Spin-orbit-coupled depairing of a dipolar biexciton superfluid

We consider quantum phase transitions in a system of bright dipolar excitons which can form bound pairs (dipolar biexcitons). The biexciton energy is tuned from negative to positive values through the scattering threshold. At sufficiently large density an exciton superfluid transforms into a superfluid of biexcitons. With the average relative momenta of excitons in the pairs being beyond the light cone, the transition is accompanied by a reduction of the photoluminescence intensity. Effective magnetic fields due to the long-range exchange splitting of exciton states shift the position of the gap in the elementary excitation spectrum to a circle of degenerate minima in the $\mathit{k}$ space. Closing the gap results in the formation of exciton stripes polarized linearly along the direction of their translational symmetry. In the biexciton energy vs density phase diagram the novel phase intervenes between the dark biexciton and radiative exciton superfluids. We conclude that formation of a BCS-like biexciton condensate induces correlated alignment of the effective magnetic fields and excitonic spins. We outline important differences in the predicted mechanism from the phenomenon of spin-orbit-coupled Bose-Einstein condensation.

Figure 1

Sketch of the two-body interaction potentials for excitons with parallel (dashed line) and antiparallel (solid line) spins. The latter features a biexciton resonance with energy $\varepsilon$ and natural width $\beta$ defined by tunneling of excitons under the barrier due to the dipolar repulsion. The inset in the top right corner shows orientations of the effective magnetic field $\mathit{\Omega }\left(\mathit{k}\right)$ with respect to the exciton wave vector $\mathit{k}$. The field $\mathit{\Omega }\left(\mathit{k}\right)$ couples the $|\uparrow \downarrow \rangle$ and $|\uparrow \uparrow \rangle$ (or $|\downarrow \downarrow \rangle$) scattering channels. The bottom inset shows a pair of dipolar excitons (X) formed of spatially separated electrons ($e$) and holes (h).

Figure 2

Sketches of the two possible phase diagrams of a resonantly paired exciton condensate with $\beta =0$ in the radiative regime ($\left|\varepsilon \right|<1$ in units of ${\hslash }^2{q}_0^2/m$). Depending on whether $\sqrt{g_{\mathrm{X}}{g}_{\mathrm{B}}}<g_{\mathrm{X}\mathrm{B}}$ or $\sqrt{g_{\mathrm{X}}{g}_{\mathrm{B}}}>g_{\mathrm{X}\mathrm{B}}$, the transition to the biexciton phase (XX) at $\varepsilon >0$ may be either of the first or the second kind, respectively. The second-order transition occurs via an intermediate mixed phase (X-XX), where a fraction of biexcitons is dissolved in the exciton superfluid (X).

Figure 3

The phase diagram of a paired exciton condensate with spin-orbit coupling. We adopt units of ${\hslash }^2{q}_0^2/m$, where $q_0$ corresponds to the boundary of the light cone. The insets show exemplary spectra of elementary excitations in the biexciton phase [$\mathit{\Omega }$-XX, dark gray; Eq. (13)]. Red dashed lines account for the exchange term in the kinetic energy in Eq. (11). Closure of the rotonlike gap defines a second-order phase boundary with a new $\mathit{\Omega }\text{−}\mathrm{X}\text{−}\mathrm{XX}$ phase (magenta) characterized by smectic order and transverse linear polarization of excitons [Eq. (16)]. The hatch in the vicinity of $\left|\varepsilon \right|=0$ marks the region where the magnitude of ${\mathit{p}}_0$ becomes inferior to $q_0$. In the region $\varepsilon >0$ the $\mu =\varepsilon /2$ line marks a boundary with the radiative exciton superfluid X (light gray). We have used the parameters typical for a ${\mathrm{MoS}}_2$ homobilayer. More details can be found in Appendixes pp2 and pp3.

Figure 4

The magnitude of the wave vector ${\mathit{p}}_0$ as a function of the chemical potential $\mu$ along the line $\varepsilon =4$ of the phase diagram shown in Fig. 3. Units of ${\hslash }^2{q}_0^2/m$ are used. In the $\mathit{\Omega }$-XX phase (dark gray area) the magnitude of ${\mathit{p}}_0$ corresponds to the rotonlike minimum in the pair-breaking excitation spectrum. In the $\mathit{\Omega }\text{−}\mathrm{X}\text{−}\mathrm{XX}$ phase (magenta) the wave vector ${\mathit{p}}_0$ defines the period and the local orientation of the exciton stripes pictured schematically in the inset. The transverse arrow depicts the linear polarization of excitons. The magnitude of ${\mathit{p}}_0$ here is a solution of Eq. (14).