness

Peizhi Mai1, Jinchao Zhao1, Benjamin E. Feldman2,3,4, and Philip W. Phillips1,*

1Department

of Physics and Institute of Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

2Geballe Laboratory of Advanced Materials, Stanford, CA 94305, USA

3Department of Physics, Stanford University, Stanford, CA 94305, USA

4Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA

ABSTRACT

In non-interacting systems, bands from non-trivial topology emerge strictly at half-ﬁlling and exhibit either the

quantum anomalous Hall or spin Hall effects. Here we show using determinantal quantum Monte Carlo and an

exactly solvable strongly interacting model that these topological states now shift to quarter ﬁlling. A topological

Mott insulator is the underlying cause. The peak in the spin susceptibility is consistent with a possible ferromagnetic

state at

T=0

. The onset of such magnetism would convert the quantum spin Hall to a quantum anomalous

Hall effect. While such a symmetry-broken phase typically is accompanied by a gap, we ﬁnd that the interaction

strength must exceed a critical value for this to occur. Hence, we predict that topology can obtain in a gapless

phase but only in the presence of interactions in dispersive bands. These results explain the recent quarter-ﬁlled

quantum anomalous Hall effects seen in moir´

e systems.

Introduction

Although topological insulators

1–16

represent a new class of bulk insulating materials with gapless

conducting edges, their physics is completely entailed by the band theory of non-interacting electrons.

The new twist is that should two atoms reside in each unit cell, the standard insulating gap that obtains at

half-ﬁlling, full lower band, does not tell the whole story when spin-orbit coupling

1–3

is present. As long as

time-reversal invariance is maintained, two spinful counter-propagating edge modes exist and exhibit a

quantized conductance proportional to

e2/h

, thereby giving rise to a quantum spin Hall (QSH) effect in two

dimensions. Within the Kane-Mele (KM)

1,2

and Bernevig-Hughes-Zhang (BHZ)

models, the QSH effect

obtains only at half-ﬁlling. In a general non-interacting system, this physics obtains at a ﬁlling equal to the

inverse number of atoms per unit cell,

1/q

. This physics is robust to perturbations that yield only smooth

deformations

of the Hamiltonian. Additionally, the quantum anomalous Hall (QAH) effect, that is, the

existence of a quantized Hall conductance with zero net magnetic ﬁeld, also requires half-ﬁlling of the

Haldane model

. As the QAH effect breaks time-reversal symmetry while the QSH effect does not, it is

difﬁcult for them to be realized in the same material.

However, recently, both effects

18,19

have been observed in the same material in direct contrast to

predictions of standard non-interacting models

1–3

. In the AB-moir

e-stacked transition metal dichalcogenide

(TMD) bilayer MoTe

/WSe

218,19

, the QSH insulator is observed at

ν=2

with the QAH effect residing at

ν=1

. To date, this constitutes the ﬁrst observation of the intertwining of these effects in the same material

and hence the question of the minimal model required to explain the conﬂation of both is open. In terms

of the 4-band KM/BHZ model,

ν=2

and

ν=1

correspond to half-ﬁlling and quarter-ﬁlling, respectively.

Numerous theories

20–31

have been put forth in this context, and the most recent experiment

shows

that both valleys contribute to the QAH effect and hence valley coherence rather than valley polarization

is the operative mechanism. The striking deviation from the standard theory raises the question: can

interactions drive either of these transitions away from half- to quarter-ﬁlling in the KM/BHZ models?

arXiv:2210.11486v5 [cond-mat.mes-hall] 5 Sep 2023

It is this question that we address here. We show quite generally that at a temperature above any

ordering tendency, strong interactions shift the QSH effect to quarter ﬁlling with a decrease of the spin

Chern number by a factor of two. However, the spin susceptibility exhibits a peak indicating a tendency

to ferromagnetism as the temperature is lowered. Such an ordered ground state would be consistent

with the Lieb-Schultz-Mattis

33,34

(LSM) theorem and recent exact diagonalization

on one of the models

treated here. Whether or not such a ground state is gapped depends on the ﬂatness of the band and

interaction strength. In the ﬂat-band limit, the ferromagnetic ground state is always gapped whereas for a

dispersive band, the interactions must exceed a critical value for a gap to obtain. These results raise the

possibility of a gapless topological semi-metallic state with non-trivial temperature corrections to the Hall

conductance

. Generally, We argue that when the interactions dominate, the QSH must give way to a

ferromagnetic QAH state at

T=0

1/4

-ﬁlling. Since this is a generic conclusion on the most general

models proven to undergird the QSH effect, we analyze the experiments

19,32

in this context. Our model

yields a quarter-ﬁlled QAH effect which coexists with a QSH effect at half-ﬁlling as is seen experimentally,

in the presence of a ﬂat lower band and intermediate interaction.

