Quadrupolar and dipolar phases of excitons in transition-metal dichalcogenide trilayer heterostructures

TL;DR

This work addresses quantum phases of trilayer TMD excitons with fluctuating dipole moments induced by interlayer charge tunneling. It builds an effective bosonic model including an Ising dipole degree of freedom per exciton and a dipole-dependent interaction, analyzed by numerically exact path-integral QMC to map the low-temperature phase diagram as a function of density $n$ and out-of-plane field $E_z$, with scales set by the dipole length $d$ and Coulomb energy $\varepsilon=e^2/(\kappa d)$. The study reveals a quadrupolar superfluid in the dilute, high-tunneling regime, a self-bound droplet stabilized by antiparallel attraction at reduced tunneling, a partially fragmented high-density superfluid, and, at large mass, a staggered dipolar crystal that can be melted into a polarized superfluid by $E_z$. The dependence of exciton energy shifts $\mu_X(E_z)$ on density and field provides direct experimental fingerprints and guides future searches for these quantum phases in trilayer TMDs, including Moiré-engineered platforms where heavier effective masses may stabilize the crystal.

Abstract

Recent experiments on trilayer transition-metal dichalcogenide heterostructures have revealed the rich behavior of dipolar excitons. Motivated by these experimental observations, we investigate the collective dynamics of planar quantum dipoles whose orientation fluctuates due to charge tunneling between the outer layers. Using large-scale quantum Monte Carlo simulations, we map out the low-temperature phase diagram as a function of experimentally tunable parameters. We uncover a diverse landscape of phases driven by dipolar correlations. Under strong dipole fluctuations, a quadrupolar superfluid emerges. Suppressing charge tunneling nucleates a droplet state stabilized by the attractive interaction between antiparallel dipoles. At high exciton densities, the system gives way to a partially fragmented condensate, characterized by competing quadrupolar and dipolar superfluid states. Furthermore, at a large exciton mass and high density, we find a staggered dipolar crystal. Our detailed study of the dependence of exciton energy shifts on an external electric field directly interprets existing experimental data and underscores the crucial role of the antiparallel dipolar configuration. Our results provide a guide for future experimental explorations of quantum phases of trilayer excitons.

Figures (20)

