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Inhomogeneous superfluids

We show examples of a striped superfluid in a simple $λφ^4$ model at finite velocity and chemical potential with a global $U(1)$ or $U(2)$ symmetry. Whenever the chemical potential is large enough we find flowing homogeneous solutions and static inhomogeneous solutions at any arbitrary small velocity. For the $U(1)$ model the inhomogeneous solutions found are energetically favourable for large enough superfluid velocity and the homogeneous and inhomogeneous phases are connected via a first order phase transitions. On the other hand, the $U(2)$ model becomes striped as soon as we turn on the velocity through a second order phase transition. In both models increasing the velocity leads to a second order phase transition into a phase with no condensate.