Multiferroicity and Topology in Twisted Transition Metal Dichalcogenides Ahmed Abouelkomsan1 2Emil J. Bergholtz1and Shubhayu Chatterjee3 4 1Department of Physics Stockholm University AlbaNova University Center 106 91 Stockholm Sweden

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Multiferroicity and Topology in Twisted Transition Metal Dichalcogenides

Ahmed Abouelkomsan,1, 2, ∗Emil J. Bergholtz,1and Shubhayu Chatterjee3, 4

1Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden

2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

3Department of Physics, University of California, Berkeley, California 94720, USA

4Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA

(Dated: July 15, 2024)

Van der Waals heterostructures have recently emerged as an exciting platform for investigating the effects of

strong electronic correlations, including various forms of magnetic or electrical orders. Here, we perform an

unbiased exact diagonalization study of the effects of interactions on topological ﬂat bands of twisted transition

metal dichalcogenides (TMDs) at odd integer ﬁllings. For hole-ﬁlling νh= 1, we ﬁnd that the Chern insulator

phase, expected from interaction-induced spin-valley polarization of the bare bands, is quite fragile, and gives

way to spontaneous multiferroic order — coexisting ferroelectricity and ferromagnetism, in presence of long-

range Coulomb repulsion. We provide a simple real-space picture to understand the phase diagram as a function

of interaction range and strength. Our ﬁndings establish twisted TMDs as a novel and highly tunable platform

for multiferroicity, and we outline a potential route towards electrical control of magnetism in the multiferroic

phase.

Introduction. In recent years, moir´

e materials have emerged

as highly versatile Van der Waals heterostructures [1,2].

Through a wide range of experimental tuning knobs, the in-

teraction strength can be enhanced relative to the bandwidth.

Consequently, a variety of correlated phases have been ob-

served in two separate classes of moir´

e materials, graphene-

based heterostructures such as twisted bilayer graphene [3–

8] and heterostructures made from semiconductor transition

metal dichalcogenides (TMDs) [9–19] either by using the

same TMDs (homobilayers) or different TMDs (heterobi-

layers). These correlated phases encompass unconventional

forms of magnetism driven by both orbital and spin degrees

of freedom [20–27] or different forms of electric order such

as ferroelectricity, charge density waves and Wigner crystals

[12–14,28]. However, multiferroicity — the simultaneous

presence of magnetic and electric order, is highly desirable,

as it enables electrical manipulation of magnetism, and vice

versa [29]. The high tunability of moir´

e materials combined

with prospects of spontaneous multiferroicity would therefore

not only sharpen our understanding of strong electronic corre-

lations, but also open the door to next-generation spintronics

devices [30–32].

In this Letter, we introduce twisted TMD bilayers as a can-

didate for interaction-driven spontaneous multiferroic order.

To this end, we consider the ﬂat bands of TMD homobilayers

at small twist angles, and present an unbiased exact diagonal-

ization study of the interacting phase diagram at odd integer

hole-ﬁllings (νh= 1,3). The key feature of the phase diagram

at νh= 1 (Fig. 1(a)) is the robust presence of multiferroic

order — ferromagnetism from spin-valley polarization and

ferroelectricity from layer polarization, driven by long-range

Coulomb repulsion. Decreasing the range of the Coulomb in-

teraction by tuning the gate-distance drives the system through

a topological phase transition from the multiferroic phase to a

ferromagnetic Chern insulator. We provide an intuitive un-

derstanding of the competition between the two phases via a

simple real space description of the topological ﬂat bands. Fi-

Chern

Multiferroic

Ferroelectric ?

Metallic

Interaction Strength

Screening Length

FIG. 1. (a) Schematic phase diagram at hole integer ﬁlling νh= 1 as

a function of screening length and interaction strength of Coulomb

interactions. (b) Moir´

e TMD homobilayer in real space with two in-

terpenetrating triangular lattices from the two layers forming a hon-

eycomb lattice that carries most of the electronic spectral weight in

the active bands. (c) Schematic real-space description of the multi-

ferroic and Chern insulator phases.

nally, leveraging multiferroicity, we propose a route towards

electrical control of magnetism in twisted TMDs.

