Nonbosonic Moiré Excitons

Optical excitations in moiré transition metal dichalcogenide bilayers lead to the creation of excitons, as electron-hole bound states, that are generically considered within a Bose-Hubbard framework. Here, we demonstrate that these composite particles obey an angular momentum commutation relation that is generally nonbosonic. This emergent spin description of excitons indicates a limitation to their occupancy on each site, which is substantial in the weak electron-hole binding regime. The effective exciton theory is accordingly a spin Hamiltonian, which further becomes a Hubbard model of emergent bosons subject to an occupancy constraint after a Holstein-Primakoff transformation. We apply our theory to three commonly studied bilayers (${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$, ${\mathrm{WSe}}_2/{\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2/{\mathrm{MoS}}_2$) and show that in the relevant parameter regimes their allowed occupancies never exceed three excitons. Our systematic theory provides guidelines for future research on the many-body physics of moiré excitons.

Figure 1

(a) Diagram for the charge exchange scattering between two excitons $\widehat{x}$. The strength is captured by the exchange integral $\mathrm{\Lambda }$. In such a process, two incoming excitons (purple shaded) swap their electrons ($c$ and $c^{\prime }$, blue dots) while keeping their holes ($v$ and $v^{\prime }$, red dots). Note that the hole-exchanged diagram is topologically equivalent. (b) Disconnected diagram for a two-exciton process reminiscent of two free bosons. (c) Illustration of moiré excitons on top of a superlattice potential (yellow) when charge exchange is suppressed ($\mathrm{\Lambda }=0$), allowing for arbitrary exciton occupation. This situation occurs at $a_W^x\gg a_B$, where $a_W^x$ is the center-of-mass width of the exciton Wannier orbital (green) and $a_B$ is the Bohr radius. $a_M$ is the superlattice constant. (d) Moiré excitons with the strongest charge exchange ($\mathrm{\Lambda }=1$), occurring at $a_W^x\ll a_B$. Double occupancies (per supersite and valley) are prohibited in this case.

Figure 2

Exchange integral $\mathrm{\Lambda }$ and the corresponding occupancy upper bound ${\nu }_{\max }$ for interlayer excitons in several $R$-stacked TMD bilayers at various twisting angles (see Supplemental Material [39] for details of the derivation based on parameters of Refs. [6, 8, 10, 17, 46, 47]). Dashed vertical lines correspond to twisting angles realized in literature [8, 10, 17].

Figure 3

Exchange integral $\mathrm{\Lambda }$ at various $a_B/a_W^x$ for interlayer distance $d_z=0$ (or equivalently intralayer excitons). $a_W^x$ is tuned by $|Z|$, considered as a free parameter in this plot. Data shown utilize parameters (except $|Z|$) from ${\mathrm{WSe}}_2/{\mathrm{WS}}_2$ [39] at $a_M=8\mathrm{nm}$. Red and blue curves are from perturbative wave functions in the strong Coulomb and deep moiré regimes, respectively [39], with values in their regimes of validity (indicated by opacity) perfectly matching the numerics.