# Moiré-Bose-Hubbard model for interlayer excitons in twisted transition metal
dichalcogenide heterostructures

Niclas Götting  Frederik Lohof  Christopher Gies  Institute for Theoretical
Physics, University of Bremen, Bremen, Germany

###### Abstract

In bilayers of semiconducting transition metal dichalcogenides, the twist
angle between layers can be used to introduce a highly regular periodic
potential modulation on a length scale that is large compared to the unit
cell. In such structures, correlated states can emerge, in which excitons in
the heterostructure are strongly localized to the potential minima due to
exciton-exciton interactions. We explore the transition between Mott and
extended exciton phases in terms of a moiré-Bose-Hubbard Hamiltonian. Hopping
and on-site interaction parameters are obtained from a Wannier representation
of the interlayer-exciton wave functions, and a non-local Rytova-Keldysh model
is used to attribute for the dielectric screening of excitons in the two-
dimensional material. For sufficiently small exciton concentrations and
substrate screening our model predicts the emergence of Mott-insulating
states, establishing twisted TMD heterostructures as possible quantum
simulators for bosonic many-body systems.

moiré-Bose-Hubbard physics, moiré heterostructures, Bose-Hubbard model,
interlayer excitons, correlated exciton states

##  I Introduction

Figure 1: (a) Real space lattices of top and bottom monolayers. The periodic
variation of the stacking registry defines the moiré lattice vectors
$\mathbf{a}^{\text{M}}_{i}$ that span the moiré unit cell. (b) Illustration of
the moiré heterostructure. The two constituting TMD monolayers are embedded
between top and bottom dielectric materials. (c) Schematic arrangement of the
monolayer band gaps to form a type-II band alignment. Generation of electron-
hole pairs, fast interlayer charge transfer, and IX formation are indicated.
(d) A relative twist of the monolayer Brillouin zones allows the construction
of a subordinate moiré lattice and corresponding unit cell. The moiré lattice
vectors
$\mathbf{G}^{\text{M}}_{i}=\mathbf{b}_{\mathrm{b},i}-\mathbf{b}_{\mathrm{t},i}$
are generated from differences of lattice vectors of the top and bottom
layers.

Twisted heterobilayers of transition metal dichalcogenides (TMDs) offer unique
opportunities to study many-body and quantum phenomena, such as strongly
correlated states of matter and unusual topological phases [1, 2]. The twist
angle between two adjacent layers of a heterostructure gives rise to
periodically changing stacking order – a so-called moiré pattern (Fig. 1(a)) –
resulting in a regular variation of the band gap due to different interlayer
hybridization strengths [3, 4, 5, 6]. Charge carriers and excitonic complexes
confined to the bilayer material experience an effective potential with a
periodicity of the arising superlattice, which, for small twist angles, is
much larger than the unit cell size of the individual monolayers. Crucially,
the moiré superlattice defines a periodic array of potential minima, in which
charge carriers can be trapped and interact with each other – a behavior
reminiscent of Hubbard systems [7, 8]. Only recently, studies on gated van der
Waals heterostructures have demonstrated the extent of this analogy by
revealing correlated hole-Mott phases at integer and fractional filling
factors of charge carriers, i.e. a realization of an effective Fermi-Hubbard
system [9, 10]. Theoretical models have also predicted topological phases,
charge density waves, and generalized Wigner crystals [11, 12, 13, 14].

While the Fermi-Hubbard physics of twisted moiré heterostructures is still in
its infancy, even less studied is the Bose-Hubbard physics of moiré excitons
and other bosonic complexes. In TMD heterobilayers with type-II band
alignment, intra- and interlayer excitons can exist [15, 16]. We focus on
interlayer excitons (IX), which are energetically favorable and dominate the
emission properties. They exhibit strong dipole alignment due to the
separation of electrons and holes in the two constituting planes of the
bilayer material. The strong Coulomb interaction in atomically thin materials
gives reason to treat IXs as composite bosonic particles, so that their mutual
interaction in the presence of the regular moiré potential realizes a
generalized Bose-Hubbard (BH) system. On the one hand, TMD moiré systems may
thus serve as a promising semiconductor platform for simulating many-body
effects and quantum phase transitions in addition to dilute ultracold atomic
gases in optical lattices [17, 18]. On the other hand, predictions of the
moiré-Bose-Hubbard model can provide insight into the widely unexplored phase
boundaries of IX. First studies have already investigated the possibility of
superfluid phase formation in TMD heterobilayers [19, 20], exploring the
transition between superfluid and Mott state. Realizations of BH systems have
also been studied with IX in gated double quantum-well structures [21, 22],
where the lattice potential was realized by patterned, externally applied
electric fields. In comparison, in twisted TMD heterobilayers the moiré
potential emerges naturally from the varying stacking order in the twisted
material layers, where the degree of localization can be altered via the twist
angle.

In the present work, we start from an effective description of IX in twisted
heterobilayers and calculate the characteristic on-site and hopping parameters
directly from the IX wave functions in a localized Wannier basis. It is well
known that excitons in atomically thin van der Waals layers are highly
susceptible to screening by the dielectric environment [23, 24, 25, 26]. We
account for dielectric screening by using a non-local Rytova-Keldysh potential
[27, 28, 29, 30] and explore the effect of different dielectric environments
on the phase boundary between the Mott and superfluid state of IXs in twisted
TMD heterobilayers for the typical material combinations MoS2/WS2 and
MoSe2/WSe2 [31, 4, 3, 16, 15]. We begin by discussing the emergence of the
moiré potential and its influence on the energy structure of IXs in Section
II, while in Section III we introduce a single-band Wannier basis to map the
IX’s dynamics and their dipolar interaction onto a generalized BH model,
extracting hopping amplitudes as well as on-site and nearest-neighbor
interaction parameters. Section IV lays out the model used to account for
screening of the dipolar interaction due to the dielectric environment.
Finally, in Section V we discuss implications for IX phases that arise in
MoS2/WS2 and MoSe2/WSe2 moiré-Bose-Hubbard systems.

##  II Interlayer Excitons in a Moiré superlattice

We first focus on twisted heterobilayers of MoS2/WS2, embedded in a dielectric
environment, as depicted in Fig. 1(b). For negligible mismatch in lattice
constants, the emerging moiré lattice is only determined by the twist angle.
The moiré lattice constant can be approximated by
$|\mathbf{a}^{\text{M}}|\approx a_{0}/\theta$, with $a_{0}$ the lattice
constant of the TMD monolayers. The spatial twist of the constituent
monolayers translates to an offset of the K points in reciprocal space. It
shifts the K-valley extrema of each respective monolayer relative to each
other to form a type-II band alignment that is indirect in momentum space with
the conduction-band minimum in the MoS2 and the valence-band maximum in the
WS2 layer as seen in Fig. 1(c). As illustrated in Fig. 1(d) (black), an
effective smaller moiré Brillouin zone (MBZ) emerges that is due to the long-
range periodicity of the real lattice. Outer corners of the MBZ are denoted by
$\kappa$. The resulting moiré lattice vectors are given by the differences of
the reciprocal lattice vectors of the top (t) and bottom (b) monolayers
$\mathbf{G}^{\text{M}}_{i}=\mathbf{b}_{\mathrm{b},i}-\mathbf{b}_{\mathrm{t},i}$
with
$(\mathbf{b}_{\mathrm{t}/\mathrm{b},1},\mathbf{b}_{\mathrm{t}/\mathrm{b},2})=2\pi\left((\mathbf{a}_{\mathrm{t}/\mathrm{b},1},\mathbf{a}_{\mathrm{t}/\mathrm{b},2})^{\intercal}\right)^{-1}$
and $\mathbf{a}_{\mathrm{t}/\mathrm{b},i}$ the real space lattice vectors.

Electron-hole pairs generated in the heterostructure quickly separate due to
the type-II band alignment and fast carrier scattering [32, 33, 34], leaving
electrons residing in the conduction band of the MoS2 layer and holes in the
valence band of the WS2 layer. In the presence of the strong Coulomb
interaction, momentum-indirect IXs form that possess long radiative lifetimes
[35], allowing them to thermalize before recombining. Furthermore, with charge
carriers separated to different layers, the IX dipole moments align
perpendicular to the in-plane direction, an arrangement that is in close
analogy to spatially-indirect excitons in coupled quantum wells, and which
results in a pronounced dipolar repulsion [36, 37, 38].

Figure 2: (a), (b) IX dispersion at twist angles $\theta=1^{\circ}$ and
$4^{\circ}$ along a linear cut through the MBZ
($\kappa,\gamma,\kappa^{\prime}$). Smaller twist angles lead to stronger
localization of the IX wave functions at lowest energies (flat dispersion).
(c)–(d) IX Bloch-wave functions (colors). The wave function’s localization is
reflected in the spatial extent relative to the moiré potential (contours).
(e)–(f) Single-band IX Wannier functions are localized at individual potential
minima. (g) First absolute moment
$\langle|\mathbf{r}|\rangle_{w_{\mathbf{R}}}$ and relative extent as a
function of moiré lattice length and twist angle $\theta$.