Model. Our starting point is continuum models [33,34] de-

veloped for the valence bands of twisted TMD homobilay-

ers. We are particularly interested in the regime where the

active bands are topologically non-trivial and carry opposite

Chern numbers in a given valley, together with their time-

reversed partners in the opposite valley. To be speciﬁc, we

focus on MoTe2where this regime is accessible at twist an-

gles θ= 1◦−1.5◦[33] as shown in Fig. 2(a), although our re-

sults apply more generally to other twisted TMD homobilay-

ers such as twisted WSe2[34] or when the low-lying energy

bands admit a speciﬁc real space description as we elaborate

below. The strong spin-orbit coupling in TMD valence bands

[35] leads to an effective spin-valley locking where the spin

up (down) is tied to valley K+(K−).

The active pair of bands in each valley are very close in

arXiv:2210.14918v4 [cond-mat.str-el] 11 Jul 2024

FIG. 2. (a) Band structure [33] of twisted MoTe2at twist angle

θ= 1.2◦. (b) A schematic represnetation of the problem considered

in both band and layer basis.

energy while there is a much bigger gap to the remaining va-

lence bands (Fig. 2(a)). Therefore we project interactions [36]

to the active bands, leading to the following Hamiltonian:

H=X

kατ −ϵατ (k)c†

ατ (k)cατ (k)+ 1

NcX

qτ1

τ2

U(q):ρτ1(q)ρτ2(−q):

(1)

where α= 1 (2) denotes the top (bottom) band and τ=±

denotes the valley (equivalently the spin Szbecause of the ef-

fective spin-valley locking). c†

ατ are hole band creation oper-

ators, and ϵατ (k)is the non-interacting band dispersion. The

second term represents density-density interactions between

the holes with the projected density operator in one valley

given by ρτ(q) = Pkαβ λαβ

τ(k+q,k)c†

ατ (k+q)cβτ (k)

where λαβ

τ(k+q,k)≡ ⟨uατ (k+q)|uβτ (k)⟩are the form

factors calculated from the Bloch eigenstates uατ (k)of the

bands. Ncdenotes the number of moir´

e unit cells. We

model the interaction U(q)as the dual-gated Coulomb repul-

sion: U(q)=2πU0tanh(dg|q|)/(√3|q|aM)where aMis

the moir´

e lattice constant. The screening length dgrepresents

the distance between the gates and the moir´

e superlattice and

U0= 1/(4πϵϵ0aM)is the strength of interaction, ϵbeing the

dielectric constant. The Hamiltonian (1) is normal-ordered

with respect to charge neutrality. In addition, it has the fol-

lowing key symmetries: a π-rotation about an in-plane axis

—C2ythat swaps the two layers, time reversal symmetry T

that ﬂips the spin-valley degree of freedom, and U(1) ×U(1)

symmetry corresponding to independent charge conservation

within each valley.

To diagnose the possible competing phases, it is useful con-

struct a maximally layer-polarized bases for the ﬂat bands.

To this end, we deﬁne ˜c†

βτ (k) = PαUβα(k)c†

ατ (k), where

Uβα(k)is a 2×2unitary matrix that is optimized to max-

imize the layer polarization in each valley [37]. The layer

basis vectors within a single valley are related to each other

by a combined C2yTsymmetry (Fig.2(b)), thus spontaneous

ferroelectric order can be conveniently detected numerically

through C2ybreaking in the ground state.

Results. We perform an unbiased momentum space exact

diagonalization (ED) study [36] of the Hamiltonian (1) focus-

Chern

Multiferroic

Ferroelectric?

Metal

Chern

Ferroelectric?

