Quantum Monte Carlo sign bounds topological Mott insulator and thermodynamic transitions in twisted bilayer graphene model Xu Zhang1Gaopei Pan2 3Bin-Bin Chen1Heqiu Li4Kai Sun5and Zi Yang Meng1

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Quantum Monte Carlo sign bounds, topological Mott insulator and thermodynamic

transitions in twisted bilayer graphene model

Xu Zhang,1Gaopei Pan,2, 3 Bin-Bin Chen,1Heqiu Li,4Kai Sun,5, ∗and Zi Yang Meng1, †

1Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics,

The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China

2Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,

Chinese Academy of Sciences, Beijing 100190, China

3School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

4Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada

5Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA

(Dated: October 24, 2022)

We show that for magic-angle twisted bilayer graphene (TBG) away from charge neutrality, al-

though quantum Monte Carlo (QMC) simulations suﬀer from the sign problem, the computational

complexity is at most polynomial at certain integer ﬁllings. For even integer ﬁllings, this polyno-

mial complexity survives even if an extra inter-valley attractive interaction is introduced, on top of

Coulomb repulsions. This observation allows us to simulate magic-angle twisted bilayer graphene

and to obtain accurate phase diagram and dynamical properties. At the chiral limit and ﬁlling ν= 1,

the simulations reveals a thermodynamic transition separating metallic state and a C= 1 correlated

Chern insulator – topological Mott insulator (TMI) – and the pseudogap spectrum slightly above

the transition temperature. The ground state excitation spectra of the TMI exhibit a spin-valley

U(4) Goldstone mode and a time reversal restoring excitonic gap smaller than the single particle

gap. These results are qualitatively consistent with the recent experimental ﬁndings at zero-ﬁeld

and ν= 1 ﬁlling in h-BN nonaligned TBG devices.

Introduction.— Magic-angle twisted bilayer graphene

(TBG) has attracted great attention in recent years,

as it hosts a variety of nontrivial phases beyond semi-

classical or band-theory description [1–41]. To theoreti-

cally characterize these ﬂat bands and correlated quan-

tum phases, tight-binding [2, 5] and continuous (BM) [3]

models has been developed. In this study, we focus on

the continuous-model approach, which avoids the chal-

lenge to construct localized orbitals that preserves all the

symmetries [42–47]. By projecting long-range Coulomb

interactions onto the moir´e ﬂat bands with proper quan-

tum metric, such projected Hamiltonian has been stud-

ied using mean-ﬁeld approximations [20, 31, 48, 49]. At

certain limits, exact analytical solution has also been ob-

tained [50–54]. On the numerical side, the charge neu-

trality point has been studied using sign-problem-free

momentum-space quantum Monte Carlo (QMC) simula-

tions [55–57]. However, away from the charge neutrality

point, due to the arising of the sign problem, such simu-

lations have not yet been performed.

Although the sign problem often implies exponential

computational complexity, it is worthwhile to emphasize

that not all sign problems cause such severe damage.

Very recently, a much more mild type of sign problem

has been demonstrated, where the computation complex-

ity scales as a polynomial function of the system size,

known as polynomial sign problem [58, 59].

In this Letter, we study the sign problem in TBG ﬂat

bands. Utilizing the sign bound theory [60], we prove

that TBG ﬂat bands at even integer ﬁllings, or arbitrary

integer ﬁllings in the chiral limit, exhibit (at most) poly-

nomial sign problem. This observation allows us to utilize

QMC methods to study ﬁllings away from charge neutral-

ity, away from the chiral limit, and/or in the presence of

extra attractive interactions on top of the Coulomb re-

pulsion (See Tab. I for details).

To demonstrate this new approach, we performed

large-scale QMC simulations to examine the chiral limit

at ﬁling ν= 1 (we denote the fully empty/ﬁlled ﬂat bands

as ν=−4/+ 4 and charge neutrality as ν= 0). At

T= 0, this model can be solved exactly [51, 53, 56],

and the exact solution reveals that at T→0, the sys-

tem is a correlated Chern insulator – a topological Mott

insulator (TMI) [61–66] – with Chern number C= 1.

