  We consider holographic superconductors whose bulk description consists of
gravity minimally coupled to a Maxwell field and charged scalar field with
general potential. We give an analytic argument that there is no "hard gap":
the real part of the conductivity at low frequency remains nonzero (although
typically exponentially small) even at zero temperature. We also numerically
construct the gravitational dual of the ground state of some holographic
superconductors. Depending on the charge and dimension of the condensate, the
infrared theory can have emergent conformal or just Poincare symmetry. In all
cases studied, the area of the horizon of the dual black hole goes to zero in
the extremal limit, consistent with a nondegenerate ground state.