A brief survey of interacting topological systems is in order as our key result hinges on the interplay

between the two. Most studies on the KM-Hubbard

37–40

and the BHZ-Hubbard

41–44

models focused on the

half-ﬁlled system and found a transition from a QSH insulator to a topologically trivial anti-ferromagnetic

Mott insulator as the interaction strength

increases. In addition, for models more relevant to ﬂat-band

twisted bilayer graphene, Ref. [45,46] have provided a strong-coupling analysis and a density-matrix-

renormalization group study

has found that the gapless state at half-ﬁlling in the spinless (and hence

Mottless) Bisritzer-MacDonald (BM) model

yields a quantum anomalous Hall state in the presence

of Coulomb interactions. In an extensive

exact diagonalization study on an 8-band BM model,

U(4)

ferromagnets were observed always with the onset of a gap. Quantum Monte Carlo

50,51

on the spinful

model reveals a series of insulating states at half-ﬁlling. In the mean-ﬁeld context, models focused on

layered graphene systems have addressed the origin of quantum Hall ferromagnetism in the interacting

BM model

45,52,53

while others have argued that a topological Mott insulators (TMI) emerges at half-ﬁlling

in the presence of on-site and nearest neighbor interactions in the tight-binding model (with only nearest-

neighbor hopping) on a honeycomb lattice

. However, the latter proposal has not been substantiated by

subsequent numerical studies

55–58

that have found half-ﬁlling to be a trivial Mott insulator when interactions

are sufﬁciently large. Interactions also lie at the heart of fractional topological insulators

4,10,59–61

built from

fractional Chern insulators

62–66

which resemble the fractional quantum Hall effect but with no net magnetic

ﬁeld. Such phases appear at a fractional ﬁlling in a ﬂat-band

∆0≫W0

(where

∆0

is the non-interacting

topological gap and

is the bandwidth) and require nearest-neighbor interactions. A recent study on

the strongly interacting spinful Haldane model

demonstrates that a Chern Mott insulator originates at

quarter-ﬁlling with Chern number

C=±1

. This physics arises as a general consequence of an interplay

between Mottness and topology.

Motivated by Ref. [19,32,67], we explore the general phenomena that emerge from the interplay

between Mottness and the QSH effect in the context of the KM and BHZ models. To demonstrate that

the quarter-ﬁlled state is a TMI with a strongly correlated QSH effect, we numerically solve both the

KM-Hubbard and BHZ-Hubbard Hamiltonians using determinantal quantum Monte Carlo (DQMC) as well

as dynamical cluster approximation (DCA) and construct an analytically solvable Hamiltonian for a general

interacting QSH system and obtain consistent results for sufﬁciently large interactions.

Results

Hubbard interaction

The DQMC simulation results for the generalized KM-Hofstadter-Hubbard (KM-HH) model (see Methods)

on a honeycomb lattice at

ψ=0.81

and

t′/t=0.3

are shown in Fig. 1. For this choice of parameters, the

non-interacting lower band is rather ﬂat with bandwidth

W0−≈0.28

and the topological gap is

∆0≈1.62

2/20

the upper bandwidth is

W0+≈4.37

, where the subscript

indicates non-interacting. This mimics the

ﬂat-bands in moir

e TMD experiments. The tunability of bandwidths in the KM model (unlike the bands in

the BHZ model which are always dispersive

W0+=W0−≥∆0

) makes the KM model ideal for studying both

ﬂat-band and dispersive physics.

