Magnetism and Quantum Melting in Moir e-Material Wigner Crystals Nicol as Morales-Dur an1Pawel Potasz2and Allan H. MacDonald1 1Department of Physics The University of Texas at Austin Austin Texas 78712 USA

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Magnetism and Quantum Melting in Moir´e-Material Wigner Crystals

Nicol´as Morales-Dur´an,1Pawel Potasz,2and Allan H. MacDonald1

1Department of Physics, The University of Texas at Austin, Austin, Texas, 78712, USA

2Institute of Physics, Faculty of Physics, Astronomy and Informatics,

Nicolaus Copernicus University, Grudziadzka 5, 87-100 Toru´n, Poland

(Dated: July 21, 2023)

Recent experiments have established that semiconductor-based moir´e materials can host incom-

pressible states at a series of fractional moir´e-miniband ﬁllings. These states have been identiﬁed as

generalized Wigner crystals in which electrons localize on a subset of the available triangular-lattice

moir´e superlattice sites. In this article, we use momentum-space exact diagonalization to investigate

the many-body ground state evolution at rational ﬁllings from the weak-hopping classical lattice

gas limit, in which only spin degrees-of-freedom are active at low energies, to the strong-hopping

metallic regime where the Wigner crystals melt. We speciﬁcally address the nature of the magnetic

ground states of the generalized Wigner crystals at ﬁllings ν= 1/3 and ν= 2/3.

I. INTRODUCTION

It is now several years since Wu et al. [1] pointed out

that the Hamiltonian of interacting holes in the moir´e

bands of transition metal dichalcogenide (TMD) heter-

obilayers can be mapped to the triangular lattice Hub-

bard model. Experiments quickly conﬁrmed the validity

of this assertion by observing Mott insulating states at

band ﬁlling ν≡N/M = 1 of the moir´e superlattice [2–

5], where Nis the number of holes and Mthe number of

moir´e unit cells in the system. Subsequent experiments

have established that TMD-based moir´e materials also

exhibit correlated insulating states at a discrete series

of fractional ﬁllings of the lowest moir´e miniband [5–10].

These insulating states form because electrons localize on

a subset of moir´e sites in order to minimize strong long-

range Coulomb interactions. Because they break transla-

tional symmetry, they are reminiscent of the Wigner crys-

tals expected to appear in the two-dimensional electron

gas (2DEG) at very low densities [11]. There are how-

ever some qualitative diﬀerences between Wigner crystals

formed in an electron gas with continuous translational

symmetry, and the incompressible states at fractional ﬁll-

ings in moir´e materials, which have only discrete trans-

lational symmetry. Most importantly, the moir´e super-

lattice potential narrows bands and reduces the relevant

single-particle energy scales, making interactions domi-

nant in much of the available phase space.

The incompressible states in moir´e superlattices are

commonly referred to as generalized Wigner crystals and

we adopt that terminology in this paper. The ubiquity of

robust crystalline states at fractional ﬁllings in the moir´e

material platform opens up a new thread in the study

of strongly interacting electrons in low dimensions and

promises to reveal new physics. Given the abundance

of distinct moir´e semiconductor heterostructures, even

within the group VI transition metal dichalcogenide fam-

ily alone, and the ability to tune samples through large

ranges of ﬁlling factor by varying gate voltage, it seems

likely that it will be possible to realize a rich variety of

generalized Wigner crystal states with distinct structural

and magnetic properties in the coming years.

The emergence of incompressible states at non-integer

partial band ﬁlling can be explained only if inter-site

electron-electron interactions are included. Recent ex-

periments have therefore established moir´e TMDs as a

platform to simulate extended Hubbard models whose

Hamiltonians have tunable hoppings tn, on-site inter-

action U0, and long-range interaction strengths Un(n

stands for n-th neighbor). Assisted hopping and direct

exchange non-local interaction terms can also play a cru-

cial role [12] in determining the magnetic properties of

moir´e Hubbard systems. The mapping to a Hubbard

model is a one-band approximation, whose applicability

at ν≤1 has generally been conﬁrmed by experiment. For

ﬁllings above half-ﬁlling, there is a competition between

the upper Hubbard band and remote bands; hence the

simple one-band Hubbard model is often insuﬃcient. For

that reason, in this work we focus on the regime ν < 1,

having addressed ν= 1 in a previous study [12, 13].

