Towards the Lattice Effects on the Holographic Superconductor

We study the lattice effects on the simple holographic toy model; massive U(1) gauge theory for the bulk action. The mass term is for the U(1) gauge symmetry breaking in the bulk. Without the lattice, the AC conductivity of this model shows similar results to the holographic superconductor with the energy gap. On this model, we introduce the lattice effects, which induce the periodic potential and break the translational invariance of the boundary field theory. Without the lattice, due to the translational invariance and the mass term, there is a delta function peak at zero frequency on the AC conductivity. We study how this delta function peak is influenced by the lattice effects, which we introduce perturbatively. In the probe limit, we evaluate the perturbative corrections to the conductivities at very small frequency limit. We find that the delta function peak remains, even after the lattice effects are introduced, although its weight reduces perturbatively. We also study the lattice wavenumber dependence of this weight. Our result suggests that in the U(1) symmetry breaking phase, the delta function peak is stable against the lattice effects at least perturbatively.