A key quantity that helps discern the topology in the presence of a probe magnetic ﬁeld is the charge

compressibility,

χ=β χc=β

N∑

i,j⟨ninj⟩−⟨ni⟩⟨nj⟩,(1)

where the sublattice and spin summations are implied in

. Regardless of density, the inverse slope of

the leading straight-line incompressible valley that extends to the zero-ﬁeld limit

provides the Chern

number. As a probe, this ﬁeld does not alter our claim of a QSH phase at zero ﬁeld. In the non-interacting

case (Fig. 1a) at

β=7

, there is a short middle vertical straight line at low ﬁelds which indicates a Chern

number

C0=0

⟨n⟩=2

. This state bifurcates into two lines or equivalently two Landau levels (LLs) at

higher magnetic ﬂux. This crossing pair of zero-mode LLs is a reliable ﬁngerprint for the QSH effects

observed in experiments

. Note the asymmetry around

⟨n⟩=2

arises entirely because the lower band is

ﬂat while the upper band is dispersive. In this regime, the lines with ﬁnite slopes all represent the standard

integer quantum Hall states.

The second quantity we calculate is the spin susceptibility deﬁned as

χs=∑

S(r)−Nm2

z=1

N∑

i,r

[⟨Sz

iSz

i+r⟩−⟨Sz

i⟩⟨Sz

i+r⟩],(2)

where

mz=∑i⟨Sz

i⟩/N

is the magnetization per spin. The non-interacting spin susceptibility is related to

the compressibility by

χs=χ/(4β)

as shown in Fig. 1(b) with reverse color scale. Fig. 1(c) shows the

magnetization. Even though the Zeeman ﬁeld is absent, a non-zero Peierls ﬂux can magnetize the system

since the spin-up and -down electron bands have different Chern numbers. The non-interacting results at

lower temperatures can be found in the supplement. What we alert the reader to is the absence of any

topologically non-trivial states at ⟨n⟩=1.

In the presence of interactions

U=3t

(already strongly correlated for the lower band), the new feature

and hence prediction is the emergence of a topologically non-trivial state at

⟨n⟩=1

. In Fig. 1d, the inverse

slope of the trace extending to

⟨n⟩=1

±1

and thus gives the Chern number. The absence of the

right-moving counterpart signiﬁes a QAH effect rather than a QSH effect. At

⟨n⟩=2

, the standard QSH

effect remains. Consequently, we have a system in which both the QAH and QSH effects obtain simply by

changing the ﬁlling. For

⟨n⟩>2

, the physics is weakly interacting as

U<W0+

. The bright peak in the spin

susceptibility in Fig. 1e indicates a possible tendency for ferromagnetism at

⟨n⟩=1

. This is supported

by the asymmetry in the dotted lines that cross at zero ﬁeld and

⟨n⟩=1

in the magnetization in Fig. 1f.

Such asymmetry signiﬁes that an inﬁnitesimal ﬁeld would lead to a polarization of the spins and hence

ferromagnetism.

We then further increase the interaction strength but have to raise the temperature to

β=3

due to the

Fermion sign problem in DQMC (see supplement). In the ﬁnal row of Fig. 1for the compressibility when

U=12t

, which far exceeds

W0−+W0++∆0≈6

, the non-interacting QSH Landau fan vanishes for

⟨n⟩=2

turning into a trivial Mott insulator and most strikingly, a new LL emerges corresponding to the mirror

image of the QAH state that terminates at

⟨n⟩=1

. The presence of both Landau components completes

the high-temperature QSH features at quarter ﬁlling. The magnetization (Fig. 1(i)) shows a more dramatic

change than does the compressibility; namely it vanishes at

⟨n⟩=2

as a result of the anti-ferromagnetic

Mott insulator. Further, the magnetization splits into peaks on either side of

⟨n⟩=1

that continues to be

asymmetrical and hence is consistent with a tendency for spontaneous Ising ferromagnetism despite the

presence of both LLs. This physics in Fig. 1(d-f) is only present in the ﬂat-band limit when

is much

3/20

larger than the bandwidth but comparable to the topological gap. Consequently, our theoretical work here

is consistent with the sudden onset of the QAH state. Since the temperature for Fig. 1(g-i) is higher than

the previous row, their features are softer.