In moir´e superlattices, localization of electrons in an

insulating state is expected [1] in the long-moir´e-period

narrow-moir´e-band limit. In this regime, the dominant

energy scale is U0at ν= 1 and U1for 1/3≤ν < 1. When

the twist angle is increased and the moir´e period de-

creased, or a displacement ﬁeld is applied to decrease the

moir´e potential strength, the eﬀective hopping parame-

ters tbetween moir´e lattice sites increase and eventually

become comparable to inter-site interaction strengths U1,

complicating the electronic properties. The interplay be-

tween spin and charge degrees of freedom can give rise

to diﬀerent magnetic orders at each ﬁlling factor. For

example, recent experiments have reported that some of

the crystal states are striped phases [8], and that antifer-

romagnetic interactions are frustrated for ν= 2/3 band

ﬁlling factor [14]. When hopping is strong enough to

overcome the near-neighbor interaction, the Wigner crys-

tal will melt into a liquid state – the Mott-Wigner tran-

sition. Interestingly a recent experiment performed on

MoSe2/WS2observed that the charge gap continuously

vanishes as the superlattice potential is weakened [10].

Further experiments have shown that the charge gaps

of the generalized Wigner crystal states are asymmetric

arXiv:2210.15168v2 [cond-mat.str-el] 20 Jul 2023

with respect to half-ﬁlling (ν= 1) of a single spinful

band, and also with respect to quarter-ﬁlling (ν= 1/2)

[6] and demonstrate the role of quantum ﬂuctuations in-

volving remote bands, as we will show in this work, before

and across the Mott-Wigner transition.

The magnetic order of the generalized Wigner crys-

tal phases, as well as the nature of their bandwidth

and density-tuned quantum melting transitions are still

a matter of debate. Previous theoretical eﬀorts to un-

derstand moir´e Wigner crystals and their evolution with

interaction strength have focused on the deep crystalline

regime [15–17], where classical Monte Carlo simulations

can be used to investigate the ground state charge order

at diﬀerent ﬁllings. Hartree-Fock [18–20] and classical

Monte Carlo studies [21] have addressed the competition

between diﬀerent charge and spin orders in the crystal

phase, and its transition to metal when bandwidth or

density are tuned. An analysis that includes quantum

ﬂuctuations and goes beyond mean-ﬁeld is needed, how-

ever, since mean-ﬁeld theory approximations are known

to favor ferromagnetic groundstates and to overestimate

the stability of insulating states in the proximity of metal-

insulator transitions.

In this work, we report on a ﬁnite-system exact di-

agonalization study of semiconductor moir´e materials at

fractional ﬁlling factors. Starting from the continuum

model description [1], we add relevant electron-electron

interactions projected to the topmost moir´e band and ob-

tain the many-body spectrum. We ﬁnd, as already sug-

gested by experiment, that a rich set of fractional band

ﬁllings νsupport correlated insulating states with tun-

able magnetic properties. We show that there is an over-

riding competition between antiferromagnetism and fer-

romagnetism particularly at ν= 2/3 band ﬁlling, which

we explain using a low-energy eﬀective spin model de-

scription and relate to a recent experiment [14]. We also

address the Mott-Wigner transitions at fractional ﬁllings,

ﬁnding that as in the ν= 1 case they are not strongly

ﬁrst order.

II. MOIR´

E MATERIAL MODEL

A. Continuum model

Our starting point is the continuum model description

of TMD heterobilayers with type-I or type-II band align-

ment [1]. In this case the topmost moir´e band is concen-

trated in one of the layers, which we refer to as the active

layer, depicted in red in Fig. 1(a)-(b). The inﬂuence of

the second layer, shown in blue in Fig. 1(a)-(b), is re-

sponsible for a moir´e potential that aﬀects charge carriers

in the active layer. The moir´e pattern can be induced by

a small twist angle θor a lattice mismatch δbetween

the two layers. The moir´e lattice constant is given by

aM=a0/(θ2+δ2)1/2, with a0the lattice constant of

the active TMD layer. To date most heterobilayer ex-

periments have focused on unrotated WSe2/WS2with a

moir´e lattice constant of aM≈8.2 nm, or MoSe2/WS2,

with a moir´e lattice constant of aM≈7.5 nm. Because

the moir´e lattice constant reaches a maximum, the sys-

tem is expected to be less sensitive to twist angle disorder

at zero twist angle.

