Correlated phases of moat-band excitons in two-dimensional systems

Abstract

We study two-dimensional systems of interacting excitons with a moat dispersion, for which the ground-state energy manifold presents a ring of discrete or continuously degenerate minima around a single point in momentum space. At low densities and for an idealized, perfectly degenerate moat, we show that the excitons undergo statistical transmutation and stabilize a chiral spin liquid. At higher densities, the moat dispersion favors Bose-Einstein condensation into states occupying multiple momenta, leading to inhomogeneous condensate phases and potentially supersolidity. We discuss the impact of band-structure warping present in realistic systems and argue that it generically stabilizes Bose-condensed phases over the chiral spin liquid, and analyze the superfluid response of the former which is unconventional due to the moat band. We also demonstrate that a proper renormalization of the exciton-exciton interaction is essential for describing these phases, and show that even purely repulsive interactions can favor inhomogeneous condensates. To further explore inhomogeneous condensate phases, we employ a Gross-Pitaevskii framework with a pseudopotential approximation and map out the resulting phase diagram. We show that the presence of degenerate dispersion minima can drive supersolidity already at weak coupling, in contrast to systems with a standard parabolic dispersion. Finally, we discuss our results in the context of real excitonic systems and argue that moat-band-induced supersolidity can be within experimental reach for realistic values of the model parameters.

Figures (9)

• Figure 1: Diagrammatic Lippmann--Schwinger equation for the many-body $T$ matrix. The solid lines correspond to many-body propagators and the squiggly lines represent the bare interaction.
• Figure 2: Energy difference $\delta \equiv (\tilde{E}^{\mathrm{(LO)}} - \tilde{E}^{\mathrm{(FF)}}) / N$ on the $\tilde{q}_{\mathrm{m}}$--$\gamma$ plane for a quadratic moat (top) and a quartic one (bottom) for two different values of the small fixed parameter $\Lambda \equiv 2 \rho^{2} \gamma$. In this regime, the ordering vector has the magnitude of the moat momentum and the two phases have vanishing kinetic energy, so the energies coincide with the interaction energies. The dashed black line marks the phase boundary between the homogeneous and inhomogeneous phases. As $\rho = \sqrt{\Lambda / 2 \gamma}$, one observes that a smaller $\rho$ favors the formation of an LO phase at fixed $\tilde{q}_{\mathrm{m}}$ and $\gamma$, as the transition line shifts to smaller $\gamma$ and smaller $\tilde{q}_{\mathrm{m}}$. The calculation takes into account the full $T$ matrix associated with the bilayer exciton--exciton potential of Eq. \ref{['eq:BilayerXXInt']} and shows that the existence of a moat can give rise to translationally broken phases even in the case of weak interactions and dilute exciton densities.
• Figure 3: (a) Landau levels $\tilde{E}^{(l)}_{\mathrm{LL}} = [2(2l + 1) \rho^{2} + \lambda]^{2}$ as a function of the density parameter $\rho$ for $\lambda = {-}1$ and various values of $l$. Below $\rho = \sqrt{-\lambda/2} \approx 0.707$, the system can always perfectly quench the mean-field kinetic energy by distributing particles between two adjacent levels. This is not possible when $\rho$ is larger than this value, and the kinetic energy of the system will increase with the density. (b) Phase diagram in the $n$--$d$ plane showing the competition between the Fulde--Ferrel phase corresponding to a homogeneous Bose--Einstein condensate and the chiral spin liquid phase of fermionized excitons in a bilayer system with the exciton--exciton interaction of Eq. \ref{['eq:BilayerXXInt']}. As a result of the eventual increase of the kinetic energy with the density, at large $n$ the chiral spin liquid disappears in favor of the condensate at a single momentum on the moat. We have used the experimentally reasonable values $q_{\mathrm{m}} = 0.1nm\tothe{-1}$, $\epsilon_{\mathrm{m}} = 10meV$, and $\epsilon_{\mathrm{r}} = 12$ for the relative permittivity of the surrounding medium.
• Figure 4: (a) Particle-hole continuum of the bulk-projected Bernevig--Hughes--Zhang model for bismuth selenide of Ref. liu2010model in the presence of hexagonal warping. The projection onto the low-energy bulk bands follows the procedure of Ref. maisel2023single, and we use the same numerical values for the parameters. Due to cubic terms in $k_{x}$ and $k_{y}$, the dispersion reflects the $C_{3}$ symmetry of the underlying lattice, breaking the perfect rotational symmetry of the quadratic model and featuring six minima around the $\Gamma$ point. The red blobs mark regions that differ from the minimum by at most $1meV$. (b) Horizontal and vertical slices of the dispersion in (a) showing the energy in the $k_{x}$ and $k_{y}$ directions, which contain the largest and smallest minima. The maximum splitting between the two dispersions is about $2meV$ (corresponding to approximately $23.2K$) for realistic warping parameters $R_{1} = 0.05eVnm\tothe{3}$ and $R_{2} = -0.1eVnm\tothe{3}$ (cf. Ref. liu2010model).
• Figure 5: Phase diagram of 1D moat-band excitons with a soft-core interaction $T^{\mathrm{sc}}(x) = U_{\mathrm{sc}} \Theta(1 -|x| / r_{\mathrm{sc}})$. The parameter $\lambda$ is related to the moat size (when $\lambda < 0$) and $\Lambda$ is an effective interaction strength. The left plot shows the result of the full numerical GPE calculation, while the right plot shows the result of the analytical Landau theory. The color coding represents the length of the momentum vector $G$ characterizing each phase, made dimensionless via $\tilde{G} = G r_{\mathrm{sc}}$. The BEC has a constant wave function, $\phi_{\mathrm{BEC}}(x) = \sqrt{n}$ with $n$ the exciton number density, while the FF phase has $\phi_{\mathrm{FF}}(x) = \sqrt{n} \space \mathrm{e}^{\mathrm{i} (G / 2) x}$ with $G = 2 q_{\mathrm{m}}$. These two phases give a homogeneous density profile. In the LO phase, the order parameter near the transition behaves as $\phi_{\mathrm{LO}}(x) \propto \cos (G x / 2)$, leading to a lattice constant $2 \pi / G$ for the density profile $|\phi(x)|^{2}$. The transition between the BEC and FF phases at $\lambda = 0$ is continuous, while that between the homogeneous phases and the LO phase is discontinuous. For a large moat, i.e., negative enough $\lambda$, an inhomogeneous BEC exists even in the limit of weak interactions and low exciton density. The simple Landau theory is in excellent quantitative agreement with the full numerical result in the region of this plot. Deviations start to occur at large $\Lambda$, where the density peaks localize and the order parameter cannot be written as a simple cosine.