Moiré exciton condensate: Nonlinear Dirac point, broken-symmetry Bloch waves, and unusual optical selection rules

Moir\'e excitons feature tunable Dirac dispersion and spatially dependent optical selection rules. With its long lifetime due to its interlayer nature, it is promising to realize a Bose-Einstein condensation of moir\'e excitons. Here we study the properties of moir\'e exciton condensate within the mean-field theory, with special focus on the exciton-exciton interaction effect on the nonlinear Bloch band and Bloch waves of exciton condensate in moir\'e potential. We find the nonlinear dispersion of the moir\'e exciton condensate exhibits a complex loop structure induced by exciton-exciton interaction. A nonlinear Dirac cone emerges at the $\mathbit{\ensuremath{\Gamma}}$ point of the Bloch band, with a threefold degenerate nonlinear Dirac point. Each degenerate Bloch wave at the nonlinear Dirac point spontaneously breaks $C3$ rotational symmetry of the moir\'e potential and itself differs by $C3$ rotations. Since they reside in the light cone, these nontrivial Bloch band structures and broken-symmetry Bloch waves can be experimentally detected by examining light emission from those states. Symmetry breaking of Bloch states implies unusual optical selection rules compared with the single-exciton case: moir\'e exciton condensate at the $\mathbit{\ensuremath{\Gamma}}$ point emits light with all three components of polarization instead of only left or right circular polarization. We further propose that with a force applied on the exciton center of mass by engineering the potential energy of excitons to drive an initially optically dark exciton condensate towards the light cone, the light polarization of final states as well as their dependence on force direction serves as the smoking gun for experimental observation.