For small twist angles, the periodic, continuous spatial variation of the
interlayer band gap acts as an effective potential landscape
$V^{\mathrm{M}}(\mathbf{r})$ for the IXs. Although also the binding energy
varies with the moiré periodicity, this variation is small, and we neglect it
here [4]. The effective potential generally modifies the movement of IXs and
can give rise to states localized at the potential minima when the potential
is sufficiently strong. In a first-order plane-wave expansion, the potential
can be written as [4]

$V^{\mathrm{M}}(\mathbf{r})=\sum_{j=1}^{6}V_{j}\,e^{i\mathbf{G}^{\text{M}}_{j}\mathbf{r}},$
(1)

where $\bm{G}_{j}$ are the first-shell moiré reciprocal lattice vectors.
Because of the threefold rotational symmetry of the heterobilayer
$V_{1}=V_{3}=V_{5}$, $V_{2}=V_{4}=V_{6}$, and $V_{1}=V_{2}^{\ast}$. Using the
Bohr radius of intralayer excitons as a rough estimate, we note that for twist
angles $<4^{\circ}$, the moiré periodicity is sufficiently large in comparison
to the in-plane extent of IXs [4]. Therefore, in the limit of small densities,
we regard the IX as composite bosonic particles that move in the moiré
potential landscape, neglecting their inner structure. Considering only the
center-of-mass (c.o.m.) kinetic energy of an IX and the influence of the moiré
potential, the Hamiltonian reads

$H_{0}=-\frac{\hbar^{2}}{2M}\Delta_{\mathbf{r}}+V^{\mathrm{M}}(\mathbf{r})$
(2)

where $M=m_{\mathrm{e}}+m_{\mathrm{h}}$ is the total mass of the IX which we
solve in a plane-wave basis of the IX Bloch states. The resulting IX
dispersion for twist angles 1° and 4° is shown in Fig. 2(a)–(b) and parameters
used are listed in Table 1. The strong localization of the IX is reflected in
the vanishing curvature of the lowest IX energy bands. The resulting IX Bloch
functions for states with in-plane quasimomentum $\mathbf{Q}$ in the branch
$\alpha$ of the dispersion are given by

$\chi_{\mathbf{Q}}^{(\alpha)}(\mathbf{r})=\frac{1}{\sqrt{V}}\sum_{\mathbf{G}^{\text{M}}}c^{(\alpha)}_{\mathbf{Q}-\mathbf{G}^{\text{M}}}\text{e}^{\text{i}(\mathbf{Q}-\mathbf{G}^{\text{M}})\mathbf{r}}.$
(3)

Table 1: Parameters of two material systems for the calculation of the IX dispersion and wave functions. Total IX mass $M_{\mathrm{IX}}=m_{\mathrm{e}}+m_{\mathrm{h}}$ is the sum of electron and hole masses given in units of the free electron mass $m_{0}$. Lattice constants $a_{0}$ of the respective monolayers (assumed equal for top and bottom layer) and interlayer distance $d$ are in units of nm. For stacking configurations AA the parameters are given for the moiré potential according to $V=|V|\text{e}^{i\psi}$ with $V$ in units of meV and $\psi$ in degree (°). Parameters are taken from [4]. Material combination | $M_{\mathrm{IX}}$ | $a_{0}$ | $d$ | Stacking | $|V|$ | $\psi$
---|---|---|---|---|---|---
MoS2/WS2 | 0.76 | 0.319 | 0.615 | AA | 12.4 | 81.5
|  |  |  |  |  |
MoSe2/WSe2 | 0.84 | 0.332 | 0.647 | AA | 11.8 | 79.5

In Fig. 2(c)–(d) we show the moiré potential (contours) superimposed over the
IX Bloch wave functions (colors) corresponding to the lowest energy states at
the $\kappa$ point of the moiré BZ. While still periodic in real space, the
wave functions become more localized around the potential minima as the
dispersion flattens with decreasing twist angle.

The IXs feel an interaction potential [39, 40] that can be obtained from a
point-charge treatment [37] of two spatially-indirect excitons. The resulting
repulsive dipole-dipole interaction is given by

$U(r)=\frac{e^{2}}{4\pi\epsilon_{\mathrm{r}}\epsilon_{0}}\left(\frac{2}{r}-\frac{2}{\sqrt{d^{2}+r^{2}}}\right)~{},$
(4)

where $d$ is the mean distance between the layers (i.e. the length of the
dipole). Due to its 2D nature, in the heterobilayer the effect of dielectric
screening is particularly important as it strongly modifies the Coulomb
interaction between particles in the layers [23, 24, 25, 26]. The simplest
method to treat this effect is to account for the constant relative
permittivity $\epsilon_{\mathrm{r}}$ as the (equally) weighted average of the
relative permittivities (see Table 2) of the two constituting monolayers and
the top and bottom sub- and superstrates. An improved model of the of
dielectric screening by the heterobilayer and its environment is discussed in
Section IV.

Given the dipolar repulsion, we estimate under which conditions one can regard
the IXs as composite bosonic particles. We assume that this assumption holds
as long as the mean IX distance $\bar{r}=n^{-1/2}$ is much larger than the
interlayer distance $d$, given an IX density $n$. The distance $d$ defines the
characteristic length scale of the potential, beyond which the IXs see each
other as point-like dipoles. This results in the condition $nd^{2}\ll 1$,
which in our case amounts to $n\ll 2\times 10^{14}\,\mathrm{cm}^{-2}$ for the
MoS2/WS2 heterostructure, and smaller densities if additional dielectric
screening layers are placed between the TMD monolayers. In the limit of freely
moving particles, we need sufficiently low temperatures, such that the minimal
distance that IXs can have due mean kinetic energy is large, and they do not
witness each other’s internal structure. This is provided if the
characteristic length scale $d$ of the dipolar potential is small in
comparison to the minimal distance $r_{E}$ of two IXs with thermal energy
$E=k_{\mathrm{B}}T$, where $r_{E}$ is determined by $U(r_{E})=E$. From this we
obtain
$k_{\mathrm{B}}T\ll\frac{e^{2}}{4\pi\epsilon_{\mathrm{r}}\epsilon_{0}d}$,
which is met for all experimental conditions. In the case of IXs confined to
minima of the moiré potential, we consider a density regime that results in at
most one IX per potential minimum. This leads to densities
$n<\frac{2}{\sqrt{3}|\mathbf{a}^{\text{M}}|}$ as a function of the length of
the moiré lattice vector, see also Appendix A for more details.

##  III Moiré Bose-Hubbard Model.

Figure 3: Schematic view of the interaction processes represented by the
Hubbard parameters. On-site and nearest-neighbor interaction strengths $U_{0}$
and $U_{1}$ are indicated as well as hopping to nearest ($t_{1}$) and next-
nearest ($t_{2}$) lattice sites. Figure 4: Moiré Bose-Hubbard parameters. (a)
The (next-) nearest-neighbor hopping amplitudes $t_{1}$, $t_{2}$. Increasing
localization of the IX wave functions with decreasing twist angle $\theta$
result in suppressed hopping. (b)–(c) On-site and nearest-neighbor interaction
$U_{0}$ (circles) and $U_{1}$ (squares). Constant screening (orange) is
compared with screening given by the non-local Keldysh potential (purple). For
both dielectric embeddings (freestanding, hBN/hBN) the Keldysh model predicts
weaker on-site interactions $U_{0}$ (except in the freestanding case for
larger angles) and an increased role of nearest-neighbor interactions $U_{1}$.
With decreasing twist angles, the dipolar interaction weakens as the moiré
unit cell becomes large and IXs occupy a larger area in absolute terms.

The localization of lowest-energy moiré IXs in the effective potential minima
allows to map the system to an extended Bose-Hubbard model on a triangular
lattice. We use a basis of Wannier functions, constructed from the moiré IX
wave functions in the lowest band

$w_{\mathbf{R}}(\mathbf{r})=\frac{1}{\sqrt{N}}\sum_{\mathbf{Q}}\text{e}^{-\text{i}\mathbf{Q}\mathbf{R}}\chi_{\mathbf{Q}}(\mathbf{r}),$
(5)

where $\chi_{\mathbf{Q}}(\mathbf{r})$ are the IX c.o.m. wave functions for in-
plane momenta $\mathbf{Q}$ in the two-dimensional MBZ, and from here on we
suppress the band index $\alpha$. Note that the $\mathbf{R}$ do not correspond
to the moiré lattice vectors but to the array of moiré potential minima around
which the Wannier functions $w_{\mathbf{R}}(\mathbf{r})$ are centered. The
applicability of single-band Wannier functions relies on the energetic
separation of the lowest IX state as seen in Fig. 2(a)–(b). This can be
assumed to be a good approximation for sufficiently low exciton densities,
where higher energy bands are largely empty. The $w_{\mathbf{R}}(\mathbf{r})$
are centered around individual minima $\mathbf{R}$ of the moiré potential as
shown in Fig. 2(e)–(f). The ability to map the system dynamics to that of a
Bose-Hubbard model relies on choosing the $w_{\mathbf{R}}(\mathbf{r})$ as
basis functions that are strongly localized at discrete lattice sites. For an
intuitive way to quantify the localization of the IXs we calculate the first
absolute moment of the distribution defined by $w_{\mathbf{R}}(\mathbf{r})$
centered around its mean:

$\langle|\mathbf{r}|\rangle_{w_{\mathbf{R}}}=\int|\mathbf{r}-\mathbf{R}||w_{\mathbf{R}}(\mathbf{r})|^{2}\mathrm{d}^{2}r$
(6)

As seen in Fig. 2(g), $\langle|\mathbf{r}|\rangle_{w_{\mathbf{R}}}$ actually
increases with decreasing twist angle. Therefore, we note that localization of
IX states at lattice sites occurs only relative to the extent of the moiré
unit cell, which can be inferred from the same panel, in which also the
_relative_ extent
$\langle|\mathbf{r}|\rangle_{w_{\mathbf{R}}}/|\mathbf{a}^{\text{M}}|$ is
shown, $|\mathbf{a}^{\text{M}}|$ being the moiré lattice vector defining the
length scale of the moiré lattice.