Valley and spin are locked in TMD heterobilayers and

the valley (or spin) projected continuum Hamiltonian is

given by [1]

H0=−ℏ2

2m∗k2+ ∆(r),(1)

∆(r) =2VmX

j=1,3,5

cos(bj·r+ψ).(2)

The ﬁrst term in Eq. (1), which corresponds to the kinetic

energy of carriers in the top moir´e band, is diagonal in

momentum space and the second term, the moir´e poten-

tial, is diagonal in coordinate space. The moir´e potential

depends on only two parameters (Vm, ψ) because [1] of

the system’s C3symmetry. The phase ψﬁxes the ge-

ometry of the moir´e superlattice, which we take to be

triangular as it is in most of the TMD heterostructures,

and the strength of the moir´e potential Vmcan be re-

lated to the experimentally tunable displacement ﬁeld.

For concreteness, throughout this work we take the ef-

fective mass m∗= 0.45 m0and ψ= 45◦, corresponding

to WSe2/WS2[22]. We take the modulation potential

strength Vmas an experimentally controllable parameter

since it has been demonstrated to be sensitive to the dis-

placement ﬁeld D. The form we have used for the moir´e

modulation parameter assumes that it varies smoothly

with position on the moir´e scale; higher harmonics in the

plane-wave expansion are more important in longer pe-

riod moir´es in which relation relative to rigidly twisted

bilayers is stronger.

An example of the bandstructure obtained from di-

agonalizing Eq. (1) is shown in Fig. 1(c). We label

the band energies of Eq. (1) as ϵn

k, while the eigen-

states can be written in a plane wave expansion as

|n, ψk⟩=PGzn

k,G|k+G⟩, where nis a band index

and Gare reciprocal lattice vectors. In order to study

the emerging many-body phases we consider the interact-

ing Hamiltonian projected to the topmost moir´e valence

band (hence we omit band index)

H=X

ϵkc†

kσckσ+X

i,j,k,l

σ,σ′

Vσ,σ′

i,j,k,l

2c†

kiσc†

kjσ′cklσ′ckkσ,(3)

where the Coulomb matrix elements are given by

Vσ,σ′

i,j,k,l =1

Gi,Gj

Gk,Glz∗

ki,Giz∗

kj,Gjzkk,Gkzkl,Gl2πe2

ε q .(4)

The summation involving the term in brackets results

from the projection of interactions to a single band and

can be rewritten in terms of the form factors, as described

elsewhere [23]. In Eq. (4), q=|ki+Gi−kk−Gk|is

the momentum transfer, Ais the area of the system,

εthe eﬀective dielectric constant, and total momentum

conservation is implicit.

FIG. 1. (a) Schematics of a TMD heterobilayer with metallic

gates at distance dfrom the sample. By varying gate voltages

a displacement ﬁeld is applied. (b) Schematics of the heter-

obilayer in momentum space. Charge carriers populate the

active WSe2band while the presence of a WS2layer generates

the moir´e potential, Eq. (2), whose strength is modiﬁed by

the displacement ﬁeld D. (c) Example of WSe2/WS2moir´e

minibands obtained from Eq. (1), with Vm= 40 meV. (d)

Evolution of the energy scales of the problem as Vmis varied:

the interaction strength UM(green), the kinetic energy scale

WM(black), the bandwidth B(blue) and the gap to the ﬁrst

remote ∆R(brown).