Upon raising the temperature, the insulating state shall

“melt” into a metal and the time-reversal symmetry shall

be recovered. However, our knowledge about this ﬁnite-

temperature phase transition is very limited. Because

the sign problem is only polynomial, QMC simulations

become a highly eﬃcient tool to study this system, from

which three diﬀerent phases/states are observed (1) a

metal phase with time-reversal symmetry at high tem-

perature T > T ?, (2) a time-reversal invariant pseudo-

gap phase at intermediate temperature Tc< T < T ?

and (3) a low-temperature TMI phase at T < Tc. Here,

T?is a crossover temperature scale and Tcis the crit-

ical temperature of a second order phase transition, at

which the time-reversal symmetry is spontaneously bro-

ken. We further show that the TMI phase only breaks

the time reversal symmetry, while spins remain disor-

dered due to thermal excitations of gapless spin ﬂuctua-

tions. This absence of spin order/polarization is in direct

contrast to quantum Hall states where electron spins are

polarized due to Zeeman splitting and thus spin ﬂuctua-

arXiv:2210.11733v1 [cond-mat.str-el] 21 Oct 2022

tions are gapped. In the discussion section, we establish

the connection and consistency between our simulations

and recent experimental studies on TMI phases in h-BN

nonaligned TBG devices [41].

BM model and projected interaction.— We utilize the

BM model and project interactions between fermions to

the moir´e ﬂat bands. The BM model Hamiltonian [3]

for the τvalley takes the following form Hτ

k,k0=

−~vF(k−K1)·σδk,k0V

V†−~vF(k−K2)·σδk,k0

where K1and K2mark the two Dirac points

in the τvalley from layers 1 and 2 respectively.

V=U0δk,k0+U1δk,k0−G1+U2δk,k0−G1−G2and

V†=U†

0δk,k0+U†

1δk,k0+G1+U†

2δk,k0+G1+G2

are the inter-layer tunnelings with matrixes

U0=u0u1

u1u0,U1=u0u1e−i2π

u1ei2π

3u0and

U2=u0u1ei2π

u1e−i2π

3u0where u0and u1are the

intra- and inter-sublattice inter-layer tunneling am-

plitudes. G1= (−1/2,−√3/2),G2= (1,0) are the

reciprocal vectors of the moir´e Brillouin zone (mBZ),

and K1= (0,1/2√3)|G1,2|,K2= (0,−1/2√3)|G1,2|.

For control parameters, we set (θ, ~vF/a0, u0, u1) =

(1.08◦,2.37745 eV,0 eV,0.11 eV) for the chiral limit, and

for non-chiral model, we set u0= 0.06 eV following

Refs. [50, 55, 57, 67]. Here, θis the twisting angle and

a0is the lattice constant of monolayer graphene.

For interactions, in addition to the Coulomb repulsion,

here we have the option to include one more interaction

term and the sign problem will still remain polynomial.

HI=1

2Ω X

(V1(q)δρ1,qδρ1,−q+V2(q)δρ2,qδρ2,−q)

δρ1,q=X

k,α,τ,s

(c†

k,α,τ,sck+q,α,τ,s −ν+ 4

8δq,0)

δρ2,q=X

k,α,s

(c†

k,α,τ,sck+q,α,τ,s −c†

k,α,−τ,sck+q,α,−τ,s)

(1)

The ﬁrst term in HI(V1>0) is the Coulomb interac-

tions, and the second term V2≥0 introduces repulsive

interactions for fermions in the same valley and attrac-

tions between the two valleys, which can be introduced

as a phenomenology term describing inter-valley attrac-

tions [68–72]. At V2= 0, this model recovers the stan-

dard TBG model with Coulomb repulsions. When V2is

turned on, the inter-valley attraction favors inter-valley

pairing and could stabilize a superconducting ground

state. The normalization factor in HIis Ω = L2√3

2a2

with Lbeing the linear system size of the system. kand

qcover the whole momentum space, νis the ﬁlling factor

and α, τ, s represent layer/sublattice, valley, spin indices,

respectively. The mometum dependence for non-negative

V1and V2is unimportant to polynomial sign problem.