Having established the c.o.m. Hamiltonian of the IXs and their dipolar
interaction in the previous sections, we now seek a description of IX states
in the TMD moiré heterostructure in terms of a generalized Bose-Hubbard model,
which includes nearest-neighbor IX repulsion to account for the long-range
interaction mediated by the dipolar potential $U(r)$. In general, we write the
BH-Hamiltonian

$H=\sum_{\mathbf{R},\mathbf{R}^{\prime}}t(\mathbf{R}-\mathbf{R}^{\prime})c_{\mathbf{R}}^{\dagger}c_{\mathbf{R}^{\prime}}+\frac{1}{2}\sum_{\mathbf{R},\mathbf{R}^{\prime}}U(\mathbf{R}-\mathbf{R}^{\prime})c_{\mathbf{R}}^{\dagger}c_{\mathbf{R}^{\prime}}^{\dagger}c_{\mathbf{R}^{\prime}}c_{\mathbf{R}}$
(7)

with $c_{\mathbf{R}}^{\dagger}$, $c_{\mathbf{R}}$ the creation and
annihilation operators for IXs at lattice position $\mathbf{R}$. The hopping
amplitudes $t_{n}=-t(\mathbf{R}-\mathbf{R}^{\prime})$ and the interaction
strengths $U_{n}=U(\mathbf{R}-\mathbf{R}^{\prime})$ are obtained by projecting
the free IX Hamiltonian and the potential $U(r)$ onto the first-band Wannier
basis states in equation (5). Here, a two-center approximation was applied
that relies on the spatially fast decaying basis functions
$w_{\mathbf{R}}(\mathbf{r})$. With this notation, $U_{0}$ corresponds to on-
site interaction with $\mathbf{R}=\mathbf{R}^{\prime}$, while $U_{1}$ denotes
nearest-neighbor interaction. Generally, $U_{n}$ is the
$n^{\mathrm{th}}$-nearest-neighbor interaction, and similarly for $t_{n}$. We
define the hopping parameters $t_{n}$ with a negative sign in accordance with
the literature on BH systems. A sketch illustrating the relevant processes is
shown in Fig. 3. To obtain $t_{n}$ we use the projection of $H_{0}$ onto the
Wannier basis

$t_{n}=-\int_{\mathbb{R}^{2}}w^{*}_{\mathbf{R}}(\mathbf{r})H_{0}w_{\mathbf{R}^{\prime}}(\mathbf{r})\text{d}^{2}r=-\frac{1}{N}\sum_{\mathbf{Q}}\text{e}^{\text{i}(\mathbf{R}-\mathbf{R}^{\prime})\mathbf{Q}}E_{\mathbf{Q}},$
(8)

where $E_{\mathbf{Q}}$ are the eigenvalues of $H_{0}$ for the lowest
dispersion branch. We note that $t_{0}$ establishes a constant energy offset
given by the mean energy of the band $E_{\mathbf{Q}}$ that we neglect in the
following. The algebraic form of the $t_{n}$ parameter gives interesting
insight: Effectively, $t_{n}$ is a Fourier decomposition of the lowest energy
band with respect to the lattice site distance
$\mathbf{R}-\mathbf{R}^{\prime}$. Information about the parameter can be
directly read off of the band structure, i.e. flat bands result in small
hopping terms, while curved bands result in larger hopping terms. We show
$t_{n}$ in Fig. 4(a) as a function of twist angle. Indeed, for small twist
angles the localization of IXs becomes stronger, effectively inhibiting
hopping from one lattice site to another.

Similarly to $t_{n}$, the coefficients of the two-particle interaction $U_{n}$
can be calculated using the Wannier basis in second quantization using Eq. (4)
as

$U_{n}=\iint_{\mathbb{R}^{2}}|w_{\mathbf{R}}(\mathbf{r})|^{2}|w_{\mathbf{R}^{\prime}}(\mathbf{r}^{\prime})|^{2}U(|\mathbf{r}-\mathbf{r}^{\prime}|)\,\text{d}^{2}r\,\text{d}^{2}r^{\prime}.$
(9)

More details on the derivation are given in Appendix B. Fig. 4(b)–(c) shows
the on-site interaction $U_{0}$ (orange circles) and nearest-neighbor
interaction $U_{1}$ (orange squares) as function of twist angle for two
different dielectric environments, i.e. freestanding and encapsulated in
hexagonal boron nitride (hBN). The on-site interaction $U_{0}$ clearly
dominates over the (next-) nearest-neighbor interactions. However, while
hopping of IXs between lattice sites is quenched for small twist angles as
expected due to increasing localization, we also find the on-site interaction
$U_{0}$ to decrease with the twist angle, which might seem counterintuitive.
This is because only the relative extent gets smaller for small twists,
meaning stronger localization. At the same time the absolute spatial extent of
the IX wave function, as given by
$\langle|\mathbf{r}|\rangle_{w_{\mathbf{R}}}$ in Eq. (6) actually increases,
thereby weakening the on-site interaction.

Table 2: Parameters for the dielectric screening. Relative permittivities $\epsilon_{\mathrm{r}}$ taken from [41] are given by the geometric mean of in-plane and out-of-plane dielectric constants. Values of the 2D polarizability $\chi_{\mathrm{2D}}$ are in units of nm. Material | MoS2 | WS2 | MoSe2 | WSe2 | hBN | SiO2 | air
---|---|---|---|---|---|---|---
$\epsilon_{\mathrm{r}}$ | 9.69111Ref. [41] | 9.24111Ref. [41] | 11111Ref. [41] | 10.64 111Ref. [41] | 5.1111Ref. [41] | 3.9 | 1
$\chi_{\mathrm{2D}}$ | 0.711222Ref. [25] | 0.639222Ref. [25] | 0.846222Ref. [25] | 0.757222Ref. [25] | - | - | -

##  IV Dielectric screening

In Eq. (4) the dipolar interaction potential contains the dielectric
environment of IXs solely as a constant relative permittivity
$\epsilon_{\mathrm{r}}$ that is obtained by averaging the permittivities of
the two constituent layers and super- and substrates [11, 19]. However, a
constant permittivity is better suited to capture effects in homogeneous
media, while in heterostructures of atomically thin materials, the dielectric
environment changes on small length scales, and in many situations, a more
detailed modeling of screening effects is required [23, 42, 24, 43, 44]. To
improve on the constant-screening approach, we take into account non-local
screening effects via a non-local Rytova-Keldysh potential. This approach was
successfully used to model the Coulomb interaction of excitons, trions, and
other complexes in TMD monolayers and heterostructures [45, 30, 25, 26, 29,
46, 47]. We apply this description by modifying $U(r)$ in momentum space. The
Fourier-transformed dipolar potential that determines the IX’s interaction is
given by

$U(q)=\int_{\mathbb{R}^{2}}U(r)\text{e}^{-\text{i}\mathbf{r}\cdot\mathbf{q}}\,\text{d}^{2}r=\frac{e^{2}}{\epsilon_{\mathrm{r}}\epsilon_{0}}\frac{1}{q}(1-\text{e}^{-dq}).$
(10)

Eq. (10) enables us to replace the constant permittivity
$\epsilon_{\mathrm{r}}\epsilon_{0}$ by the non-local
$\epsilon(q)=\epsilon_{0}\kappa(1+r_{0}q)$, where
$\kappa=(\epsilon_{\mathrm{r}}^{\mathrm{t}}+\epsilon_{\mathrm{r}}^{\mathrm{b}})/2$
is the mean relative permittivity of the dielectric environment around the
heterostructure, $r_{0}=2\pi\chi_{\mathrm{2D}}/\kappa$ defines a length scale
on which the potential is modified, and $\chi_{\mathrm{2D}}$ is the 2D
polarizability of the heterostructure [30, 25]. We average the 2D
polarizabilities of the two materials constituting the bilayer to find an
effective description of the system. In real space, the modified dipolar
potential becomes

$\tilde{U}(r)=\frac{e^{2}}{2\pi\kappa\epsilon_{0}}\int_{0}^{\infty}\frac{1-\text{e}^{-dq}}{1+r_{0}q}J_{0}(rq)\,\text{d}q$
(11)

with the Bessel function of the first kind of order zero, $J_{0}$. The
integral is evaluated numerically and the non-local dipolar potential is shown
in Fig. 5 (purple). A comparison with the potential based on the constant
screening (orange) reveals that the non-local screening results in lower
dipolar potential strengths at small distances, changing the $1/r$ dependence
to a logarithmic divergence (see inset in Fig. 5) from which we expect reduced
on-site interactions. For larger distances the modified potential retains the
long-range $1/r^{3}$ dependence.

