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Exciton fractional Chern insulators in moir´
e heterostructures
Raul Perea-Causin ,1, ∗Hui Liu ,1 , †and Emil J. Bergholtz 1, ‡
1Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden
Moir´
e materials have emerged as a powerful platform for exploring exotic quantum phases. While recent
experiments have unveiled fractional Chern insulators exhibiting the fractional quantum anomalous Hall effect
based on electrons or holes, the exploration of analogous many-body states with bosonic constituents remains
largely uncharted. In this work, we predict the emergence of bosonic fractional Chern insulators arising from
long-lived excitons in a moir´
e superlattice formed by twisted bilayer WSe2stacked on monolayer MoSe2. Per-
forming exact diagonalization on the exciton flat Chern band present in this structure, we establish the existence
of Abelian and non-Abelian phases at band filling 1
2and 1, respectively, through multiple robust signatures in-
cluding ground-state degeneracy, spectral flow, many-body Chern number, and particle-cut entanglement spec-
trum. The obtained energy gap of ∼10 meV for the Abelian states suggests a remarkably high stability of this
phase. Our findings not only introduce a highly tunable and experimentally accessible platform for investigating
bosonic fractional Chern insulators but also open a new pathway for realizing non-Abelian anyons.
I. INTRODUCTION
Fractional Chern insulators (FCIs)—lattice analogs of the
fractional quantum Hall effect that remain robust in the ab-
sence of a magnetic field—hold large potential for the study
of fundamental quantum phenomena and for the develop-
ment of novel quantum technologies [1–3]. Pioneering ex-
periments [4, 5] and predictions [6–8] of FCIs in twisted van
der Waals heterostructures, followed by recent realizations
at absent magnetic field [9–12], have established moir´
e ma-
terials as an accessible and versatile platform for exploring
strongly-correlated topological phases. This breakthrough has
stimulated abundant efforts in the pursuit of exotic phases
beyond the conventional paradigm of Laughlin and hierar-
chy fractional quantum Hall states. In particular, predictions
of Moore–Read [13–18] and Read–Rezayi [19] phases host-
ing non-Abelian anyon excitations are especially promising
for fault-tolerant topological quantum computing [20]. So
far, however, research on moir´
e FCIs has focused only on
the approach of doping the system with electrons or holes—
resulting in correlated topological phases with fermionic con-
stituents and leaving their bosonic counterpart largely unex-
plored.
Excitons, i.e. Coulomb-bound pairs of conduction-band
electrons and valence-band holes [21], are obvious candi-
dates for realizing correlated bosonic phases in moir´
e mate-
rials [22, 23]. Concretely, interlayer excitons with charge-
carriers located in different layers are particularly promis-
ing due to their long lifetime, which can reach hundreds of
nanoseconds [24, 25] and even microseconds [26]. These
species typically appear in semiconducting van der Waals
heterostructures, where an optical excitation generating in-
tralayer excitons (with electrons and holes in the same layer)
is shortly followed by tunneling of either electrons or holes
∗raul.perea.causin@fysik.su.se
†hui.liu@fysik.su.se
‡emil.bergholtz@fysik.su.se
(a)
WSe2
X1
X2
WSe2
MoSe2
θ
FIG. 1. (a) Schematic illustration of twisted bilayer WSe2on top of
monolayer MoSe2. The interlayer excitons X1and X2are formed by
an electron in MoSe2and a hole in either of the two WSe2layers. (b)
Exciton band structure for ∆ = −3.8meV and θ= 1.95◦, where
the lowest band is flat and has a non-zero Chern number. The color
represents the contribution |χlQ|2from each exciton species to the
band. (c) Berry curvature ΩkABZ/2πand (d) Fubini-Study metric
tr[gk]ABZ/2πin the moir´
e Brillouin zone for the flat band. ABZ is
the Brillouin zone area.
into another layer [27]. The resulting interlayer excitons pos-
sess a permanent dipole moment, which makes them highly
tunable by an out-of-plane electric field [28, 29]. Moreover,
repulsive dipolar interactions together with the presence of
flat bands in moir´
e semiconductors lead to correlated exci-
ton physics [30–33]. In addition, the large degree of control
via twist angle, dielectric environment, and electric field can
be exploited to achieve topological bands [34] hosting long-
lived interlayer excitons [35]. Thus, moir ´
e systems enable
the realization of topological exciton flat bands, providing a
promising route for the exploration of exciton physics with
intertwined correlations and topology.