• Figure 1: Trilayer system, model and phases. (a) Side-view schematic of a representative WSe$_2$/MoSe$_2$/WSe$_2$ trilayer heterostructure. The two dipolar exciton eigenstates carry opposite dipole moments, pointing up or down, based on the hole location. The quadrupolar exciton state is an equal superposition of the hole states across the outer layers. The energy splitting between the symmetric quadrupolar ground state and the excited dipolar states is given by the hole tunneling rate, $\Delta$. An experimental bias is introduced via an external out-of-plane electric field, $E_z$, applied by gating the outer layers. (b) In the dipolar basis representation, $\hat{\sigma}^{x}$ generates dipole flips corresponding to hole tunneling events, and $\hat{\sigma}^{z}$ describes the coupling of dipoles to the external electric field. The quadrupolar state is the low-energy eigenstate of $\hat{\sigma}^{x}$, thus given by the symmetric superposition of the two dipolar states. (c) For a realistic exciton mass, $m_X=m_0$, a quadrupolar superfluid phase is realized for the currently studied experimental systems. At reduced tunneling rates, an exciton droplet with significant antiparallel dipolar correlations emerges in the dilute-density limit. At higher densities, the droplet melts into a partially fragmented superfluid, characterized by a non-vanishing condensate weight in the two dipolar states. (d) At larger exciton masses, $m_X\gtrsim 4m_0$, and densities, a staggered dipolar crystal appears. The crystal melts under the application of an external out-of-plane electric field due to dipole moment polarization. The resulting phase is a dipolar superfluid.
• Figure 2: Quadrupolar superfluid regime. (a) A typical QMC world-line snapshot illustrating the quadrupolar superfluid state at temperature $T=6\,\textrm{K}$. The black frame denotes the simulation box edges, over which periodic boundary conditions apply. Blue (orange) color represents the spatial positions of $\ket{\uparrow}$($\ket{\downarrow}$) excitons with projected imaginary time evolution, while white regions indicate empty spaces within the box. (b) The exciton chemical potential as a function of the out-of-plane electric field in trilayer (red) and bilayer (blue and turquoise) geometries, shown at $T=24\,\textrm{K}$, which probes ground-state properties (see \ref{['app:QMC']}). (c) A finite-size analysis of the normalized superfluid stiffness in the quadrupolar phase, showing a BKT transition at the critical temperature $T_{\mathrm{BKT}} \approx 17\,\textrm{K}$. Data points represent different system sizes, and the black line shows the linear function $y=2 k_B T/\pi\rho_0$, divided by the normalizing factor $\rho_0$. In all panels, the exciton density is fixed at $n=1.4\cdot10^{12}\,\textrm{cm}^{-2}$, with panels (a) and (b) computed for $N=64$ excitons.
• Figure 3: Build up of antiparallel correlation as a function of density. (a) Exciton chemical potential, offset by its unbiased value, as a function of the out-of-plane electric field, shown for $\Delta=26\,\textrm{meV}$ and several exciton densities spanning the low- and high-density regimes. Calculations were performed for $N=64$ excitons and in the low-temperature limit, see \ref{['app:QMC']}. (b) The spatial correlation function between opposite dipoles, computed at $T=6\,\textrm{K}$ and for the same exciton densities and system size as in (a). (c) $\tilde{\mu}_X(E_z)$ obtained for a static two-body model of opposite dipoles, using $\Delta=0.2\varepsilon$. Each color corresponds to a fixed in-plane inter-exciton distance $\Delta r$.
• Figure 4: Droplet and partially fragmented superfluid in the low $\Delta$ limit. Panels (a) and (d) show typical QMC world-line snapshots of a droplet and a partially fragmented superfluid, obtained at exciton densities $n=\{0.14,28\}\cdot10^{12}\,\textrm{cm}^{-2}$, temperatures $T=\{1.5,24\}\,\textrm{K}$, and system sizes $N=\{32,256\}$, respectively. Panels (b) and (e) display spatial pair correlation functions $g_{\uparrow\downarrow}(r)$ between excitons with opposite dipole moments, for the droplet and partially fragmented superfluid states, respectively, at several densities within each phase. The curves are normalized to their maxima and computed for $N=64$ and $T=1.5\,\textrm{K}$. (c) The location of the main peak of $g_{\uparrow\downarrow}(r)$ as a function of the density (blue dots), in a semi-logarithmic scale, compared with the mean inter-exciton distance $1/\sqrt{n}$ (red line). The green line indicates the inter-particle separation obtained from the exact two-body solution for opposite dipoles. (f) The thermodynamic limit of the eigenvalue ratio $\lambda_ 2/\lambda_1$, corresponding to the dipole-orientation-resolved superfluid fraction at high densities within the partially fragmented superfluid phase, shown for $T=24\,\textrm{K}$.
• Figure 5: The staggered crystal and its electric-field-induced melting. (a) World-line snapshot of the crystal at exciton density $n=1.4\cdot10^{14}\,\textrm{cm}^{-2}$. The staggered pattern is identified from the momentum-space maps of (b) $S(\vb{k})$ and (c) $S_{\sigma}(\vb{k})$, which peak at distinct Bragg vectors, $\vb{Q}$ and $\vb{Q}'$, corresponding to two square lattices with a relative $\pi/4$ rotation. (d) Exciton chemical potential within the crystalline phase (\ref{['app:QMC']}), plotted as a function of density on a logarithmic scale. The red line shows a linear fit indicating a power law scaling, $\mu_X\propto n^{\gamma}$, with $\gamma=1.40\pm0.03$. (e) World-line snapshot showing the melting of the staggered crystal in (a) into a dipolar superfluid under a strong electric bias $E_z=2\,\textrm{V}\,\textrm{nm}^{-1}$. (f) Exciton chemical potential and (g) superfluid stiffness (normalized by $\rho_0$, blue) together with the Bragg peak amplitude $S(\mathbf{Q})$ (red), plotted as a function of $E_z$. The inset of (f) highlights the onset of melting at low $E_z$. Orange and blue backgrounds in (f)–(g) indicate the staggered crystal and superfluid regions, respectively. (h) Spatial correlation function between parallel dipoles aligned with $E_z$, shown for increasing electric field values from $E_z=0$ (red) to $E_z=3.54\,\mathrm{V}\,\mathrm{nm}^{-1}$ (blue). The black dashed line marks the mean inter-exciton distance $a=1/\sqrt{n}$. Panels (a)-(f), as well as the $\rho_s$ curve in (g), were computed for $N=40$ and $T=48\,\textrm{K}$, where the data are converged with respect to both system size and temperature. The $S(\vb{Q})$ values in (g) were extrapolated to the thermodynamic limit, see \ref{['app:QMC']}. Panel (h) was calculated for $N=64$, where the crystal is commensurate with square boundary conditions and exhibits several higher-order correlation peaks.