Figs. 4(b)–(c) (purple lines) show calculations of $U_{n}$ with the modified
dipolar potential for two different dielectric environments. In general, the
modified potential predicts smaller on-site interaction strengths. The amount
of the reduction obtained from the non-local screening model depends on the
actual dielectric environment as well as on the twist angle. For freestanding
monolayers, the modified potential even predicts a stronger on-site repulsion
than the local case, while for larger twist angles the effect of dielectric
screening becomes more prominent. Furthermore, while the influence of on-site
interaction $U_{0}$ clearly remains dominant, with the non-local screening
model we find the strength of nearest-neighbor interactions $U_{1}$ strongly
increased relative to on-site repulsion. This has implications for the
occurrence of correlated IX phases as will be discussed in the following
Section.

Figure 5: Dipolar potential as a function of IX distance $r$, comparing the
effect of screening by a constant permittivity
$\epsilon_{\mathrm{r}}\epsilon_{0}$ and non-local screening according to Eqs.
(4) and (11). For small distances the potential is reduced, changing from a
$1/r$ dependence to a $\log(1/r)$ dependence (illustrated in the inset on a
semilogarithmic scale). For larger distances the $1/r^{3}$ dependence remains.

##  V Phases of interlayer excitons

Having established a connection between the moiré IX system and the
generalized BH model, we now discuss the emergence of correlated states of IXs
and possible signatures of these phases in experimental setups. The on-site
interaction $U_{0}$ and the nearest-neighbor hopping $t_{1}$ dominate the
behavior, and we consider as a first approximation the standard BH Hamiltonian

$H_{0}=-t_{1}\sum_{\langle
i,j\rangle}c_{i}^{\dagger}c_{j}+\frac{U_{0}}{2}\sum_{i}n_{i}(n_{i}-1),$ (12)

where $\langle i,j\rangle$ denotes summation only over nearest neighbors, and
$n_{i}$ is the Boson-number operator at site $i$. This Bose-Hubbard model in
two dimensions is often treated in a mean-field approximation and predicts two
distinct phases: For dominating on-site interactions ($t_{1}/U_{0}\ll 1$) the
system is found to be in a Mott insulating phase at densities that correspond
to commensurate fillings of the lattice with zero compressibility and
suppressed particle movement [8, 11, 19]. If hopping dominates, a superfluid
phase is found, which is a coherent IX state delocalized over the lattice. The
superfluid phase is also found in the Mott regime at non-commensurate
fillings, where an excess of particles moves freely over a commensurate,
insulating background. A variety of other methods have been employed to treat
the BH model beyond the mean-field level including Monte-Carlo [48, 49, 50]
and path-integral methods [51] as well as matrix-product-state [52, 53] and
density-matrix renormalization-group approaches [54] in order to extract
finite temperature properties of the system.

We first discuss results for the MoS2/WS2 material combination. In Fig. 6(a)
we show the ratio $t_{1}/U_{0}$ obtained from our calculations as a function
of twist angle for different commonly encountered combinations of sub- and
superstrates. The shaded areas (violet/green) indicate the phase transition
from the Mott insulating to the superfluid state for a filling factor of $n=1$
that we obtain from numerically solving the mean-field BH model. We would like
to emphasize that most theoretical analysis treat the BH model in a grand-
canonical ensemble description that allows for fluctuations in the particle
number with an effective Hamiltonian $H_{0}-\mu N$. Here, we consider
situations, in which a fixed density of IXs is excited in the TMD
heterobilayers that is allowed to equilibrate before radiative recombination,
which corresponds to the experimental situation following optical excitation.
In this spirit, we evaluate the mean-field model for a fixed filling factor of
$n=1$, which corresponds to a canonical ensemble picture [55, 56]. The results
in Fig. 6(a) suggest that the MoS2/WS2 heterobilayer system will be deeply in
the Mott regime for most considered twist angles and dielectric environments.
The signature of this regime is a strongly reduced IX diffusion whenever
excitation powers create IX densities that correspond to a filling of $n=1$, a
change of system behavior that has only recently been observed experimentally
in [20] for a MoSe2/WSe2 system. Numerical results for MoSe2/WSe2 are shown in
Fig. 6(b) and indicate that the selenide bilayer system remains in the Mott
phase for all examined twist angles owing to a reduced mobility of the IXs
caused by a larger lattice constant.

Figure 6: (a) Bose-Hubbard parameters $t_{1}/U_{0}$ for MoS2/WS2 as a function
of twist angle and for different combinations of dielectric environment. The
shaded areas indicate Mott and superfluid IX phases as predicted by a mean-
field treatment of the BH system for a unit filling factor ($n=1$). Inset:
Parametric plot of $U_{1}/U_{0}$ and $t_{1}/U_{0}$ as function of twist angle.
The values for intermediate to large angles indicate the possibility of
density-wave phases at fractional fillings. (b) Parameters $t_{1}/U_{0}$ for
MoSe2/WeS2 as a function of twist angle. In comparison to (a), for all
dielectric environments the system remains in the Mott phase for all examined
twist angles.

For larger twist angles and substantial dielectric screening, as realized by
hBN encapsulation or capping on a silicon substrate, also the superfluid
regime is accessible for the MoS2/WS2 heterostructure. Potentially, as the
initially excited IX density decays due to radiative recombination, transient
commensurate fillings can be revealed in the coherence of emitted light that
allows to detect the presence or absence of an excess coherent superfluid
phase on top of the commensurate, Mott-insulating background. As recently
demonstrated, in more complex experimental configurations the presence of
insulating IX phases can be detected by altered reflectivity and a blueshift
of the IX photoluminescence due to changes in the dielectric screening in the
heterobilayer system depending on the IX phase [57, 58]. We point out that the
trapping of IXs in the energetic minima of the moiré potential as discussed in
Section II is distinct from the localization of IXs in the Mott phase, which
relies on the mutual repulsion of IXs and can arise also for weak moiré
potentials.

For the hBN-encapsulated bilayer, we have added the local screening result as
discussed in Section II as a gray curve to Fig. 6(a) for comparison. It
becomes clear that the increased screening provided by the non-local potential
has a significant impact on the phase boundaries. In our case, the associated
underestimation of the dielectric screening would predict a Mott-phase even at
the highest considered twist angles. Lagoin _et al._ [19] have discussed the
possibility to insert layers of insulating hBN between the monolayers of the
heterostructure to reduce the Coulomb interaction between IXs, making the
superfluid regime more accessible. While it is an interesting avenue for
further investigations, this approach also changes the strength of the moiré
potential itself that is the prerequisite for a Bose-Hubbard description of
the system. We refrain from considering filling factors with more than one IX
per moiré unit cell, as the role of many-particle effects, such as biexciton
formation, may play an important role that is not captured in our current
approach. Further details on the validity regime are discussed in Appendix A.

Our approach to calculate the two-particle interaction beyond the on-site
interaction matrix element $U_{0}$ allows drawing conclusions towards more
exotic correlated bosonic states. Previous studies have reported density-wave
phases for particular fractional fillings ($n=1/3,\,1/2,\,2/3,\,\ldots$) in
the presence of non-negligible $U_{1}$ [59, 1, 56, 60, 10]. In the inset of
Fig. 6 we show a parametric plot of $U_{1}/U_{0}$ over $t_{1}/U_{0}$ for
varying twist angle, where values close to the origin correspond to smaller
angles. We see, especially in the case of weaker dielectric screening, that
the nearest-neighbor interaction can indeed become comparable to the on-site
interaction $U_{0}$ and the hopping amplitude $t_{1}$, opening up the
possibility for the formation of more complex phases beyond the superfluid and
Mott-insulating phases. While this work is mainly directed at establishing a
connection between the moiré physics of IXs and the Bose-Hubbard model, we do
not delve further into the phase diagram at fractional fillings. We have,
however, verified on a mean-field level that the results shown in Fig. 6
remain valid even in the presence of next-nearest neighbor hopping. Our first
results indicate that interlayer excitons in the tunable potential landscape
of moiré TMD heterostructures provide a fascinating gateway to study phase
transitions in flat-band systems [61, 62], and more complex correlated phases
like density wave and supersolids. We are convinced that the underlying
physics will be addressed both in theoretical and experimental work in the
near future.

