Quantum Phase Transitions of Trilayer Excitons in Atomically Thin Heterostructures

We determine the phase diagram of excitons in a symmetric transition-metal dichalcogenide 3-layer heterostructure. First principles calculations reveal interlayer exciton states of a symmetric quadrupole, from which higher energy asymmetric dipole states are composed. We find quantum phase transitions between a repulsive quadrupolar and an attractive staggered dipolar lattice phases, driven by a competition between interactions and single exciton energies. The different internal quantum state of excitons in each phase is a striking example of a system where single-particle and interacting many-body states are coupled.

Figure 1

(a) Calculated GW quasiparticle band structure of the ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ heterostructure. The inset shows the valence energy split at the K region (${\mathrm{\Delta }}^{\pm }$). (b) Quasiparticle wave functions of the two split valence bands at K, and the corresponding schematic representation of a double-well symmetric and antisymmetric wave functions. (c) Exciton transition energies diagram for the two low lying excitons with energies $E_x^{+}$ and $E_x^{-}$, composed of $v\to c$ and $v-1\to c$ transitions, with binding energies $E_b^{+}$ and $E_b^{-}$, respectively. (d) Spatial cross section along the $z$ direction of the probability density of the hole wave function in the quadrupolar (${|{\psi }_h^{+(-)}|}^2$) and dipolar exciton states (${|{\psi }_h^{u(d)}|}^2$). (e) Schematic illustration of dipolar ($p\parallel z$, $p\parallel -z$) and quadrupolar ($p=0$) exciton states.

Figure 2

(a) Electrostatic interaction energy $E_i^{\mathrm{int}}$ vs the density of excitons $n$. stg represents a staggered square lattice of dipolar excitons, where nearest neighbors have antiparallel dipole moments. sym represents a triangular lattice of quadrupolar excitons. The minimum energy (stg) appears at $n_d\approx 0.12d^{-2}$. Dashed black represents an uncorrelated exciton gas (see Ref. [41]). (b) Total energy per particle $E_i^t$ (where $i=\mathrm{sym}/\mathrm{stg}$) vs the density of excitons $n$. GS represents the many body ground state (black lines). Three different phase transitions are displayed: stg $\mathrm{droplet}\to \mathrm{stg}$ lattice ($R_1=0.01$), sym $\mathrm{lattice}\to \mathrm{stg}$ $\mathrm{droplet}\to \mathrm{stg}$ lattice ($R_2=0.05$), sym $\mathrm{lattice}\to \mathrm{stg}$ lattice ($R_3=0.1$). (c) Phase diagram. $R$ represents the ratio of the energy difference between a dipolar and a quadrupolar single exciton state and the potential energy scale $e^2/\kappa d$. Dashed gray lines represent the upper and lower estimates for $R_{\mathrm{WMW}}$, the special case of a ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ trilayer. (d) Schematic illustration of the competing phases in a three-layer structure with dipolar and quadrupolar excitons.