In this work, we unveil the many-body topological exciton
phases emerging in a moir´
e heterostructure. First we show
arXiv:2504.08026v1 [cond-mat.mes-hall] 10 Apr 2025
2
that, besides being topologically non-trivial, the lowest ex-
citon band in twisted bilayer WSe2stacked on monolayer
MoSe2can simultaneously be nearly flat and exhibit an al-
most ideal quantum geometry—posing this system as a strong
candidate for realizing exciton FCI phases. Employing exact
diagonalization, we show that the ground state at half filling
with contact interactions is analogous to the bosonic Laughlin
state in the lowest Landau level. Concretely, the state is char-
acterized by a twofold degeneracy as well as an approximately
zero (interaction) energy, and its nature is further confirmed
by the many-body Chern number, spectral flow, and entangle-
ment spectrum. Importantly, the Laughlin states remain robust
and exhibit a large gap (∼10 meV) when replacing the ideal-
ized contact potential by realistic long-range interactions. Fi-
nally, the many-body calculations at filling 1, again consider-
ing long-range interactions, provide compelling evidence for a
stable bosonic version of the non-Abelian Moore–Read state.
Overall, our findings predict the existence of exciton FCIs in
an experimentally accessible moir´
e heterostructure and pave
the way for the realization of Abelian and non-Abelian topo-
logical phases with bosonic constituents in moir´
e materials.
II. MODEL
A. Topological exciton flat band
We consider a van der Waals heterostructure consisting of a
twisted WSe2bilayer stacked on top of a single MoSe2layer.
An optical excitation and subsequent tunneling result in the
formation of long-lived interlayer excitons composed of a hole
in either of the two WSe2layers and an electron in MoSe2, cf.
Fig. 1(a). We assume spin-valley polarization of electrons and
holes, which can be achieved by a circularly-polarized optical
excitation if the valley decoherence time is sufficiently long
and, in some cases, occurs spontaneously due to interactions.
In order to describe moir´
e excitons in this structure, we follow
the procedure introduced in Ref. [35].
First, we set up the exciton basis, |XlQ⟩=
Pkϕlke†
keh†
lkh|0⟩, which accounts for the binding of
an MoSe2electron to a WSe2hole in the layer l= 1,2.
Details of the exciton basis, including the relation between
the electron/hole momentum ke/h and the relative and center-
of-mass momenta kand Qcan be found in Appendix A 1.
The exciton’s wave function ϕlkand binding energy Ex
l
are obtained by solving the Schr¨
odinger equation for an
electron–hole pair interacting via the interlayer 2D Coulomb
potential. Here, we describe electrons and holes in an
effective-mass approximation, where the exciton eigenvalue
problem takes the form of the Wannier equation. In this
case, ϕlkis well described by the 1s wave function of 2D
hydrogen, and variational minimization considering material-
specific parameters yields the exciton binding energies
Ex
1= 230 meV, E x
2= 140 meV in close agreement with the
exact solutions.
Next, we incorporate the effect of the moir´
e potential and
tunneling that holes in the WSe2layers experience, which is
described by the moir´
e exciton Hamiltonian [35, 36]
Hx,m =X
ll′Qq
Mll′QqX†
lQ+qXl′Q,(1)
where mixing with higher exciton states (e.g. 2p) is ex-
pected to be weak and has therefore been disregarded. We
note that this approach naturally accounts for the mixing be-
tween all moir´
e hole bands forming the exciton state. Here,
MllQq =Ex
lQδq,0+Fll,αeqUqcontains the exciton disper-
sion Ex
lQ=Egap
l+Ex
l+ℏ2Q2/2Mxwith the exciton mass
Mx, the hole moir´
e potential Uq=U0P6
n=1 eiσnlφδq,gn
with σnl = (−1)n+l−1,gn=Cn
6g0,g0= 4π/√3amˆx,
the moir´
e lattice constant amand the exciton form factor
Fll′,q=Pkϕ∗
lkϕl′k+q. Tunneling between opposite lay-
ers is contained in Mll′Qq =Fll′,αeqTq(l=l′), where
Tq=T0P3
n=1 δq,κn,κn=Cn
3(K2−K1)and Klis the K-
point in layer l. The relevant parameters for holes in twisted
bilayer WSe2are φ= 128◦,U0= 9 meV, and T0= 18
meV [35]. Diagonalizing Eq. (1) yields the band structure,
which in this system is characterized by topological bands for
a specific range of the twist angle and the offset between the
two exciton species, ∆ = Ex
2−Ex
1. The latter can be experi-
mentally controlled by an out-of-plane electric field.
