
2
Fermi superfluids [18–20] and has allowed for interpola-
tion between different regimes, including the BEC-BCS
crossover for varying densities. To get a better under-
standing of the variational subspace, we analyze the el-
ementary neutral excitations of the EXI; we show that
even though the gapless Goldstone branch is not cap-
tured at the single-excitation level, all other collective
modes, including the ones referred to in the literature as
Higgs [21] or Bardasis-Schrieffer [22] modes, are well re-
produced. In computing the elementary excitations, we
extend the geometric approach of [23, 24] to BCS the-
ory, where the BCS state is treated as a Gaussian state
and the collective modes as fluctuations on the Gaussian
manifold. Moreover, we do not assume contact interac-
tions, but rather work with realistic bilayer Keldysh [25–
27] and exciton-electron [28] interactions, marking a key
difference with respect to atomic superfluids where in-
teractions are contact-like and the polaron behavior was
studied as a function of the 3D scattering length tuned
through a Feschbach resonance.
II. FEW-BODY BINDING ENERGIES
Experiments in charge-tunable TMD monolayers have
established that the dominant resonances in the optical
excitation spectra can be identified as attractive and re-
pulsive polarons (AP and RP). In the limit of vanish-
ing doping, the AP resonance energy approaches that
of a T [12]. Similarly, since the ground-state of the bi-
layer system in the small density BEC regime consists
effectively of tightly bound IXs, we expect that the X-IX
energy determines the position of the AP resonance. De-
termining the binding energies of X, IX, T and X-IX, is
therefore key to understand the optical excitation spectra
as a function of interlayer separation.
We compute the binding energies of different com-
plexes in a MoS2/hBN/WSe2heterostructure using the
diffusion QMC method as implemented in the casino
package [29]. prisedcomIn particular, as sketched in
Fig. 1a, we calculate the binding energies of the MoS2
X, the IX made of an electron in MoS2and a hole in
WSe2, the T in MoS2and the X-IX. Our calculation is
an extension of Refs. 30 and 31 to a bilayer system; im-
portantly, the trial wavefunctions decay exponentially at
large separations and satisfy the cusp conditions for the
Keldysh potential [31]. The Jastrow function includes
smoothly truncated polynomial expansions in interparti-
cle distances [32].
The QMC approach can only treat the intraband sec-
tors of the Coulomb operator. Even though the electron-
hole exchange terms do modify the binding energy of T,
and X-IX [31, 33], their contribution should be small as
compared to the actual binding energy. Based on that
we expect QMC calculation to provide a good estimate
of the binding energies.
We take me=mh= 0.55m0for the CB and VB masses
in MoS2, and mh= 0.40m0for the VB in WSe2. These
values match the reduced masses reported in the quan-
titative investigation by Goryca et al. [34]. We neglect
the less known mass ratio imbalance, having verified that
the binding energies are not very sensitive to it [33]. As
detailed in SI.A, we use the bilayer Keldysh potential [27]
for electron-hole and electron-electron interactions: We
take the relevant screening lengths from Ref. [34], and
use = 4.5for the dielectric constant of the hBN envi-
ronment. With these parameters, for a MoS2monolayer,
we obtain 217 meV for the X binding energy and 18 meV
for the trion, which are in good agreement with the ex-
perimental values in encapsulated samples [35].
Figure 1.b shows the calculated energy of the few-
body complexes as a function of the interlayer distance
d, having set as reference to zero the energies of the band
edges. We express din units of the thickness of a single
hBN layer (L1= 0.33 nm), and plot X, T, IX,X-IX for
d= 1,2,4,...,128 ×L1. We note that the MoS2exci-
ton and trion energies also depend on d, since the WSe2
monolayer contributes to screening. The binding energies
of T and X-IX, extracted from the energies depicted in
Fig. 1b are plotted in Fig. 1.c; here the binding energy Eb
for X-IX is given by Eb=EX+EIX −EX−IX. For large
interlayer distances yielding EXEIX , the X-IX state
can be described as a strongly bound negatively charged
intralayer trion T loosely bound to a hole in the WSe2
layer (see SI.B for the probability distribution functions).
III. MANY-BODY MODEL
To analyze optical excitation spectrum in the presence
of an EXI, we assume that the X can be treated as a
point-like quantum impurity with no internal degrees of
freedom. We also assume that IXs are spin-valley polar-
ized and a few-layer-thick hBN separates MoS2and WSe2
layers, ensuring that there is no electronic moiré poten-
tial. For simplicity, we consider the particle-hole sym-
metric case, with equal electron and hole masses (me=
mh=m) and set chemical potentials µe=µh=µ. In
practice, µe, µhcan be tuned either by connecting the
two layers to different reservoirs to form a biased junc-
tion or by applying a normal electric field Ez. In a recent
experiment [10] both mechanisms have been employed to
overcome the semiconductor band gap. Our theory can
be equally applied to both scenarios.
The total system Hamiltonian is given by
H=Hel +Himp +HW.(1)
Denoting CB and VB electrons respectively as a≡
ac, b ≡av, the electronic Hamiltonian reads
Hel =X
k
εk(a†
kak+bkb†
k)+ 1
2AX
kpq
U(q)(a†
k+qa†
p−qapak+
+bk+qbp−qb†
pb†
k)−1
AX
kpq
V(q)a†
k+qbp+qb†
pak(2)