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arXiv:2602.18194

Correlated phases of moat-band excitons in two dimensions

L. Maisel Licerán, S. H. Boeve, H. T. C. Stoof

cond-mat.quant-gascond-mat.mes-hallcond-mat.othercond-mat.str-elquant-phSubmitted Feb 20, 2026• Announced Feb 23, 2026• Revised Jul 14, 2026 (v4)
v1 · Feb 20, 2026v2 · Mar 16, 2026v3 · Jun 6, 2026v4 · Jul 14, 2026
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Abstract

We study dilute two-dimensional systems of interacting excitons with a moat dispersion, whose ground-state manifold consists of a set of discrete or continuously degenerate energy minima. At low densities and in the presence of contact interactions, it is known that the bosons can undergo statistical transmutation and give rise to a chiral spin liquid. Here, we show that long-range interactions such as those expected in excitonic systems introduce a rich competition between the chiral spin liquid and different kinds of Bose-Einstein condensates. The moat dispersion can favor Bose-Einstein condensation into states occupying multiple momenta, leading to inhomogeneous and supersolid phases with a highly anisotropic superfluid response. We demonstrate that a proper $T$-matrix renormalization of the exciton-exciton interaction is essential for describing these phases and show that they can arise even from purely repulsive interactions. This formalism is employed to obtain the phase diagram of competing homogeneous and stripe condensates with the chiral spin liquid in an electron-hole bilayer model. In addition, we show how the $T$ matrix enters the familiar Gross-Pitaevskii framework and map out extended phase diagrams within a pseudopotential approximation. We place our findings in the context of real excitonic systems by discussing the roles of finite lifetimes, disorder, band-structure warping, and nonzero temperatures. We conclude that moat bands can drive Bose-Einstein condensation and supersolidity already at weak coupling, in contrast to the case of a standard parabolic dispersion.

Journal reference
Physical Review B 114, 065409 (2026)
DOI
10.1103/fdnd-hdrv
Comments
Final published version. 26 pages, 11 figures