In this work, we consider the twist angle θ= 1.95◦and the
offset ∆ = Ex
2−Ex
1= 3.8meV, where the lowest band is flat
and has a Chern number C= 1, cf. Fig. 1(b). The non-zero
Chern number can be understood in terms of a pseudospin,
describing the superposition between the two interlayer exci-
ton species |X1Q⟩and |X2Q⟩, which wraps around the Bloch
sphere once as Qtraverses the moir´
e Brillouin zone. The
varying pseudospin is encoded in the color of the flat band in
Fig. 1(b), where the weight of the state |XlQ⟩, i.e. |χlQ|2with
χlQbeing the moir´
e exciton eigenfunction, is shown. Note
that χlQis a vector where each element accounts for the mo-
mentum Q+gwith gpointing to the outer moir´
e cells up to
a certain cutoff. Interestingly, the quantum geometry of the
band is nearly ideal [37, 38], i.e. tr[gk]≈ |Ωk|where gkis
the quantum (Fubini–Study) metric and Ωkis the Berry cur-
vature, cf. Fig. 1 (c), (d). This property strongly suggests the
emergence of zero-energy ground states at even-denominator
filling of the band with pseudopotential interactions—in anal-
ogy to Laughlin states of bosons in Landau levels, albeit now
in the absence of a magnetic field. A nearly ideal quantum
geometry was also found to be the precursor of electron FCIs
in twisted bilayer graphene [8, 39] despite the fact that fluctu-
ations of the metric induce new competing states [40].
B. Many-body exciton Hamiltonian
The aim of this work is to unveil the phases emerging in
a many-body system of excitons in the topological moir´
e flat
band. To that end, we set up the many-body exciton Hamilto-
nian, Hx=Hx,m +Hx-x, where
Hx-x =1
2X
ll′q
Vll′(q) :ρl(q)ρl′(−q) : (2)
3
describes the exciton-exciton interaction, ρl(q) =
PQX†
lQ+qXlQis the exciton density operator, :: denotes
normal order, and
Vll′(q) =Ve-e
ll′(q) + Vh-h
ll′(q)−Ve-h
ll′(q)−[Ve-h
l′l(−q)]∗
Ve-e
ll′(q) =Fll′(αhq, αhq)e2
0
2Aϵϵ0|q|
Vh-h
ll′(q) =Fll′(αeq, αeq)e2
0
2Aϵϵ0|q|e−|q|d|l−l′|
Ve-h
ll′(q) =Fll′(αeq, αhq)e2
0
2Aϵϵ0|q|e−|q|dl (3)
is the interaction potential, which contains individual contri-
butions accounting for interactions between the charges in dif-
ferent excitons, and where Fll′(q1,q2) = Fll,q1Fl′l′,−q2has
been introduced. The derivation of the exciton–exciton inter-
action potential is outlined in Appendix A 2. Here, we have
focused on the direct interaction, which at long distances be-
haves as the interaction between two dipoles with lengths dl
and dl′, i.e. Vll′(r)∼(dl)(dl′)/|r|3. Exchange interactions
typically play a minor role in the context of interlayer exci-
tons [41–44].
Importantly, the exciton operators X†
Qand XQobey
bosonic commutation relations. Corrections to the bosonic
character arising from the exciton’s fermionic substructure are
typically expected in the regime where the distance between
neighboring excitons is comparable to the size of an exci-
ton [45], i.e. nxa2
B∼1where nxis the exciton density and
aBthe exciton Bohr radius. In typical moir´
e systems where
there is one exciton per moir´
e site, nxa2
B∼0.01 and thus the
bosonic description is appropriate.
The many-body problem of interacting excitons is solved
via exact diagonalization. Concretely, we project the exciton
Hamiltonian Hxinto the topological flat band and we diag-
onalize Hxin a finite-size system consisting of Nxexcitons
in Nsmoir´
e sites such that the band filling is ν=Nx/Ns.
We distinguish exciton FCIs from competing orders by as-
sessing the ground state degeneracy, many-body Chern num-
ber (defined in Appendix B 1), and quasi-hole excitations.
The latter can be accessed via the particle-cut entanglement
spectrum (PES) [46, 47], where the many-body system is
partitioned into NAand NB=Nx−NAparticles. The
set of eigenvalues {ξ}of −logρAconsitute the PES, where
ρA=trB[1
NdPNd
i|Ψi⟩⟨Ψi|]is the reduced density matrix
of the subsystem A,|Ψi⟩is the i-th ground state, and Ndis
the ground state degeneracy. Writing the many-body ground
state as |Ψ⟩=Pjexp(−ξj/2) |ΨA
j⟩⊗⟨ΨB
j|, one can see that
lower values of ξjindicate a large weight of a specific sub-
space configuration |ΨA
j⟩—meaning that such configuration
is allowed. Thus, the PES probes quasi-hole excitations that
obey the generalized statistics of a specific quantum phase.
III. ABELIAN STATES AT HALF FILLING
Based on the criteria for how closely a moir´
e band can
mimic a Landau Level—(i) nontrivial topology, (ii) flat dis-
persion, and (iii) ideal quantum geometry [3, 37, 38]—the ex-
citon flat band shown in Fig. 1(b) appears to be an excellent
candidate for realizing FCI phases analogous to the fractional
quantum Hall effect of bosons. In this context, Laughlin states
at half filling constitute a prototypical phase [48, 49]. They
emerge as exact zero-energy states in a system of bosons inter-
acting via a delta-function potential (i.e. contact interaction)
in the lowest Landau level and their elementary excitations
behave as Abelian anyons. We note that bosonic Laughlin
states have also been proposed in a different setting, where
the exciton-like bosons correspond to low-energy excitations
of a fully-filled valley-polarized electron band [50–52]. In the
following, we provide numerical evidence showing that the
nearly ideal exciton Chern band indeed supports zero-energy
Laughlin states for contact interactions. Subsequently, we will
demonstrate that this FCI phase survives when considering the
realistic long-range interactions introduced in Eq. (3).
