Inhomogeneous Phases in Holographic Superfluids

We discuss inhomogeneous solutions of a gravitating system consisting of two $U(1)$ gauge fields and a real scalar field. One of the $U(1)$ gauge fields determines the chemical potential, whereas the other one corresponds to a magnetic field interacting with the spin in the boundary theory. We solve the field equations and find a second-order phase transition to an inhomogeneous phase at a critical temperature which we compute. Below the critical temperature, the equations are solved perturbatively, and a spatially dependent charge density is generated. This is compatible with the generation of a charge density wave in condensed matter systems.