##  VI Conclusion

In conclusion, we have examined the moiré-Bose-Hubbard physics of IXs in
MoS2/WS2 and MoSe2/WSe2 heterobilayers. A connection between the properties of
IXs traversing the long-range potential landscape created by the moiré pattern
and the Bose-Hubbard model is established by explicitly calculating the two-
particle interaction and hopping parameters from a Wannier representation of
the IX wave functions. Screening from the dielectric environment is treated
within a Rytova-Keldysh-model approach, from which we find a significant
reduction of the repulsive dipolar potential in comparison to the approach
with constant permittivity that considers averages of dielectric constants of
the different layers. Mean-field treatment of the Bose-Hubbard Hamiltonian
predicts the existence of a Mott phase at unit fill factor and small twist
angles due to dominating on-site interaction strength $U_{0}$. However, a
transition to the superfluid state appears possible in the sulfur-based system
for the relevant cases of hBN-encapsulated bilayers, and hBN capped bilayers
on a silicon substrate, in which case the screening sufficiently lowers the
dipolar interaction strength.

While our model approach offers first glimpses into the exciting correlated-
state properties of excitons in van der Waals heterostructures, in which the
twist angle can be used as a tuning knob for the Bose-Hubbard parameters, it
also reveals limitations that are inherent to current models, such as multi-
band and biexcitonic effects that will likely alter the physics at larger
filling factors. A particularly promising platform to further explore
correlated phases in moiré-Bose Hubbard systems are cavity-embedded bilayers,
in which IX and photons form IX polaritons with even greater control over the
ratio between dipolar interaction and hopping. Furthermore, it will be
interesting to study effects of local reconstructions, which have been shown
to strongly impact the moiré physics at low twists angles [63].

## Acknowledgements

We would like to thank Alexander Steinhoff for many useful discussions.
Funding is acknowledged from the Deutsche Forschungsgemeinschaft (German
Research Foundation) via the priority program SP2244 (project Gi-1121/4-1) and
the graduate school 2247. F.L. further acknowledges funding by the central
research development fund (CRDF) of the University of Bremen.

##  Appendix A Estimates of IX Densities

We provide estimates under which conditions the assumption that the IXs can be
treated as composite particles is valid. For Bosons not confined to individual
lattice sites, we note that particles with average thermal energy
$E=k_{\mathrm{B}}T$ can approach each other due to their mutual dipolar
repulsion up to a distance $r_{E}$ that is determined by $U(r_{E})=E$. We aim
at densities and temperatures such that IXs only see each other’s dipole field
($\sim 1/r^{3}$) and not their inner structure. This is satisfied if we
require $d<r_{E}$ from which we find (with
$c=e^{2}/4\pi\epsilon_{\mathrm{r}}\epsilon_{0}$)

$U(r_{E})=c\left(\frac{2}{r_{E}}-\frac{2}{\sqrt{d^{2}+r_{E}^{2}}}\right)\approx\frac{cd^{2}}{r_{E}^{3}}.$
(13)

From $U(r_{E})=k_{\mathrm{B}}T$ follows

$\displaystyle\dfrac{k_{\mathrm{B}}Td}{c}=\dfrac{d^{3}}{r_{E}^{3}}\ll 1$
$\displaystyle\Longrightarrow\qquad$ $\displaystyle
k_{\mathrm{B}}T\ll\dfrac{e^{2}}{4\pi\epsilon_{\mathrm{r}}\epsilon_{0}d}.$ (14)

With exemplary values $d=0.647\,\mathrm{nm}$ and $\epsilon_{\mathrm{r}}=10$
this results in a temperature limit of $T\ll 2500\,\mathrm{K}$, which is met
for all experimental conditions.

In the case of IXs confined to the potential minima in each moiré unit cell,
Fig. 7 shows an upper density limit that corresponds to a maximum number of
one IX per unit cell as a function of the moiré lattice constant
$|\mathbf{a}^{\text{M}}|$. The estimate can easily be determined from the unit
cell geometry and is given by
$n\leq\frac{2}{\sqrt{3}|\mathbf{a}^{\text{M}}|}$.

Figure 7: Maximal density as function of moiré lattice constant to have
maximal one IX per potential minimum.

##  Appendix B Calculation of the Hubbard parameters

The Hubbard parameters are the coefficients of the second quantization
operators derived from $H_{0}$ and $H_{1}$. For the one-particle operator
$H_{0}$, these coefficients are

$t_{\mathbf{R},\mathbf{R}^{\prime}}=\braket{\mathbf{R}}{H_{0}}{\mathbf{R}^{\prime}}.$
(15)

Using the localized Wannier functions from Eq. (5) as the basis, this
expression becomes

$\displaystyle t_{\mathbf{R},\mathbf{R}^{\prime}}$
$\displaystyle=\int_{\mathbb{R}^{2}}w^{*}_{\mathbf{R}}(\mathbf{r})H_{0}w_{\mathbf{R}^{\prime}}(\mathbf{r})\text{d}^{2}r$
$\displaystyle=\frac{1}{N}\sum_{\mathbf{Q},\mathbf{Q}^{\prime}}\int_{\mathbb{R}^{2}}\text{e}^{\text{i}\mathbf{Q}\mathbf{R}}\chi^{*}_{\mathbf{Q}}(\mathbf{r})H_{0}\text{e}^{-\text{i}\mathbf{Q}^{\prime}\mathbf{R}^{\prime}}\chi_{\mathbf{Q}^{\prime}}(\mathbf{r})\text{d}^{2}r.$
(16)

As the Bloch functions $\chi_{\mathbf{Q}}(\mathbf{r})$ by definition are
eigenfunctions of $H_{0}$ with eigenvalues $E_{\mathbf{Q}}$, we obtain

$\displaystyle t_{\mathbf{R},\mathbf{R}^{\prime}}$
$\displaystyle=\frac{1}{N}\sum_{\mathbf{Q},\mathbf{Q}^{\prime}}\text{e}^{\text{i}(\mathbf{Q}\mathbf{R}-\mathbf{Q}^{\prime}\mathbf{R}^{\prime})}E_{\mathbf{Q}^{\prime}}\int_{\mathbb{R}^{2}}\chi^{*}_{\mathbf{Q}}(\mathbf{r})\chi_{\mathbf{Q}^{\prime}}(\mathbf{r})\text{d}^{2}r$
$\displaystyle=\frac{1}{N}\sum_{\mathbf{Q},\mathbf{Q}^{\prime}}\text{e}^{\text{i}(\mathbf{Q}\mathbf{R}-\mathbf{Q}^{\prime}\mathbf{R}^{\prime})}E_{\mathbf{Q}^{\prime}}\delta_{\mathbf{Q},\mathbf{Q}^{\prime}}$
$\displaystyle=\frac{1}{N}\sum_{\mathbf{Q}}\text{e}^{\text{i}(\mathbf{R}-\mathbf{R}^{\prime})\mathbf{Q}}E_{\mathbf{Q}}=:t(\mathbf{R}-\mathbf{R}^{\prime}),$
(17)

which is the result used in Eq. (8). The coefficients of the two-particle
operator $H_{1}$ are

$U_{\mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3},\mathbf{R}_{4}}=\bra{\mathbf{R}_{1}}^{(i)}\bra{\mathbf{R}_{2}}^{(j)}H_{1}\ket{\mathbf{R}_{3}}^{(j)}\ket{\mathbf{R}_{4}}^{(i)}.$
(18)

With the Wannier functions as the basis and the vectors $\mathbf{r}$ and
$\mathbf{r}^{\prime}$ being the positions of particles $(i)$ and $(j)$,
respectively, the coefficients become

$U_{\mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3},\mathbf{R}_{4}}=\iint_{\mathbb{R}^{2}}w^{*}_{\mathbf{R}_{1}}(\mathbf{r})w^{*}_{\mathbf{R}_{2}}(\mathbf{r}^{\prime})U(|\mathbf{r}-\mathbf{r}^{\prime}|)\\\
w_{\mathbf{R}_{3}}(\mathbf{r}^{\prime})w_{\mathbf{R}_{4}}(\mathbf{r})\,\text{d}^{2}r\,\text{d}^{2}r^{\prime}.$
(19)

Due to the strong localization of the Wannier functions, the product
$w_{\mathbf{R}}(\mathbf{r})w_{\mathbf{R}^{\prime}}(\mathbf{r})$ is non-
negligible only if $\mathbf{R}=\mathbf{R}^{\prime}$. This justifies the
approximation
$\mathbf{R}_{1}=\mathbf{R}_{4}:=\mathbf{R},~{}\mathbf{R}_{2}=\mathbf{R}_{3}=:\mathbf{R}^{\prime}$
and results in the two-particle coefficient