Metal

Chern

Multiferroic

Metallic

Ferroelectric

FIG. 3. ED phase diagram on cluster C6 [36,38] for (a-b) hole-

ﬁlling νh= 1 and (c-d) hole-ﬁlling νh= 3, as a function of (i) the

ratio of the gate-induced screening length to the moir´

e lattice con-

stant dg/aM, and (ii) the inverse of the dielectric constant ϵ−1.PV

and PLare the valley polarization and layer polarization densities

respectively, as deﬁned by Eq. (2). A multiferroic phase, with both

Pv, PL̸= 0, dominates the phase diagram for νh= 1.

ing on odd integer ﬁlling νh= 1 (one hole per moir´

e unit

cell) and νh= 3 (three holes per moir´

e unit cell) [36]. To

characterize the obtained many-body ground states, we deﬁne

the following two observables:

PV=1

NeX

kα⟨nα+(k)⟩−⟨nα−(k)⟩

PL=1

NeX

kτ⟨˜n2τ(k)⟩−⟨˜n1τ(k)⟩

(2)

where nατ (k) = c†

ατ (k)cατ (k)is the αband occupation,

˜nβτ (k) = ˜c†

βτ (k)˜cβτ (k)is the βlayer occupation and Neis

the number of electrons. The quantities in Eq. (2) probe dis-

tinct properties of the many-body ground state. PVmeasures

spin-valley polarization and detects ferromagnetism, while PL

measures layer polarization and therefore probes ferroelectric-

ity. Simultaneous non-zero values for both PVand PLthus

indicate multiferroic order.

In Figs. 3(a) and 3(b), we present the phase diagram at

ﬁlling νh= 1 as a function of two experimentally tunable pa-

rameters: (i) the ratio of the screening length of the dual-gated

Coulomb interaction to the moir´

e lattice constant dg/aMand

(ii) the inverse of the dielectric constant ϵ−1. When ϵ−1is

extremely small, we ﬁnd the expected metallic phase. The

metallic phase has no spin-valley polarization (c.f Fig. 3(a))

and the many-body ground state occupation is found to be

concentrated in the lower band in each valley, indicating that

the system simply minimizes the non-interacting kinetic en-

ergy in this regime. As ϵ−1is increased, we ﬁnd ubiquitous

FIG. 4. (a) ED spectrum from cluster C6 [36] in the multiferroic

phase for ϵ−1= 0.287,dg/aM= 1 at ﬁlling νh= 1.NVlabels

the different spin-valley sectors, NV= (N+−N−). The inset

shows the low-lying manifold of states. There are 84 states below

the dashed line matching a Hilbert space dimension dimH= 2 ×

PNV≥0Nc

NV/2= 84 for Nc= 6 unit cells where the factor of 2

accounts for the two different sublattices (b) Energy splitting of the

ground states of the different phases upon the application of a small

displacement ﬁeld Vzthat explicitly breaks the C2ysymmetry.

ferromagnetic order due to spin-valley polarization, as evident

in ﬁgure 3(a). The spin-valley ferromagnetism is of the Ising

type and corresponds to spontaneous symmetry breaking of

time-reversal symmetry. In addition to ferromagnetism, we

ﬁnd the ground state in the majority of the phase diagram to

show strong layer polarization, i.e, ferroelectricity, by sponta-

neously breaking the C2ysymmetry. To characterize this, we

compute the layer polarization PLdeﬁned in Eq. (2) and ﬁnd

that it is non-zero, as shown in ﬁgure 3(b). The simultaneous

occurence of spin-valley ferromagnetism and ferroelectricity

corresponds to the multiferroic phase which is observed for

dg/aM≥0.2and ϵ−1≥0.11 [39].

Further evidence that the layer-swapping C2ysymmetry is

broken in the multiferroic phase may be gleaned from the re-

sponse of the ground states to a symmetry breaking pertur-

bation in the form of a tiny out of plane displacement ﬁeld

Vz. When Vz= 0, the many-body multiferroic ground state

manifold is two-fold degenerate (see inset of Fig. 4(a)) if we

ﬁx spin-valley polarization. The two degenerate states are re-

lated by C2y, and correspond to larger carrier density in the

top or bottom layer. The displacement ﬁeld Vzis odd under

C2y, and therefore splits this degeneracy. Accordingly, as we

increase Vz, we ﬁnd the two-fold degenerate ground states of

the multiferroic phase split linearly in Vzand ﬂow in opposite

directions as shown in Fig. 4(b) — providing comprehensive

evidence of C2ybreaking. This behavior of the multiferroic

ground state manifold stands in sharp contrast to the ground

states of the competing Chern insulator phase, that is insen-

sitive to small displacement ﬁelds. Lastly, we note that the

parts of the phase diagram where we observe layer polariza-

tion also exhibit signiﬁcantly large off-diagonal components

in the band basis [36]. This is consistent with our intuition

that the layer bases - an approximate eigenbasis for the multi-

ferroic - strongly mix the top and bottom bands. Indeed, such

band-mixing is expected for large values of ϵ−1where multi-

ferroicity is observed — interactions overcome the small band

gap between the two active bands.

