Holographic Pair and Charge Density Waves

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We examine a holographic model in which a U(1) symmetry and translational invariance are broken spontaneously at the same time. Our construction provides an example of a system with pair-density wave order, in which the superconducting order parameter is spatially modulated but has a zero average. In addition, the charge density oscillates at twice the frequency of the scalar condensate. Depending on the choice of parameters, the model also admits a state with co-existing superconducting and charge density wave orders, in which the scalar condensate has a uniform component.