In order to strengthen the ideal conditions of the system
and establish the emergence of Laughlin states in the ex-
citon Chern band, we initially consider contact interactions
Vll′(q) = V0and disregard the impact of the kinetic en-
ergy. With these assumptions, exact diagonalization yields
two many-body ground states with approximately zero energy,
cf. Fig. 2(a). Importantly, the ground states are separated
from excited states by a large energy gap, which reflects the
expected topological protection. The twofold degeneracy and
the total momentum of each state are characteristic of Laugh-
lin states and can be understood with the aid of Landau level
physics in the thin-torus limit [53]. In particular, the exclu-
sion principle that arises in the thin-torus limit for bosons in
the lowest Landau level dictates that a zero-energy state con-
tains, at most, one particle in two consecutive sites [54]. The
two states that fulfill this principle and therefore constitute
the two degenerate ground states are those with a Fock-space
configuration 101010 ·· · and its translational-invariant part-
ner 010101 ·· ·, whose total momentum matches that of our
numerical ground states. Furthermore, upon threading a mag-
netic flux (corresponding to twisted boundary conditions), the
ground states evolve into each other after one flux quantum
and return to their original states after the insertion of two flux
quanta (Fig. 2(b)), indicating that the two states are adiabati-
cally connected and that the Hall conductivity is quantized to
1
2. The latter aspect is further confirmed by a direct calculation
of the many-body Chern number, which yields Cavg =1
2for
each ground state and arises from a homogeneous many-body
Berry curvature, cf. Fig. 2(d).
Next, we calculate and analyze the PES, which reveals
the nature of the elementary quasi-hole excitations of ground
states. In particular, the states with lowest eigenvalues in the
PES typically correspond to hole excitations that fulfill the
generalized exclusion principle and therefore serve as a fin-
gerprint for distinguishing a specific phase from competing
orders. In Fig. 2(c), we display the PES where the subsys-
tem Ais taken to consist of NA= 4 particles. Here, states
with low eigenvalues are well separated from higher eigen-
values by a large entanglement gap. Importantly, the number
of states below the gap exactly matches the analytical count-
ing of quasi-hole excitations in the ν=1
2Laughlin states.
4
0
5
10
E
/
V
0
avg
= 1/2
(a)
Contact
6
3
0
3
E
/
V
0
E
0
(b) x10
10
20
30
PES (
NA
= 4)
(c)
0.0
0.5
1.0
Flux
y
/0
(d)
0 6 12 18
Momentum sector
0
10
20
E E
0 (meV)
avg
= 1/2
(e)
Long-range
012
Flux / 0
5
0
5
E E
0 ( eV)
(f)
0 6 12 18
Momentum sector
10
20
30
PES (
NA
= 4)
(g)
0.0 0.5 1.0
Flux
x
/0
0.0
0.5
1.0
Flux
y
/0
(h)
0.498
0.500
0.502
Berry curvature
0.498
0.500
0.502
Berry curvature
FIG. 2. Laughlin states at half filling. (a) Many-body energy spectrum containing the 10 lowest energies for each momentum sector, (b) spectral
flow, (c) particle-cut entanglement spectrum, and (d) many-body Berry curvature for the two ground states considering contact interactions
without kinetic energy effects at ν=1
2. The respective data considering the long-range interaction Vll′(q)and the kinetic energy is shown in
(e)-(h). The ground states in (a) and (e) are marked in red and have an average many-body Chern number Cavg = 1/2. In the PES, the number
of states below the first entanglement gap (denoted by the red solid line) is 1287, matching the number of quasi-hole excitations in the ν=1
2
Laughlin states. The considered system has Ns= 18 moir´
e sites and its spanning vector can be found in the Appendix B 2
Thus, the many-body ground state degeneracy, spectral flow,
many-body Chern number, and PES all taken together clearly
confirm the emergence of a stable FCI phase of excitons in the
half-filled moir´
e Chern band.
We now consider the realistic long-range exciton–exciton
interactions introduced in Eq. (3) and take into account the im-
pact of the small but finite kinetic energy. The results of exact
diagonalization and the subsequent analysis are remarkably
similar to those obtained for contact interactions, cf. Fig. 2(e)-
(h). The fact that the energy gap between ground and excited
many-body states (∼10 meV) persists for the considered sys-
tems ranging from Ns= 10 up to Ns= 20 sites clearly re-
flects that such phases are robust (see Appendix C 1). Thus,
our numerical calculations demonstrate that exciton FCIs are
stable in the considered moir´
e heterostructure.