$\displaystyle U_{\mathbf{R},\mathbf{R}^{\prime}}$
$\displaystyle=\iint_{\mathbb{R}^{2}}w^{*}_{\mathbf{R}}(\mathbf{r})w^{*}_{\mathbf{R}^{\prime}}(\mathbf{r}^{\prime})U(|\mathbf{r}-\mathbf{r}^{\prime}|)$
$\displaystyle\qquad\qquad\qquad
w_{\mathbf{R}^{\prime}}(\mathbf{r}^{\prime})w_{\mathbf{R}}(\mathbf{r})\,\text{d}^{2}r\,\text{d}^{2}r^{\prime}$
$\displaystyle=\iint_{\mathbb{R}^{2}}|w_{\mathbf{R}}(\mathbf{r})|^{2}|w_{\mathbf{R}^{\prime}}(\mathbf{r}^{\prime})|^{2}U(|\mathbf{r}-\mathbf{r}^{\prime}|)\,\text{d}^{2}r\,\text{d}^{2}r^{\prime},$
(20)

which is the result used in Eq. (9). Finally, we note that care must be taken
during the numerical evaluation of Eq. (20) as the term
$U(|\mathbf{r}-\mathbf{r}^{\prime}|)$ features singularities. However, as the
integration area is the two-dimensional plane, and the singularities are of
the type $1/r$ the singularities can be lifted via a transformation to polar
coordinates. For that it is useful to first express $\mathbf{r}$ in Eq. (20)
in terms of a new variable
$\tilde{\mathbf{r}}=\mathbf{r}-\mathbf{r}^{\prime}$. By furthermore shifting
the Wannier functions so that $\mathbf{R}^{\prime}$ lies in the origin and the
distance between the functions’ centers remains the same, the following
expression for $U$ is obtained:

$\displaystyle U_{\mathbf{R},\mathbf{R}^{\prime}}$
$\displaystyle=\iint_{\mathbb{R}^{2}}|w_{\mathbf{R}-\mathbf{R}^{\prime}}(\tilde{\mathbf{r}}+\mathbf{r}^{\prime})|^{2}|w_{0}(\mathbf{r}^{\prime})|^{2}U(\tilde{r})\,\text{d}^{2}\tilde{r}\,\text{d}^{2}r^{\prime}$
$\displaystyle=:U(\mathbf{R}-\mathbf{R}^{\prime}).$ (21)

Finally, the transformation to polar coordinates can be applied, yielding

$U(\mathbf{R}-\mathbf{R}^{\prime})=\frac{e^{2}}{4\pi\epsilon_{\text{r}}\epsilon_{0}}\iint_{\mathbb{R}^{2}}|w_{\mathbf{R}-\mathbf{R}^{\prime}}(\tilde{\mathbf{r}}+\mathbf{r}^{\prime})|^{2}|w_{0}(\mathbf{r}^{\prime})|^{2}\\\
\left(r^{\prime}-\frac{r^{\prime}\tilde{r}}{\sqrt{\tilde{r}^{2}+d^{2}}}\right)\,\text{d}\tilde{r}\,\text{d}\tilde{\varphi}\,\text{d}r^{\prime}\,\text{d}\varphi^{\prime},$
(22)

which is free of singularities and can be readily evaluated.

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Here we study spectral properties of the operator $K:={\cal
F}^{-1}\tilde{K}{\cal F}$, where $\tilde{K}$ denotes the operator of
multiplication by $\tilde{K}(\theta)$ in the Hilbert space
$L^{2}(\Pi^{*},{\cal X}^{0}({T^{3}}))$.

######  Lemma 5.4.1.

$K$ is a selfadjoint operator in ${\cal X}^{0}({\mathbb R}^{3})$ with a dense
domain $D(K)$.

###### Proof.

Lemma 4.6.2 ii) implies that the operator of multiplication by
$\tilde{K}^{-1}(\theta)$ is selfadjoint and injective in $L^{2}(\Pi^{*},{\cal
X}^{0}({T^{3}}))$. Hence, its inverse is densely defined selfadjoint operator
in $L^{2}(\Pi^{*},{\cal X}^{0}({T^{3}}))$ by Theorem 13.11 (b) of [70]. ∎

######  Corollary 5.4.2.

The Hamiltonian generator $A$ from (5.1.1) admits the representation

$AY=-i\Lambda^{-1}K\Lambda Y,\qquad Y\in\Lambda^{-1}D(K),$ (5.4.1)

where $\Lambda:={\cal F}^{-1}\tilde{\Lambda}(\theta):{\cal W}\to{\cal
X}^{0}({\mathbb R}^{3})$ is the isomorphism.

By (5.2.1),

$KZ(n)=|\Pi^{*}|^{-1}\int_{\Pi^{*}_{+}}e^{-in\theta}{\cal
M}(-\theta)\sum_{k}\omega_{k}(\theta)P_{k}(\theta)\tilde{Z}(\theta)d\theta,~{}~{}~{}~{}~{}~{}~{}n\in{\mathbb
Z}^{3}$ (5.4.2)

for any $Z\in{\cal X}^{0}({\mathbb R}^{3})$. Similarly

$Z(n)=|\Pi^{*}|^{-1}\int_{\Pi^{*}_{+}}e^{-in\theta}{\cal
M}(-\theta)\sum_{k}P_{k}(\theta)\tilde{Z}(\theta)d\theta,~{}~{}~{}~{}~{}~{}~{}n\in{\mathbb
Z}^{3}.$ (5.4.3)

Therefore,

$(K-\omega)Z(n)=|\Pi^{*}|^{-1}\int_{\Pi^{*}_{+}}e^{-in\theta}{\cal
M}(-\theta)\sum_{k}(\omega_{k}(\theta)-\omega)P_{k}(\theta)\tilde{Z}(\theta)d\theta.$
(5.4.4)

Hence, the discrete spectrum $\sigma_{p}(K)$ consists of constant dispersion
relations.

######  Lemma 5.4.3.

Let all conditions of Theorem 5.1.1 hold. Then
i) $\sigma_{p}(K)=\Omega^{*}$.
ii) The multiplicity of every eigenvalue is infinite.

###### Proof.

Let $\omega^{*}\in\Omega^{*}$ is a constant eigenvalue (5.2.12) corresponding
to a point $R^{*}=(\theta^{*},\omega^{*})\in{\cal R}$. Let us take any
$Z\in{\cal X}^{0}({\mathbb R}^{3})$ with the Bloch transform
$\tilde{Z}(\theta)\in{\rm Ran{\hskip 0.56905pt}}P(\omega^{*})$ for $\theta\in
V(\theta^{*})$ and $\tilde{Z}(\theta)\equiv 0$ for $\theta\not\in
V(\theta^{*})$. Then (5.2.12) and (5.4.4) imply that $(K-\omega^{*})Z=0$.
Obvioulsy, the space of such $Z$ is infinite dimensional.
Conversely, let $(K-\omega^{*})Z=0$ for some $Z\in{\cal X}^{0}({\mathbb
R}^{3})$, and $\tilde{Z}(\theta^{*})\neq 0$. Then (5.4.4) implies (5.2.12)
with some $l=1,...,L(\theta^{*},\omega^{*})$. ∎

Let us show that the continuous spectrum of $K$ is absolutely continuous.
First, (5.4.4) implies that the resolvent $R_{K}(\omega):=(K-\omega)^{-1}$ for
${\rm Im{\hskip 1.42262pt}}\omega\neq 0$ is given by

$R_{K}(\omega)Z(n)=|\Pi^{*}|^{-1}\int_{\Pi^{*}_{+}}e^{-in\theta}{\cal
M}(-\theta)\sum_{k}(\omega_{k}(\theta)-\omega)^{-1}P_{k}(\theta)\tilde{Z}(\theta)d\theta,\qquad
Z\in{\cal X}^{0}({\mathbb R}^{3}).$ (5.4.5)

Denote by ${\cal X}_{d}$ the space of discrete spectrum of $K$.

######  Lemma 5.4.4.

Let all conditions of Theorem 5.1.1 hold. Then the singular spectrum of $K$ is
empty.

###### Proof.

This follows by Theorem XIII.20 of [69]. Namely, it suffices to check the
corresponding criterion

$\sup_{0<\varepsilon<1}\int_{a}^{b}|{\rm Im{\hskip 1.42262pt}}\langle
Z,R_{K}(\omega+i\varepsilon)Z\rangle|^{p}d\omega<\infty$ (5.4.6)

with any $a,b\in{\mathbb R}$ and some $p>1$ for a dense set of $Z\in{\cal
X}_{d}^{\bot}$. For example, for the linear span of vectors $Z\in{\cal
X}^{0}({\mathbb R}^{3})$ with the Bloch transform

$\tilde{Z}(\theta)=P_{l}(\theta)D(\theta),~{}~{}~{}~{}~{}~{}~{}~{}D\in
C_{0}^{\infty}(W,{\cal X}^{0}({T^{3}})),$ (5.4.7)

as constructed in Lemma 5.2.2 for each $R^{*}=(\theta^{*},\omega^{*})\in{\cal
R}$, where $P_{l}(\theta)$ is the projection corresponding to an eigenvalue
$\omega_{l}(\theta)$ satisfying (5.2.11). It suffices to check (5.4.6) only
for the vectors of type (5.4.7). Applying Sokhotski-Plemelj’s formula, we
obtain for these vectors

$\displaystyle{\rm Im{\hskip 1.42262pt}}\langle
Z,R_{K}(\omega+i\varepsilon)Z\rangle$ $\displaystyle=$
$\displaystyle\,\,\,\int_{W}{\rm Im{\hskip 1.42262pt}}\langle
P_{l}(\theta)D(\theta),(\omega-\omega_{l}(\theta)-i\varepsilon)^{-1}P_{l}(\theta)D(\theta)\rangle_{{\cal
X}({T^{3}})}d\theta$ (5.4.8) $\displaystyle\to\\!\\!$
$\displaystyle\\!\\!-\\!\pi\\!\\!\int_{\omega_{l}(\theta)=\omega}\frac{\langle
P_{l}(\theta)D(\theta),P_{l}(\theta)D(\theta)\rangle_{{\cal
X}({T^{3}})}}{|\nabla\omega_{l}(\theta)|}d\theta,\qquad\varepsilon\to 0+,$

which implies (5.4.6) with any $p\geq 1$. ∎

In concluzion, let us prove the Limiting Absorption Principle. Let us denote
by ${\cal X}_{\alpha}$ the Hilbert space of functions with the finite norm
(5.3.12).