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We study dilute two-dimensional systems of interacting excitons with a moat dispersion, whose ground-state manifold consists of a set of discrete or continuously degenerate energy minima. At low densities and in the presence of contact interactions, it is known that the bosons can undergo statistical transmutation and give rise to a chiral spin liquid. Here, we show that long-range interactions such as those expected in excitonic systems introduce a rich competition between the chiral spin liquid and different kinds of Bose-Einstein condensates. The moat dispersion can favor Bose-Einstein condensation into states occupying multiple momenta, leading to inhomogeneous and supersolid phases with a highly anisotropic superfluid response. We demonstrate that a proper TTT-matrix renormalization of the exciton-exciton interaction is essential for describing these phases and show that they can arise even from purely repulsive interactions. This formalism is employed to obtain the phase diagram of competing homogeneous and stripe condensates with the chiral spin liquid in an electron-hole bilayer model. In addition, we show how the TTT matrix enters the familiar Gross-Pitaevskii framework and map out extended phase diagrams within a pseudopotential approximation. We place our findings in the context of real excitonic systems by discussing the roles of finite lifetimes, disorder, band-structure warping, and nonzero temperatures. We conclude that moat bands can drive Bose-Einstein condensation and supersolidity already at weak coupling, in contrast to the case of a standard parabolic dispersion.
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We discuss bosonic models with a moat spectrum, where in momentum space the minimum of the dispersion relation is on a sphere of nonzero radius. For spinless bosons with O(N)O(N)O(N) symmetry, we emphasize the essential difference between N=2N=2N=2 and N>2N > 2N>2. When N=2N=2N=2, there are two phase transitions: at zero temperature, a transition to a state with Bose condensation, and at nonzero temperature, a transition to a spatially inhomogeneous state. When N>2N > 2N>2, previous analysis suggests that a mass gap is generated dynamically at any temperature. In condensed matter, a moat spectrum is important for spin-orbit-coupled bosons. For cold nuclear or quarkyonic matter, we suggest that the transport properties, such as neutrino emission, are dominated by the phonons related to a moat spectrum; also, that at least in the quarkyonic phase the nucleons may be a non-Fermi liquid.
We experimentally engineer a moatlike dispersion in a system of weakly interacting bosons. By periodically modulating the amplitude of a checkerboard optical lattice, the two lowest isolated bands are hybridized such that the single particle energy displays a continuum of nearly degenerate minima that lie along a circle in reciprocal space. The moatlike structure is confirmed by observing a zero group velocity at nonzero quasimomentum and we directly observe the effect of the modified dispersion on the trajectory of the center of mass position of the condensate. We measure the lifetime of condensates loaded into different moat bands at different quasimomenta and compare to theoretical predictions based on a linear stability analysis of Bogoliubov excitations. We find that the condensate decay increases rapidly as the quasimomentum is decreased below the radius of the moat minimum, and argue that such dynamical instability is characteristic of moatlike dispersions, including spin-orbit coupled systems. The ground state of strongly interacting bosons in such degenerate energy landscapes is expected to be highly correlated, and our work represents a step toward realizing fractional quantum Hall-like states of bosons in an optical lattice.
Motivated by the fundamental question of the fate of interacting bosons in flat bands, we consider a two-dimensional Bose gas at zero temperature with an underlying quartic single-particle dispersion in one spatial direction. This type of band structure can be realized using the NIST scheme of spin-orbit coupling [Y.-J. Lin, K. Jim\'enez-Garcia, and I. B. Spielman, Nature (London) 471, 83 (2011)], in the regime where the lower-band dispersion has the form ${\ensuremath{\varepsilon}}_{\mathbit{k}}\ensuremath{\sim}{k}_{x}^{4}/4+{k}_{y}^{2}+...$, or using the shaken lattice scheme of Parker et al. [C. V. Parker, L.-C. Ha, and C. Chin, Nat. Phys. 9, 769 (2013)]. We numerically compare the ground-state energies of the mean-field Bose-Einstein condensate (BEC) and various trial wave functions, where bosons avoid each other at short distances. We discover that, at low densities, several types of strongly correlated states have an energy per particle ($\ensuremath{\epsilon}$), which scales with density (nnn) as $\ensuremath{\epsilon}\ensuremath{\sim}{n}^{4/3}$, in contrast to $\ensuremath{\epsilon}\ensuremath{\sim}n$ for the weakly interacting Bose gas. These competing states include a Wigner crystal, quasicondensates described in terms of properly symmetrized fermionic states, and variational wave functions of Jastrow type. We find that one of the latter has the lowest energy among the states we consider. This Jastrow-type state has a strongly reduced, but finite, condensate fraction, and true off-diagonal long-range order, which suggests that the ground state of interacting bosons with quartic dispersion is a strongly correlated condensate reminiscent of superfluid helium-4. Our results show that even for weakly interacting bosons in higher dimensions, one can explore the crossover from a weakly coupled BEC to a strongly correlated condensate by simply tuning the single-particle dispersion or density.
Strongly interacting bosons display a rich variety of quantum phases, the study of which has so far been focused in the dilute regime, at a fixed number of particles. Here we demonstrate the formation of a dense Bose-Einstein condensate in a long-lived dark spin state of 2D dipolar excitons. A dark condensate of weakly interacting excitons is very fragile, being unstable against a coherent coupling of dark and bright spin states. Remarkably, we find that strong dipole-dipole interactions stabilize the dark condensate. As a result, the dark phase persists up to densities high enough for a dark quantum liquid to form. The striking experimental observation of a step-like dependence of the exciton density on the pump power is reproduced quantitatively by a model describing the nonequilibrium dynamics of driven coupled dark and bright condensates. This unique behavior marks a dynamical condensation to dark states with lifetimes as long as a millisecond, followed by a brightening transition at high densities.
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 $\mathbit{k}$ 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.