Here, for simplicity, we set V2=γV1with γbeing a

non-negative constant and for V1, we use Coulomb inter-

action screened by single gate in our simulation V1(q) =

4πε Rd2r1

r−1

√r2+d2eiq·r=e2

2ε

|q|1−e−|q|dwhere

2= 20 nm is the distance between graphene layer and

single gate, = 70is the dielectric constant. We then

project the interactions HIto the moir´e ﬂat bands (See

SM [73]) and use the projected Hamiltonian to carry out

sign bounds analysis and QMC simulations.

Polynomial sign bounds.— In QMC simulations, the ex-

pectation value of a physical observable Ois measured as

hˆ

Oi=PlWlhˆ

Oil, where Wland hˆ

Oilare the weight and

the expectation value for the conﬁguration l. Instead of

summing over all conﬁgurations, a QMC simulation sam-

ples the conﬁguration space using the probability Wl. In

sign-problem-free QMC simulations, Wl≥0 for all land

an accurate expectation value can be obtained by only

sampling a small number of conﬁgurations – the impor-

tance sampling, and this number scales as a power-law

function of the system size. However, for many quantum

systems, Wlcan be negative or even complex, and thus

to obtain an accurate expectation value, it requires to

sample a large number of conﬁgurations, which usually

scales as an exponential function of the system size [74].

It is worthwhile to emphasize that the sign prob-

lem doesn’t always lead to an exponentially high com-

putational cost. To measure the severity of the sign

problem, here we use the average sign hsigni=

PlWl/Pl|Re(Wl)|, where |Re(Wl)|is the absolute

value of the real part of Wl. For physical partition func-

tion Z=PlWl=PlRe(Wl), this average sign is be-

tween 0 and 1, and hsigni= 1 means that the system

is sign problem free, while smaller hsignimeans sev-

erer sign problem. In a d-dimension quantum systems

that suﬀers from the sign problem, hsigni ∼ exp(−βLd)

where β= 1/T the inverse temperature, indicating that

the number of conﬁgurations needed in QMC simulations

scales as an exponential function of the space-time vol-

ume. For polynomial sign problem, although hsigni<1

(i.e., the system does suﬀer from the sign problem),

1/hsigniis a polynomial function of the system size, and

thus the number of conﬁgurations needed only scales as

a power-law function of the system size.

Although the average sign can be easily measured in

QMC simulations, it usually doesn’t have a simple an-

alytic formula. To estimate the numerical cost to over-

come the sign problem, we utilize the sign bound hsignib

deﬁned in Ref. [60]. As proved in Ref. [60], hsignibis

the lower bound of hsigni(i.e., hsignib6hsigni). Thus,

if the sign bound scale is a power-law function of the

system size, the sign problem is (at most) polynomial.

Remarkably, the low temperature sign bound in moir´e

ﬂat bands can be easily calculated by counting ground

state degeneracy, which can be obtained using SU(4) and

SU(2) Young diagram as employed in Refs. [50, 51, 56]

TABLE I. Scaling of the sign bound hsignibat low tempera-

ture and large moir´e lattice size N=L2. A power-law func-

tion of Nindicates that the sign problem is (at most) polyno-

mial. 7indicates the sign bound decays to zero exponentially.

Filling(ν) Chiral(γ= 0) Non-chiral(γ= 0) Chiral(γ > 0)

0 1 1 1

±1N−17 7

±2N−2N−1N−2

±3N−57 7

±4N−8N−4N−4

(See SM for details [73]).

Details about this calculation are shown in the SM

and the conclusions are summarized in Tab. I. At charge

neutrality (ν= 0), moir´e ﬂat bands with Coulomb inter-

actions (γ= 0) is known to be sign problem free [55, 56],

and thus the sign bound is 1. Here, we further prove

that adding inter-valley attractions (γ > 0) to the chiral

system doesn’t cause sign problem either (hsignib= 1).

Away from charge neutrality, sign problem arises, but it is

polynomial at certain integer ﬁllings. For Coulomb repul-

sion (γ= 0), the sign problem is polynomial at any (even)

integer ﬁllings at (away from) the chiral limit. When

inter-valley attractions are introduced (γ > 0), even in-

teger ﬁllings at the chiral limit also have polynomial sign

bound.

To further verify the polynomial sign problem summa-

rized in Tab. I, we directly calculate hsigniand hsignib

in QMC simulations at various ﬁlling ν, and compare

them with the exact formula of the sign bound obtained

at integer ﬁllings. As shown in Fig. 1, hsigniis always

larger or equal to hsignibas expected, and the sign bound

at integer ﬁlling indeed converges to the exact solution.

