Holographic superconductors at zero density

We construct holographic superconductors at zero density. The model enjoys a luxury property that the background geometry dual to ground state is analytically available. It has hyperscaling-violating in IR and asymptotically AdS UV. Classification by geometries gives new insights on supergravity solutions. numerically finite temperature solution of hairy black holes verify phase transition tuning double-trace deformation parameter. For superconductor from M-theory, we obtain an analytic AC conductivity, which explicitly shows superconducting delta function hard gap.