To conﬁrm the tendency for ferromagnetism, it is important to compute the temperature dependence

of the spin susceptibility. Shown in Fig. 2a is the inverse spin susceptibility as the temperature is

lowered with zero external magnetic ﬂux. Displayed clearly is a possible divergence of the susceptibility

(

1/χs→0

) consistent with ordering. With extrapolation, we ﬁnd that it supports a ﬁnite-temperature

transition to ferromagnetism. Note that this does not violate the Mermin-Wagner theorem which forbids the

spontaneous breaking of continuous symmetries at ﬁnite temperature in low-dimensional (

d≤2

) systems

with short-range interactions. In the KM-Hubbard model with spin-orbit coupling, the system no longer

has the full SU(2) symmetry but only conserves

. Then it is the Ising symmetry that is spontaneously

broken in this transition and thus allowed at a ﬁnite temperature. As this is an interaction-driven effect, we

expect an enhancement of the susceptibility as

increases. This is also borne out in Fig. 2b. Together

these ﬁgures justify our claim of interaction-driven ferromagnetism as the temperature is lowered. A

ferromagnetic QAH state will stabilize at zero temperature even though QSH features could be present at

high temperatures when

is sufﬁciently large (Fig. 1(g-i)). We also observe a similar high-temperature

phenomenon in the dispersive case ψ=0.5(see supplement).

To show the generality of the 1/4-ﬁlled topological state, we consider the BHZ-Hofstadter-Hubbard

(BHZ-HH) model (see Methods) on a square lattice. Note in this model, both bands are dispersive and

have the same bandwidth. Without loss of generality, we set

M/t=1

, then

W0−=W0+=∆0=2t

(

t=1

as the energy scale). The non-interacting

1/2

-ﬁlled system is a QSH insulator with

Cs=2

. It is the

spin Chern number that describes a QSH insulator. To measure this quantity, we use a spin-dependent

time-reversal-invariant (TRI) magnetic ﬁeld inspired by cold-atom experiments

72,73

, namely

φi,j→σφi,j

The compressibility measured in this way we refer to as TRI compressibility. The minus sign coupled

to spin-down electrons changes the corresponding Chern number

CTRI

↓=−C↓

. Thus, the “TRI” Chern

number measured in the TRI compressibility

CTRI =CTRI

↑+CTRI

↓=C↑−C↓=Cs

corresponds to the spin

Chern number in the BHZ-HH model. This method overcomes the breakdown of the simple additivity

formula for

when the spin channels are mixed and kis no longer a good quantum number in the

presence of interactions. Through this quantity, we can read the spin Chern number from the inverse

slope of the TRI compressibility (see the supplement for the non-interacting examples).

The simulation results for the BHZ-HH models at

U=8t,β=4/t

are presented in Fig. 3. In Fig.

3a, two (red) straight lines appear from the zero-ﬁeld

1/4−

and

3/4−

ﬁlled system, whose inverse slope

indicates that the corresponding zero-ﬁeld

1/4−

and

3/4−

ﬁlled BHZ-HH systems present QSH feature

with

Cs=1

while the

1/2−

ﬁlled system becomes a topologically trivial Mott insulator with

Cs=0

. This

physics becomes much clearer by studying the standard charge compressibility in Fig. 3b which reveals

identical features at

⟨n⟩=1

and

⟨n⟩=3

of left and right moving LLs indicative of the QSH effect. Also,

the spin susceptibility exhibits a peak both at

⟨n⟩=1

and

⟨n⟩=3

. The simultaneous appearance of

compressibility minima and spin-susceptibility maxima are key features of this Mottness-driven QSH effect,

in contrast to its non-interacting counterpart. The magnetization in Fig. 3d is also asymmetrical indicating

a possible tendency towards ferromagnetism at ⟨n⟩=1and ⟨n⟩=3. We return to this in a later section.

To corroborate our ﬁndings, we conducted a ﬁnite-size analysis (see supplement) and conﬁrm that the

same spin Chern number survives in system sizes as large as

Nsite =12 ×12

with insigniﬁcant ﬁnite-size

effects and hence our results are valid in the thermodynamic limit. We conclude then that the DQMC

exhibits the QSH effect at high temperatures at 1/4-ﬁlling when Uis sufﬁciently large.

Exactly solvable model for interacting quantum spin Hall insulators

The natural question arises: why is

1/4

-ﬁlling the new topologically relevant ﬁlling and can it be understood

in a simple way? The answer is yes. For a system with 2 atoms per unit cell, there should be interaction-

induced insulating states at any integer ﬁlling up to 4 charges in each unit cell. The ﬁrst such state should

4/20

be at 1/4-ﬁlling. This physics arises naturally from a momentum-space formulation of the interactions

which will result in 4-poles of the Green function, each corresponding to the four insulating states possible.