Interestingly, the calculated gap of ∼10 meV is two times
larger than in the case of hole FCIs in the similar system
of twisted bilayer MoTe2[55]. In principle, this suggests
a higher stability of exciton FCIs compared to their elec-
tron/hole counterparts, whose gap has been experimentally
estimated to be 20 K [56]. The potentially larger gap of ex-
citon FCIs might be a result of the stronger repulsive inter-
action dominated by Ve-e +Vh-h in the short range (Ve-h is
weaker since the charges are located in separate layers) in-
stead of just Ve-e or Vh-h. A larger gap for bosons can also be
expected from a pseudopotential perspective, where the inter-
action for bosons and fermions is dominated by the zeroth, v0
(δ(r)component of the interaction) and first v1(−a2
m∇2δ(r)
component) pseudopotentials, respectively, with the bosonic
interaction being stronger than the fermionic one (generally
vn> vn+1 in lowest Landau level-like bands) [8, 49]. We
note, though, that a reliable quantitative estimation of the gap
is difficult due to limitations such as the finite system size,
the use of a pure Coulomb potential instead of the thin-film
Rytova-Keldysh potential [21], and uncertainties in parame-
ters such as the dielectric constant.
IV. NON-ABELIAN STATES AT FILLING ONE
After confirming the presence of Abelian exciton FCIs,
we now seek exotic phases whose elementary excitations
obey non-Abelian anyon statistics. In this context, the most
straightforward and promising candidate is the Moore–Read
state [57], which emerges as the exact zero-energy ground
state at filling ν= 1 for bosons in the lowest Landau level
with artificial three-body contact interactions [58] and which
was predicted to appear in rotating Bose-Einstein conden-
sates [59, 60]. In the following, we show that this phase is
stable in the realistic conditions considered here, i.e. in the
moir´
e band and assuming long-range interactions.
First, we note that the most prominent feature of the
Moore–Read phase is the dependence of the ground state de-
generacy on the parity of the particle number. Here, the thin-
torus exclusion principle [61] states that, at most, two par-
ticles can occupy two consecutive orbitals [54, 62], result-
ing in three possible Fock-space configurations, 111111 ·· ·,
202020 ·· ·, and 020202 ···. As a result of periodic bound-
ary conditions, only the first configuration is allowed for an
odd number of particles—resulting in a single ground state—,
while for an even number of particles all three configurations
are allowed and the ground state is threefold degenerate.
In Fig. 3(a),(c), we show the calculated many-body spectra
for systems with 13 and 14 excitons. The single and three
lowest states in the case of odd and even number of exci-
tons, respectively, are located at the momenta expected from
5
0
10
20
E E
0 (meV)
= 1
(a)
13 particles
4
6
8
10
PES (
NA
= 3)
(b)
0 6 12
Momentum sector
0
10
20
E E
0 (meV)
avg
= 1
(c)
14 particles
0 6 12
Momentum sector
4
6
8
10
PES (
NA
= 3)
(d)
FIG. 3. Moore–Read states at filling one. Many-body spectrum (10
lowest energies) and PES at ν= 1 with long-range interactions for
(a)-(b) 13 and (c)-(d) 14 particles. The single and threefold quasi-
degenerate ground states (red dots in panels (a) and (c)) for an odd
and even number of sites are located at the momenta expected for
Moore–Read states. The red line in panels (b) and (d) indicates
the entanglement gap respecting the quasi-hole counting rules of the
Moore–Read state (416 and 518 quasihole excitations for 13 and
14 particles with NA= 3). PES for larger NAare shown in Ap-
pendix C 2.
the thin-torus exclusion rule for Moore–Read states. We note
that the spread of the ground state energies and the small gap
with respect to excited states is a common feature of finite-
size systems. We expect the states to become exactly degen-
erate in the thermodynamic limit and note that the ground and
excited states remain well separated upon threading magnetic
flux (see Appendix C 2). Moreover, the calculation of many-
body Chern number yields Cavg = 1 for each state and re-
veals a uniform Berry curvature distribution, indicating that
the Moore–Read phase is robust.
In order to unambiguously determine the nature of the
phase beyond the parity dependence of the ground-state de-
generacy, we calculate the PES. For both Nx= 13 and
Nx= 14 systems, the PES is characterized by a large gap, be-
low which the number of states exactly matches the counting
of quasi-hole excitations allowed by the Moore–Read exclu-
sion rule. Furthermore, the PES gap persists across various
system sizes and different NA(see Appendix C 2). Taken to-
gether, the ground-state degeneracy and momenta, the many-
body Chern number, and the PES for systems with different
particle-number parity constitute convincing evidence for the
presence of a robust non-Abelian Moore–Read phase with ex-
citon constituents at filling ν= 1 in the considered moir´
e
heterostructure.