The peak in hsigniat integer (or even integer) ﬁllings

indicates that the sign problem is less severe and QMC

simulations have a faster convergence at these ﬁllings.

The location of these peaks are fully consistent with the

polynomial sign problem summarized in Tab. I. The only

exceptions are ν= 3 and 4 of Fig. 1(a), where the sign

problem is polynomial but the ﬁgure doesn’t exhibit vis-

ible peaks. This is because the power-law function here

has high powers N−5and N−8, which requires higher res-

olution (longer simulation time) to show clear distinction

from exponential functions.

Chiral limit at ν= 1 and γ= 0.— With the polynomial

sign bound obtained, here we discuss the QMC results as

a function of temperature for the chiral limit ν= 1 ﬁlling

case in this section. In the QMC simulation, we observe

a thermal phase transition with pseudogap spectrum and

spontaneous time reversal symmetry breaking, which is of

immediate relevance to the recent experimental ﬁnding of

correlated Chern insulators at zero-ﬁeld and ν= 1 ﬁlling

in h-BN nonaligned TBG devices and its relatively high

Curie temperature of Tc∼4.5 K [41].

01234

0.2

0.4

0.6

0.8

01234

0.2

0.4

0.6

0.8

01234

0.2

0.4

0.6

0.8

01234

0.2

0.4

0.6

0.8

01234

0.2

0.4

0.6

0.8

01234

0.2

0.4

0.6

0.8

( )

FIG. 1. hsigniversus ﬁlling ν(a,c,e) and hsignibversus ﬁlling

ν(b,d,f) at low temperature T= 1 meV. The ﬁllings of ν < 0

are symmetric with respect to ν= 0 via the particle-hole

symmetry. (a-b) are the chiral limit γ= 0 cases, (c-d) are

the non-chiral γ= 0 cases where we take u0= 0.06 eV and

(e-f) are the chiral limit γ= 4 cases. (a,c,d) are the average

sign hsignifor L= 3(N= 9) and L= 4(N= 16) measured

from QMC for ﬁlling from ν= 0 to ν= 4. (b,d,f) are the

sign bounds hsignibfor L= 3(N= 9) and L= 4(N=

16) measured from QMC (solid line) and derived from exact

solution (ES) at the low temperature limit for ﬁlling ν=

1,2 (dash line values). The ES values in (b) are 1119/3278,

3630/111493 for L= 3 at ν= 1,2 and 945/4357, 183/14912

for L= 4 at ν= 1,2. The ES values are 1/10 for L= 3, 1/17

for L= 4 at ν= 2 in (d) and 100/2601 for L= 3, 71/5201

for L= 4 at ν= 2 in (f). (See SM for details [73])

At low temperature limit for ν= 1, exact solution

at T= 0 expects degenerate ground states with Chern

number C=±1, and ±3 [51, 53, 56]. In the large system

size limit N→ ∞, the number of ground states scales as

∝N7for C=±1 and N3for C=±3 [73]. Due to the

higher number of ground state degeneracy, thermal ﬂuc-

tuations shall stabilize the C=±1 state as the thermal

equilibrium state at low temperature via the order by dis-

order mechanism [75]. This low-temperature state breaks

spontaneously the time-reversal symmetry, but this sym-

metry breaking process as a function of temperature is

unknown.

Our QMC simulation at ﬁnite temperature reveals this

process. To probe the time-reversal symmetry breaking,

we use the Chern band polarization as the order param-

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QuantumMonteCarlosignbounds,topologicalMottinsulatorandthermodynamictransitionsintwistedbilayergraphenemodelXuZhang,1GaopeiPan,2,3Bin-BinChen,1HeqiuLi,4KaiSun,5,andZiYangMeng1,y1DepartmentofPhysicsandHKU-UCASJointInstituteofTheoreticalandComputationalPhysics,TheUniversityofHongKong,PokfulamRoad,Hon...

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Quantum Monte Carlo sign bounds topological Mott insulator and thermodynamic transitions in twisted bilayer graphene model Xu Zhang1Gaopei Pan2 3Bin-Bin Chen1Heqiu Li4Kai Sun5and Zi Yang Meng1

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