B. Exact diagonalization methodology

In this article we take the same approach as in pre-

vious work [12, 13], utilizing exact diagonalization (ED)

to solve the many-body Hamiltonian. We present our

results in phase diagrams that depend on dimensionless

parameters obtained by taking ratios of the relevant en-

ergy scales of the system. In particular, three energy

scales can be identiﬁed; the moir´e potential depth Vm,

the interaction strength UM=e2/(εaM), and the kinetic

energy scale WM=ℏ2/m∗a2

M. By varying the two ratios

of these three scales, we can simulate any heterobilayer as

long as the moir´e period is much larger than the micro-

scopic lattice constant. This ensures that our conclusions

apply for arbitrary heterobilayers as long as their low en-

ergy physics is captured by the continuum model, Eq.

(1) and that the shape of the moir´e potential is similar to

that of a triangular lattice model. The energy gap to the

remote moir´e bands ∆Rcan, in principle, be viewed as

another relevant energy scale. We take parameter values

such that this scale is always larger than all other scales

involved, as can be seen in Fig. 1(d), justifying the sin-

gle band projection of Eq. (3). As we have pointed out

above, however, interactions renormalize bands more at

higher electron densities. For this reason the single-band

approximation should be treated with caution for ﬁllings

ν > 1, where remote band mixing is often relevant.

The model we study has orbital and spin degrees of

freedom, discrete triangular lattice translational symme-

try, SU(2) spin-rotational invariance, and no spin-orbit

coupling. The Hilbert space can be divided into smaller

subspaces with total momentum Kwith discrete values

determined by the number of moir´e unit cells M, to-

tal spin S, and azimuthal spin Sz. The basis is con-

structed in an occupation number representation, dis-

tributing particles among single-particle states labeled by

Szquantum number and quasi-particle (kx, ky) momen-

tum. The total number of possible conﬁgurations Nconf

for particles distributed on Msingle particle states with

N↑spins up and N↓spins down is determined by a prod-

uct of binomial coeﬃcients, Nconf =M

N↑·M

N↑. The

many-body Hamiltonian projected to a given total mo-

mentum Kis diagonalized in Szsubspaces. We do not

rotate the Hamiltonian matrix to a Sbasis as this is

an additional computational cost, and instead determine

the ground state total total spin Sby identifying mul-

tiplets from the Sz-dependent energy eigenvalues. The

total spin Sassignments have been conﬁrmed by calcu-

lating the spin structure factor S(q= 0). For a given

momentum, the S-multiplet structure implies that the

largest subspace corresponds to the lowest possible Sz

which contains states with all values of total spin S.

All of our exact diagonalization calculations are limited

only by the maximal matrix size of a given subspace, with

the largest subspace corresponding to the lowest possible

Sz. The results presented here correspond to systems

containing M= 9,12 and 16 moir´e unit cells. Despite the

limited system sizes that can be reached with the exact

diagonalization method, important information can still

be extracted concerning the behaviors of charge gaps and

the nature of the magnetic order of insulating states. Nu-

merical non-perturbative approaches, like the one taken

in this paper or DMRG, are particularly important in the

quantum melting regime, where Hartree-Fock is known

to overestimate the stability of the insulating phase.

III. FINITE-SIZE PHASE DIAGRAMS

A phase diagram for TMD heterobilayers as a function

of ﬁlling factor and the kinetic-energy-scale to moir´e-

depth ratio W/Vmis shown in Fig. 2(a) . The color

scale speciﬁes the ratio of the charge gap to the Coulomb

interaction-energy scale ∆c/U. A ﬁnite value of the

charge gap indicates an incompressible state, while a van-

ishing charge gap indicates a metallic state. The charge

gap is extracted from the many-body spectrum via the

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MagnetismandQuantumMeltinginMoir´e-MaterialWignerCrystalsNicol´asMorales-Dur´an,1PawelPotasz,2andAllanH.MacDonald11DepartmentofPhysics,TheUniversityofTexasatAustin,Austin,Texas,78712,USA2InstituteofPhysics,FacultyofPhysics,AstronomyandInformatics,NicolausCopernicusUniversity,Grudziadzka5,87-100Toru´...

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Magnetism and Quantum Melting in Moir e-Material Wigner Crystals Nicol as Morales-Dur an1Pawel Potasz2and Allan H. MacDonald1 1Department of Physics The University of Texas at Austin Austin Texas 78712 USA

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