  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.