We now introduce the Hatsugai-Kohmoto (HK) interaction74–76 into a general QSH Hamiltonian,

H=∑

k,σ(ε+,k,σ−µ)n+,k,σ+ (ε−,k,σ−µ)n−,k,σ

+U∑

(n+,k,↑n+,k,↓+n−,k,↑n−,k,↓).(3)

Without loss of generality, we use the dispersions from the BHZ model (see Methods) setting

M=1

as an example. This interaction introduces Mottness by tethering double occupancy to k-space rather

than the usual real space as in the well-known Hubbard model. As we will show, this model yields

physics for strong interactions consistent with the Hubbard model. The reason for this consilience

is that

both models break the underlying

(distinct from the classiﬁcation scheme for topological insulators)

symmetry of the non-interacting Fermi surface

. As the interaction commutes with the kinetic term, the

original non-interacting wave function is untouched and momentum kremains a good quantum number.

Therefore, it makes sense to extract the Chern number from an integration over the Brillouin zone. The

interacting Green function can be written down analytically67,75 as

G±,k,σ(ω) = 1− ⟨n±,k¯

σ⟩

ω+µ−ε±,k,σ

+⟨n±,k¯

σ⟩

ω+µ−(ε±,k,σ+U).(4)

The Green function immediately reveals the effect of the correlations. The non-interacting lower and

upper bands which were degenerate for spin-up and -down electrons split into singly and doubly occupied

sub-bands as a result of Mottness. In the following, we use the abbreviation LSB and LDB for lower singly

and doubly occupied sub-bands respectively, and likewise USB and UDB for the upper bands. The energy

of the LSB and USB remains at the non-interacting value, while the LDB and UDB move up by a value

equal to

. For a large enough

, the quarter-ﬁlled system emerges as an insulator with a ﬁlled LSB. This

physics falls out naturally from the HK model because of the 4-pole structure of the Green function.

Since the interaction mixes the spin channels, leading to a huge degeneracy (

d=2Nc

) in the ground

state (

is the number of unit cells), we need to average over all degenerate ground states

to rigorously

calculate the spin Chern number: ¯

Cs=¯

C↑−¯

C↓. For each spin, the contribution is

Cσ=1

∑

Ω=1

2πZd2k fxy,σ⟨Ω|nk,σ|Ω⟩,(5)

where

fxy,σ

is the normal Berry curvature deﬁned with Bloch wave function

because

remains a

good quantum number in the HK model, and

(1/2π)Rd2k fxy,σ=C0σ

. When

U=0

⟨Ω|nk,σ|Ω⟩=1

below

the chemical potential. When

is ﬁnite,

⟨Ω|nk,σ|Ω⟩

can be

. We can conduct the average

ﬁrst for Eq. (5). When

is large enough to fully separate the singly and doubly occupied bands,

(1/d)∑d

Ω=1⟨Ω|nk,σ|Ω⟩=⟨nk,σ⟩=⟨nσ⟩=1/2. Then Eq. (5) becomes

Cσ=⟨nσ⟩1

2πZd2k fxy,σ=⟨nσ⟩C0σ=C0σ

2.(6)

Thus, the spin Chern number

Cs=C0s/2

(we will drop the average bar symbol in the following text.).

This result demonstrates that each momentum state is equivalently occupied by half spin-up and half

spin-down electrons on average. Similarly, the LDB has the same

, while the USB and UDB have the

opposite

. In short, the strongly correlated quarter-ﬁlled system becomes a Mott insulator with a spin

Chern number Cs=C0s/2should the interaction exceed the bandwidth.

To visualize how this phase emerges, we plot the band structure in Fig. 4for varying

. With

M=1

the bandwidth for the lower and upper BHZ bands is

W0+(−)=2

and

∆0=2

is the topological gap. The

5/20

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1/4isthenew1/2whentopologyisintertwinedwithMottnessPeizhiMai1,JinchaoZhao1,BenjaminE.Feldman2,3,4,andPhilipW.Phillips1,*1DepartmentofPhysicsandInstituteofCondensedMatterTheory,UniversityofIllinoisatUrbana-Champaign,Urbana,IL61801,USA2GeballeLaboratoryofAdvancedMaterials,Stanford,CA94305,USA3Departme...

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14 is the new 12 when topology is intertwined with Mottness Peizhi Mai1 Jinchao Zhao1 Benjamin E. Feldman234 and Philip W. Phillips1

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