Light-induced exciton spin supersolid in atomic double layers

We propose an optical scheme to realize an exciton spin supersolid in an excitonic insulator formed in atomic double layers. By coupling exciton internal levels with three spatially modulated laser fields in an inverted-tripod configuration, an effective non-Abelian gauge potential is generated for the center-of-mass motion of excitons. This gauge field shifts the energy minima to finite in-plane momenta, driving the system into two distinct exciton condensate phases: a plane-wave condensate with uniform phase coherence, and more strikingly, an exciton spin-density wave. The latter phase exhibits a hallmark of spin supersolidity, simultaneously possessing superfluid coherence and periodic modulation of the spin density. Its excitation spectrum reveals two gapless Goldstone modes and a distinct periodic structure, originating from the spontaneous breaking of the $U(1)$ phase symmetry of the condensate and continuous spatial translational symmetry.