V. CONCLUSION AND OUTLOOK
We have explored the emergence of strongly-correlated
topological phases arising from a many-body bosonic system
of long-lived moir´
e excitons in a flat Chern band. In partic-
ular, by combining exact diagonalization with many-body di-
agnosis tools, we have demonstrated the existence of robust
Abelian and non-Abelian exciton fractional Chern insulators
in a realistic model of a moir´
e heterostructure at filling ν=1
2
and ν= 1, respectively. Our work introduces a versatile and
accessible platform for investigating exciton FCIs and opens
a new avenue towards the realization of non-Abelian anyons.
Importantly, our findings provide a specific guideline for
the experimental realization of exciton FCIs in twisted bilayer
WSe2stacked on monolayer MoSe2. We note that, despite
the exciton’s neutral net charge, there are methods which en-
able exciton transport that could be utilized to detect these
phases. The most promising route is likely to be counterflow
transport measurements [63, 64]. Other methods for studying
transport of interlayer excitons involve creating spatial gradi-
ents in the out-of-plane electric field, dielectric environment,
or strain profiles [65, 66]. Future studies should also attempt
to identify the optical fingerprints of these phases.
Apart from prompting experimental efforts in a specific sys-
tem, our work motivates the search for exciton FCIs in other
moir´
e structures. Furthermore, while we have only focused on
two specific phases at fillings ν=1
2and ν= 1, other highly
nontrivial phases remain unexplored. Besides other bosonic
Laughlin and hierarchy states, the pursuit of additional non-
Abelian phases such as the Moore–Read state at ν=1
3and the
Read–Rezayi state [67] at ν=3
2is particularly interesting—
especially so for the latter as it hosts Fibonacci anyons which
hold large promise for topological quantum computing.
Finally, we note that many challenging aspects are yet to
be addressed. For instance, investigating the competition be-
tween liquid- and crystal-like exciton FCIs as well as super-
fluids and supersolids would be crucial for the understand-
ing of these phases [68, 69]. Such phase transitions could be
studied by adding hBN spacers between the electron layer and
the hole bilayer to enhance the interaction strength (by reduc-
ing the electron–hole attraction). In addition, while the spin-
valley physics in the many-body exciton system is at the edge
of current numerical capabilities, it is an important issue that
must be tackled. Moreover, theoretical efforts exploring large
filling factors might demand more advanced models consider-
ing deviations from the bosonic description of excitons [70].
Last but not least, the phase diagram can be enriched by
adding doping, which in combination with excitons consti-
tutes a realization of mixed Bose–Fermi physics [33, 71, 72].
Our work shows that an idealized model can provide a faith-
ful description of moir´
e exciton FCIs, suggesting a route to
approach these formidable problems.
ACKNOWLEDGMENTS
We acknowledge useful discussions with Daniel Erkensten,
Zhao Liu, Ahmed Abouelkomsan and Atac¸ ˙
Imamo˘
glu. This
work was supported by the Swedish Research Council (VR,
grant 2024-04567), the Wallenberg Scholars program of the
Knut and Alice Wallenberg Foundation (2023.0256) and the
G¨
oran Gustafsson Foundation for Research in Natural Sci-
6
ences and Medicine. The computations were enabled by
resources provided by the National Academic Infrastructure
for Supercomputing in Sweden (NAISS), partially funded by
the Swedish Research Council through grant agreement no.
2022-06725. In addition, we utilized the Sunrise HPC facil-
ity supported by the Technical Division of the Department of
Physics, Stockholm University.
Appendix A: Moir´
e exciton model
1. Exciton basis
Electrons and holes in each layer can be described in an
effective mass approach at the K valley with the Hamil-
tonian He-h =He,0+Hh,0+He-h, int, where He,0=
PkEe
ke†
kek,Hh,0=PlkEh
kh†
lkhlk, and He-h, int =
−Plkk′qVe-h
lqe†
k+qh†
lk′−qhlk′ek. Here, Ee
k=ℏ2k2/2me,
Eh
k=Egap
l+ℏ2(k−Kl)2/2mh,Klis the K-point for holes at
the layer l= 1,2and Ve-h
lq=e2
0
2Aϵϵ0|q|e−|q|dl is the electron-
hole interaction potential with the interlayer distance d, the
dielectric constant ϵ, and the system area A. Considering that
all electrons and holes are paired into tightly bound excitons,
the system can be described in terms of the 1s exciton state,
|XlQ⟩=Pkϕlke†
keh†
kh|0⟩, where |0⟩is the semiconductor
ground state, ϕlkis the exciton wave function with the relative
momentum k=αhke−αe(kh−Kl),αe/h =me/h/(mh+me),
and Q=kh−Kl+keis the center-of-mass momentum. The
exciton dispersion Ex
lQ=Egap
l+Ex
l+ℏ2Q2
2Mxis determined
by the mass Mx=mh+meand the offset energy Ex
lthat ful-
fills the eigenvalue problem He-h |XlQ⟩=Ex
lQ|XlQ⟩, which
corresponds to the Wannier equation
ℏ2k2
2µx
ϕlk−X
q
Ve-h
lqϕl,k+q=Ex
lϕlk,(A1)
where µx=memh/Mxis the reduced mass. The exciton
wave function is well approximated by a hydrogen ansatz,
which in real space reads ϕl(r)∝exp(−|r|/al). Taking
mh= 0.43m0, me= 0.8m0,d= 0.67 nm and ϵ= 3.8, [35]
we obtain the exciton binding energies Eb
1= 230 meV, Eb
2=
140 meV and Bohr radii a1= 1.3nm, a2= 1.8nm via varia-
tional minimization, in close agreement with the exact numer-
ical solution of Eq. (A1).