Upon decreasing the screening length dg, the range of

Coulomb interaction is shortened in real space and we observe

a transition to a Chern insulator phase obtained by hole ﬁlling

one of the upper Chern band within one valley at νh= 1.

While the Chern insulator is still ferromagnetic in both orbital

(valley) and spin sectors, as indicated in Fig. 3(a), the fer-

roelectricity disappears as the Chern bands themselves have

almost equal spectral weight in both layers.

In addition to the fully symmetric metal, spin-valley po-

larized Chern insulator and multiferroic phases, we also ﬁnd

signatures of an intermediate ferroelectric phase — character-

ized by non-zero layer polarization and vanishing valley po-

larization, between the metallic and the multiferroic phases.

In contrast to the rest of the phases, we ﬁnd the existence of

this intermediate ferroelectric phase at νh= 1 to be sensitive

to the choice of the momentum space cluster used for ED [36].

Henceforth, we focus on understanding the rest of the phase

diagram, which is robust to the choice of momentum cluster.

The observed structure of the phase diagram in Fig 3(a-b)

for νh= 1, in particular the competition between the Chern

insulator and the multiferroic, can be understood from an ap-

proximate real space description of the ﬂat bands. The layer-

projected wavefunctions of the two active bands in each spin-

valley sector are found [33,34] to be localized on atomic sites

that form a triangular lattice in each layer (Fig. 1(b)). In the

top layer, these atomic sites are denoted by RM

X, as metal (M)

atoms of the top layer are locally aligned with chalcogen (X)

atoms of the bottom layer within the moir´

e unit cell. In the

bottom layer, these sites are denoted by RX

M, as the chalco-

gen (X) atoms of the top layer are locally aligned with the

metal (M) atoms of the bottom layer. Taken together, the two

inter-penetrating triangular lattices from the two layers form a

honeycomb lattice at the moir´

e lengthscale.

This lattice structure combined with the opposite Chern

C=±1bands allows the construction of a tight-binding

model that provides a realization of Kane-Mele physics [40] at

the non-interacting level. Since states in the two layers are lo-

calized on the two sublattices RM

Xand RX

M, layer polarization

implies sublattice polarization. The reason for sublattice po-

larization for long-range repulsion is quite intuitive. For large

screening lengths dg, the interaction is essentially long range

— there is signiﬁcant nearest-neighbor repulsion between the

electrons in addition to on-site repulsion. In this regime, the

electrons can minimize both on-site and nearest neighbor re-

pulsive interactions by localizing on a single triangular sub-

lattice of the effective honeycomb lattice (Fig. 1(c)), leading

to ferroelectric order.

The aforementioned real space picture is strongly reﬂected

in the many-body spectrum of the multiferroic phase (Fig.

4(a)) where we observe a low energy manifold of states sepa-

rated by a large gap from the rest of the spectrum. The number

of the low-lying states in this manifold is always consistent

with the expected dimension of a Hilbert space that is the sum

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MultiferroicityandTopologyinTwistedTransitionMetalDichalcogenidesAhmedAbouelkomsan,1,2,∗EmilJ.Bergholtz,1andShubhayuChatterjee3,41DepartmentofPhysics,StockholmUniversity,AlbaNovaUniversityCenter,10691Stockholm,Sweden2DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139...

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Multiferroicity and Topology in Twisted Transition Metal Dichalcogenides Ahmed Abouelkomsan1 2Emil J. Bergholtz1and Shubhayu Chatterjee3 4 1Department of Physics Stockholm University AlbaNova University Center 106 91 Stockholm Sweden.pdf

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Multiferroicity and Topology in Twisted Transition Metal Dichalcogenides Ahmed Abouelkomsan1 2Emil J. Bergholtz1and Shubhayu Chatterjee3 4 1Department of Physics Stockholm University AlbaNova University Center 106 91 Stockholm Sweden

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