######  Lemma 5.4.5.

Let all conditions of Theorem 5.1.1 hold, and let $Z\in{\cal X}_{d}^{\bot}$ be
a finite linear combination of the vectors with the Bloch transform of type
(5.4.7). Then for any $\omega\in{\mathbb R}$ and $\alpha<-7/2$

$R_{K}(\omega\pm i\varepsilon)Z\stackrel{{\scriptstyle{{\cal
Y}^{0}}_{-\alpha}}}{{-\\!\\!-\\!\\!\\!\longrightarrow}}R_{K}(\omega\pm
i0)Z,~{}~{}~{}~{}~{}~{}\varepsilon\to+0.$ (5.4.9)

###### Proof.

It suffices to prove (5.4.9) for every vector of type (5.4.7). By (5.3.3) the
corresponding solution $Z(t)$ with $Z(0)=Z$ reads

$Z(n,t)=|\Pi^{*}|^{-1}\int_{W}e^{-in\theta}{\cal
M}(-\theta)e^{-i\omega_{l}(\theta)t}P_{l}(\theta)\tilde{Z}(\theta)d\theta,\quad
n\in{\mathbb Z}^{3}$

The partial integration shows the time-decay

$\|Z(n,t)\|_{{\cal X}^{0}(\Pi)}\leq C(1+|n|)^{2}(1+|t|)^{-2}.$

Hence,

$\|Z(t)\|_{{\cal X}_{\alpha}}\leq C(1+|t|)^{-2}.$

Now the convergence (5.4.9) follows from the integral representation

$R_{K}(\omega\pm
i\varepsilon)Z=\int_{0}^{\pm\infty}e^{(i\omega\mp\varepsilon)t}Z(t)dt.$

∎

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License Attribution 4.0 International (CC BY 4.0).

# How to Evaluate Coreference in Literary Texts?

Ana-Isabel Duron-Tejedor  Pascal Amsili  Thierry Poibeau  Lattice, CNRS & ENS-
PSL & Université Sorbonne Nouvelle

(2023)

###### Abstract

In this short paper, we examine the main metrics used to evaluate textual
coreference and we detail some of their limitations. We show that a unique
score cannot represent the full complexity of the problem at stake, and is
thus uninformative, or even misleading. We propose a new way of evaluating
coreference, taking into account the context (in our case, the analysis of
fictions, esp. novels). More specifically, we propose to distinguish long
coreference chains (corresponding to main characters), from short ones
(corresponding to secondary characters), and singletons (isolated elements).
This way, we hope to get more interpretable and thus more informative results
through evaluation.

###### keywords:

Coreference Evaluation Metrics Literature analysis

##  1 Introduction

Coreference resolution is a natural language processing (NLP) task that
consists in determining the linguistic expressions, also called mentions, that
refer to a same entity (Poesio et al., 2023). This task is crucial for NLP,
and constitutes a logical complement to named entity recognition. For example,
if one wants to track all the actions involving a given entity in a text, an
accurate analysis of coreference is required, as most of the references to
entities are usually done through coreference (i.e., pronouns and other
linguistic items referring to an entity).

This is in particular the case for the analysis of fictions, especially
novels: a proper coreference analysis is mandatory to analyze the plot, the
evolution, and the interactions between the different characters (Bamman et
al., 2014). Characters play a prominent roles in fiction. For this reason, we
will only consider reference to characters in this study (which means we
exclude, e.g., locations and company names, that are of course useful, but do
not play a so prominent role in fictions).

The importance of coreference in NLP has been put forward by the Message
Understanding Conferences (MUC) in the late 90ies (Grishman and Sundheim,
1996). Since then, numerous coreference resolvers have been developed for a
variety of languages and coreference is a very active domain of research in
NLP. A related issue is how to evaluate coreference. As coreference involves
sets of mentions of various size linked together (i.e., referring to a same
entity), traditional indicators like precision and recall have to be
redefined, but there is no obvious way of doing it (should entities be
weighted based on their relative importance in the text? What to do with
singletons?). This led to several measures that do not always agree (i.e. one
scoring method may indicate a good precision, whereas another method will not
be so positive).

This has been already been demonstrated in various studies on the topic
(Borovikova et al., 2022; Lion-Bouton et al., 2020; Moosavi and Strube, 2016;
Moosavi, 2020). Here we want to stress complementary, albeit different,
issues, and more specifically two of them:

  * •

Existing systems and evaluation methods are based on small texts, and are not
fully relevant when addressing long texts like novels;

  * •

Existing evaluation methods tend to consider all the entities on an equal
basis, regardless of the linguistic properties of the entity, which is not the
case for most practical applications.

We will also point out a more general issue: the fact that none of the
existing evaluation methods is really interpretable by a human.

The paper is structured as follows: we first remind the reader with state-of-
the art metrics and the main issues that have been reported about them. We
then show that these measures have been defined for small texts but long texts
like fictions tend to exacerbate the issues detailed in the previous section.
We finish with some proposals based on a typology of coreference chains, to
get more readable and actionable results.

##  2 State of the Art in Coreference Evaluation Methods

Coreference evaluation metrics can be divided in two categories: link-based
and mention-based metrics. Roughly, link-based metrics represent entities as a
set of links and mention-based represent them as a set of mentions. In this
section, we just give a brief overview of the most popular methods and of
their relative weaknesses, as more detailed overviews already exist. The
interested reader can refer to (Borovikova et al., 2022; Lion-Bouton et al.,
2020; Moosavi and Strube, 2016; Moosavi, 2020). Traditionally, in the
literature on coreference, ‘key’ refers to the reference (gold annotations),
and ‘response’ corresponds to the computed links.

MUC (Vilain et al., 1995) is only based on the minimum number of missing/extra
links in the response compared to the key entities. It does not take into
account singletons, nor the relative weight of the different entities (all the
links have the same weight).

BLANC (Recasens and Hovy, 2011) is a metric that was built in a first place to
evaluate coreference on key mentions. It was then adapted (Luo et al., 2014)
to also evaluate response mentions. Contrary to MUC, BLANC uses non-coreferent
links. The major problem with BLANC is that, when the number of gold mentions
in the response entities is larger, the number of non-coreference links is
also larger. In this case BLANC gives higher scores (for both recall and
precision) without knowing if the gold mentions were resolved (Moosavi and
Strube, 2016).

Two other measures, $B^{3}$ and CEAF, are mention-based. In the case of
$B^{3}$ (Bagga and Baldwin, 1998), the overall recall/precision is computed
based on the recall/precision of the individual mentions. For each mention m
in the key entities, $B^{3}$ recall considers the fraction of the correct
mentions that are included in the response entity of m. Similarly to MUC,
$B^{3}$ precision is computed by switching the role of the key and response
entities (Moosavi, 2020).

CEAF (Luo, 2005) relies on a one-to-one mapping, where each key entity is
mapped to only one response entity (CEAFe is based on the mapping at the
entity level, a variant CEAFm works at the mention level). Via this mapping,
CEAF uses a similarity metric $\phi$(i,j) to measure the similarity of the
mapped entities with the reference. The one-to-one mapping at the heart of
CEAF leads to ignoring other correct decisions of unaligned response entities.
Finally, CEAFe does not assign weights to the different entities: it can be
seen as a metric that gives a global score of the proportion of entities that
were found in the response (for recall) and the proportion of correct entities
in the response (for precision).

All these metrics have been studied, compared and criticized, esp. in (Moosavi
and Strube, 2016; Moosavi, 2020). They suffer from a number of known issues,
especially what (Moosavi and Strube, 2016) call the “mention identification
effect”. The authors propose the following experiment: they remove spurious
mentions from the responses entities (the ones calculated automatically). This
should automatically increase precision (since there are then less spurious
mentions), while having no effect on recall. Surprisingly, the authors observe
the following, “(i) recall changes for all the metrics except for MUC; (ii)
both $CEAF_{e}$ recall and precision significantly decrease; and (iii) BLANC
recall notably decreases so that F1 drops significantly in comparison to Base”
(Moosavi and Strube, 2016).