2. Exciton–exciton interaction
We extend the method employed to derive the single-
particle moir´
e exciton Hamiltonian [35] to obtain the Hamilto-
nian describing exciton–exciton interactions, which takes the
form
Hx-x =1
2X
QQ′q
V(q)X†
Q+qX†
Q′−qXQ′XQ
V(q) = ⟨XQ+qXQ′−q|Hint |XQ′XQ⟩.
For the sake of simplicity, we omit the layer indices and note
that the generalization to the multilayer system is straight-
forward. The matrix element V(q)contains the interaction
Hamiltonian in the electron–hole picture,
Hint =He-e, int +Hh-h, int +He-h, int
He-e, int =1
2X
kk′q
Ve-e(q)e†
k+qe†
k′−qek′ek
Hh-h, int =1
2X
kk′q
Vh-h(q)h†
k+qh†
k′−qhk′hk
He-h, int =−X
kk′q
Ve-h(q)e†
k+qh†
k′−qhk′ek,
The matrix element V(q)thus has contributions from differ-
ent interaction mechanisms. Here, we focus on the contri-
bution arising from electron–electron interactions, He-e, int, to
illustrate the procedure. First, we expand the exciton states
into the electron–hole basis and exploit the fermionic com-
mutation relations together with ek|0⟩=hk|0⟩= 0 until all
electron and hole operators in the matrix element disappear,
obtaining
V(q)|e-e =X
ki
ϕ∗
k1ϕ∗
k2ϕk3ϕk4(A−B)C
A=δ−k2−αhq,−k3δ−k1+αhq,−k4
B=δ−k2+αh(Q′−q),−k4+αhQδ−k1+αh(Q+q),−k3+αhQ′
C=⟨0|ek1+αe(Q+q)ek2+αe(Q′−q)He-ee†
k3+αeQ′e†
k4+αeQ|0⟩,
where i={1,2,3,4}. Next, we calculate C, again by making
use of ek|0⟩=hk|0⟩= 0 and the commutation relations,
and evaluating the Kronecker deltas. The resulting expression
reads
V(q)|e-e =V(q)|e-e,1 −V(q)|e-e,2 −V(q)|e-e,3 +V(q)|e-e,4
V(q)|e-e,1 =X
kk′
ϕkϕk′ϕ∗
k+αhqϕ∗
k′−αhqVe-e(q)
V(q)|e-e,2 =X
kk′
ϕkϕk′ϕ∗
k+αhqϕ∗
k′−αhq
×Ve-e(k+q−k′+αe(Q−Q′))
V(q)|e-e,3 =X
kk′
ϕkϕk′ϕ∗
k+αh(Q+q−Q′)ϕ∗
k′−αh(Q+q−Q′)
×Ve-e(k+q−k′+αh(Q−Q′))
V(q)|e-e,4 =X
kk′
ϕkϕk′ϕ∗
k+αh(Q+q−Q′)ϕ∗
k′−αh(Q+q−Q′)
×Ve-e(Q+q−Q′),
where each term accounts for direct interaction, electron–
electron exchange, hole–hole exchange, and exciton–exciton
exchange, respectively [43]. For interlayer excitons, the direct
term V(q)|e-e,1 dominates [41] and therefore we disregard the
remaining terms. Applying this method to the hole–hole and
electron–hole interaction terms yields the exciton–exciton in-
teraction potential in Eq. (3).
7
Appendix B: Additional details of numerical methods
1. Many-body Chern number
In order to further assess the topological nature of ground
states, we calculate the many-body Chern number, which
takes the form
Ci=i
2πZT2
dΦxdΦy⟨∂Ψi
∂Φx|∂Ψi
∂Φy⟩ − c.c.,(B1)
with (Φx,Φy)and |Ψi⟩being the inserted magnetic flux and
the many-body ground state i, respectively. Here, the integral
is evaluated over the 2π×2πtorus (T2). We calculate the
Chern number for all many-body ground states and obtain the
average many-body Chern number Cavg for each state [73].