As a consequence, they propose LEA, a new method supposed to overcome the
previous limitations. LEA (Moosavi and Strube, 2016; Moosavi, 2020) represents
the importance of each entity through a weight (meaning that LEA considers
that a spurious singleton is less harmful than a spurious long entity). Since
LEA is link-based, the metric is not affected by the mention-identification
effect (Moosavi and Strube, 2016), which causes mention-based metrics scores
to be less interpretative. Finally, even if LEA gives a representation to
singletons (i.e. a self-link), a singleton is considered to be found only if
the singleton is correctly resolved.

Despite these improvements, it has been demonstrated that LEA also suffers
from a number of problems (Lion-Bouton et al., 2020). More specifically, LEA
does not support specific configurations (esp. what the authors of (Lion-
Bouton et al., 2020) call ‘triangular inequality’, which is not a rare case).
Very similar configurations may lead to very different scores, which means
systems cannot be compared and results cannot be directly interpreted.

##  3 The Inadequacy of Traditional Evaluation Schemes for Literary texts

In this section, we observe the application of the previous measures (that
constitute de facto standard, despite their limitations) to the analysis of
long texts, especially fictions and novels.

###  3.1 The Inadequacy of Reference Corpora

Coreference is usually evaluated on a specific corpus called OntoNotes.
OntoNotes was built in the 2000s by a group of researchers from different
American universities and covering a number of structural annotations (Hovy et
al., 2006). This corpus has been especially popular for coreference
evaluation, as it is one of the very rare resources available for the task. It
gathers different kinds of texts (although they are mostly news) and is thus
considered quite balanced and representative.

However, the OntoNotes corpus suffers from a number of shortcomings. The
corpus is made of short texts (including short extracts of longer texts),
which means coreference chains are local and limited in size. The task is thus
limited in its complexity, for example the set of antecedents for a given
pronoun is small, given the short size of the text to consider. Coreference is
also centered around pronouns, whereas real texts provide a wider diversity of
coreferences, especially definite descriptions (”the lady”, ”the milkman”).

Long texts, especially novels, are of course very different. Table 1 reports
some numbers that show a clear difference between OntoNotes and a novel (Manon
Lescaut in this case — numbers would be different with other novels but the
same tendency could be observed).

| Ontonotes | Manon Lescaut
---|---|---
| (average over |
| 3,493 documents) |
# of mentions | 56 | 11,888
# of chains | 13 | 143
# of mentions per Chains (incl. singletons) | - | 59
# of mentions per Chains (without singleton) | 4.1 | 83
# of singletons | 0 | 56
# of tokens | 466 | 77,232
Table 1: Comparison of Ontonotes with a typical French novel, Manon Lescaut.

As one can see, the differences are huge. While OntoNotes documents are made
of a few short chains (4.1 mentions per chain on average), novels like Manon
Lescaut have 83 mentions per coreference chain on average. Moreover the
distribution of chain lengths is very skewed, as nearly half of the mentions
in Manon Lescaut correspond to one character only (the so called Chevalier Des
Grieux), see Figure 1.

figureThe distribution of coreference chains in Manon Lescaut, based on the
number of mentions per chain. As one can see, the distribution is not even
Zipfian (in which case, the curve would be straight), the number of very long
chains being extremely limited. Experiments with other novels, by Balzac among
others, show a quasi Zipfian curve, showing that more characters with first
and secondary roles appear in Balzac’s world.

Even if Manon Lecaut is thus an extreme case, most novels are centered around
a handful of characters that correspond to most of the mentions. Lastly, one
may notice differences due to the annotation schemes, as OntoNotes does not
consider singletons for the coreference task. As a conclusion, we think that
even if OntoNotes is a de facto standard for coreference evaluation, it cannot
be used as an indicator of future performance on the fiction domain.

###  3.2 The Low Readability of Existing Evaluation Methods

Besides the known limitations of existing evaluation methods that we have seen
in section 2, these metrics are not ideal in that they all give different
scores for a same input (but these differences are not directly
interpretable). For example, Pradhan et al. (Pradhan et al., 2014) describe
precisely how these metrics work on a toy example. In the end, their example
corresponds to the following scores (F-measure): 0.4 for MUC, 0.46 for B3,
0.52 for CEAF and 0.36 for Blanc (so 0.16 difference between Blanc and CEAF).
Given that different choices can be made concerning the way to calculate
precision and recall (mention-based or relation-based, taking into account
chain length or not, etc.) it is normal that each method has different scores
for precision and recall. However, this remains a major issue.

The problem is exacerbated in the case of long novels, as one can imagine.
Other issues concern the reference: some corpora/experiments consider only
pronouns, some others also definite descriptions (like “the coachman”), which
should be part of the task but are harder to annotate and evaluate. For
example, Bamman et al. (Bamman et al., 2020) say that “OntoNotes and LitBank
do not link two generic common nouns, while PreCo does” (OntoNotes, LitBank
and PreCo are three corpora used in Bamman et al.’s paper), which makes the
comparison between the different results delicate. The annotation of
singletons also differs a lot from one corpus to the other. All these are
known problems, that have reinforced the status of OntoNotes as a de facto
standard (since OntoNotes provides stability, although the corpus is limited
and not always relevant).

Given all these differences, a known strategy consists in calculating the mean
of the different evaluation scores (or of a subset of these scores – for
example (Denis and Baldridge, 2009) propose to combine MUC, CEAF and B3), a
score sometimes called the CoNLL score (since it was used at CoNLL 2012
(Pradhan et al., 2012)). In our opinion, this is probably worse than keeping
the different scores apart: since these scores have few in common, calculating
the mean does not really make any sense (like calculating the mean of an
heterogeneous set of elements).

These scores may be useful in an NLP perspective, if one really needs or wants
to compare systems (although we, and many others, have shown that this
comparison is dubious). But they are probably useless in a Digital Humanities
perspective (or just from a linguistic point of view), where one needs to have
a clear view of what is going on. In this context, the key question is to know
what kinds of coreference chains are accurately analyzed, and which ones may
suffer from errors.

##  4 Proposals for a Better Evaluation of Coreference in Long Documents

Following what we have shown in the previous sections, we think that a unique
score is unable to take into account all the complexity of coreference.
Moreover, a unique score (or even a combination of scores) is not really
useful in the context of fictions, since no score is really interpretable in
itself and scores are not comparable with each other.

It would thus make more sense to promote indicators based on the linguistic
content that is measured. In the case of fictions for example, at least three
different cases should be considered:

  * •

long coreference chains, especially those including a named entity (i.e. where
one of the mentions is a named entity). This generally corresponds to main
characters and is especially important for most analyses (esp. representing
the plot).

  * •

singletons. We have seen that most metrics have a hard time with singletons,
since by definition a singleton (a single mention) is not a chain (that
requires at least two mentions linked together). Hence the choice of MUC (that
ignores singletons) or LEA (that consider singleton as an entity with a link
to itself). Singletons are important to evaluate the “population in a novel”
(the number of characters in a novel) but should be considered in isolation
and probably not as a coreference element (of course, a singleton can be
included in a longer coreference chain by mistake, but in such a case, it
would just be a mistake that is independent from the singleton issue).

  * •

remaining coreference chains refer for the most part to secondary characters
or just ‘background characters’, who are generally not named but can be
important for the plot. Being able to evaluate the accuracy of the analysis on
this kind of character is thus key. It can also be problematic, as the issue
is not always fully decidable. For example, if there are different mentions of
‘coachman’ in a novel (referred to just as ’coachman’), should we consider
that they all refer to the same character, or not? This kind of precision is
probably not possible in a number of cases in each novel, which means that
calculating precisely the “population of a novel” is not possible with perfect
accuracy (Woloch, 2004; Lavocat, 2020).

Traditional scores can then be calculated for the different cases and thus
give a better picture of the situation at stake. This method may make it more
difficult to compare different systems, but existing measures do not allow
such a comparison anyway. A more fine-grained analysis based on the linguistic
content would probably be more useful for digital humanists than current NLP-
based way of reporting results.

For example, when analyzing the
BookNLP111https://www.lattice.cnrs.fr/projets/booknlp/ scores on the French
LitBank222https://github.com/lattice-8094/fr-litbank, it was observed that
long coreference chains were sometimes broken in different parts, thus showing
the necessity to reinforce the clustering dimension of the algorithm (this
procedure makes it possible to calculate coreference over different mentions
of a same entity, thus reinforcing the quality of the analysis despite surface
variation).

The same strategy can be used to track the analysis of singletons for example.
This is in line with the approach described in (Bamman et al., 2020), where
the authors propose some manipulations and an evaluation with the $B^{3}$
metric to study precisely the treatment of singletons with their approach
(section 6.1, page 51 of (Bamman et al., 2020)). The same approach can be done
on the French LitBank to refine this part of the analysis.

##  5 Conclusion

In this short paper, we presented known issues with evaluation scores related
to coreference, and we have shown that this inadequacy is especially
meaningful for long texts. We have proposed a new way of evaluating
coreference, taking into account the application context (digital humanities)
as well as the need to have more informative and interpretable metrics. In the
future, we will apply this approach to more novels, in different languages, to
ensure the approach gives an appropriate representation of the problem, help
the user spot weaknesses in her/his models and, above everything, ensure that
the metrics used are both readable and interpretable.

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