2. Finite system geometry
Throughout this work, we consider finite systems (with pe-
riodic boundary conditions), the geometry of which is de-
termined by the spanning vectors Tn·(a1,a2),n= 1,2,
with T1= (nx1, ny1)and T2= (nx2, ny2)and where
a1,a2are the moir´
e lattice vectors. The system encloses
Ns=|nx1ny2−nx2ny1|moir´
e unit cells [74]. To achieve a
uniform sampling of the momentum points in the moir´
e Bril-
louin zone, we select the spanning vectors for each system
size as follows:
•Ns= 10:T1= (3,1),T2= (−1,3).
•Ns= 11:T1= (3,−2),T2= (1,3).
•Ns= 12:T1= (4,0),T2= (0,3).
•Ns= 13:T1= (3,−2),T2= (2,3).
•Ns= 14:T1= (4,2),T2= (−3,2).
•Ns= 16:T1= (4,0),T2= (0,4).
•Ns= 18:T1= (4,1),T2= (−2,4).
•Ns= 20:T1= (5,0),T2= (0,4).
Appendix C: Supporting numerical evidence
1. Abelian states at half filling
The observed signatures for Laughlin states at half filling
remain robust across the wide range of system sizes explored.
In Fig. 4(a), we show that the two ground states appear in the
energy spectra also for the largest system size that is accessi-
ble in our numerical calculations, Ns= 20. These states are
present in smaller systems as well. Importantly, the gap with
respect to the excited states remains robust (∼10 meV) for
all the studied systems ranging from Ns= 10 up to Ns= 20
sites, cf. Fig. 4(b).
0 5 10 15 20
Momentum sector
0
10
20
E E
0 (meV)
(a)
gap
0.00 0.05 0.10
1/
Ns
0
5
10
15
Gap (meV)
(b)
FIG. 4. Additional data for Abelian states at half filling with
long-range interactions. (a) Many-body energy spectrum (10 low-
est states) for Ns= 20 sites. The two degenerate ground states
are marked in red, while the blue arrow denotes the energy gap.
(b) Scaling of the energy gap including calculations for Ns=
10,12,14,16,18,20 sites.
0.0 0.5 1.0
Flux / 0
0
5
10
15
E E
0 (meV)
(a) K=1
0.0 0.5 1.0
Flux / 0
0
5
10
15
E E
0 (meV)
(b) K=8
0 6 12
Momentum sector
5
10
15
PES
(c)
NA
= 4
0 6 12
Momentum sector
10
20
PES
(d)
NA
= 5
FIG. 5. Additional data for non-Abelian states at filling one with long
range interaction in a system of Nx= 14 excitons. (a)-(b) Spectral
flow for the momentum sectors K= 1 and K= 8, respectively,
showing that the ground states remain separated from excited states.
(c)-(d) PES for the three ground states with NA= 4 and NA=
5, respectively. The number of states below the red line in (c) and
(d) is 1848 and 4942, respectively, corresponding to the quasi-hole
excitation counting for Moore–Read states. The inset in (d) displays
a zoomed-in region in the PES showing the sparse distribution of
states where the entanglement gap is expected.
2. Non-Abelian states at filling one
In the case of Moore–Read states at filling 1, the gap be-
tween ground and excited states for an even number of exci-
tons (Nx= 14) is small. Nevertheless, we show in Fig. 5(a)-
(b) that ground and excited states remain well separated upon
8
0
10
20
E E
0 (meV)
(a)
11 particles
4
6
8
10
PES (
NA
= 3)
(b)
0 6 12
Momentum sector
0
10
20
E E
0 (meV)
(c)
12 particles
0 6 12
Momentum sector
4
6
8
10
PES (
NA
= 3)
(d)
FIG. 6. Moore–Read states at filling one for systems with Nx= 11
and Nx= 12 excitons. Many-body spectrum (10 lowest energies)
and PES at filling ν= 1 of the exciton Chern band with long-
range interactions for (a)-(b) 11 and (c)-(d) 12 particles. The num-
ber of states below the red line matches the quasi-hole counting for
the Moore–Read state (253 and 328) for 11 and 12 particles with
NA= 3.
threading a magnetic flux. In addition, in Fig. 5(c)-(d) we
show that the presence of a PES gap matching the count-
ing of quasi-hole excitations is a robust feature also for a
higher number of particles in the Asub-system, NA= 4,
and is still present for NA= 5, although less pronounced.
In Fig. 6 we show that the characteristic features of Moore–
Read states are present not only in the largest systems studied
(Nx= 13,14) but also in systems with a lower number of
excitons (Nx= 11,12 is shown here).
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