Alternating current conductivity and superconducting properties of a holographic effective model with broken translations

We construct a holographic effective superconducting theory by considering a special gauge-axion coupling. In the normal state, we observe that a peak emerges in the mid-IR in the AC conductivity for certain strength of broken translations and the gauge-axion coupling. It is attributed to the emergence of pseudo-Goldstone mode due to the competition of spontaneous symmetry breaking (SSB) and the explicit symmetry breaking (ESB). However, when the ESB dominates over the SSB, the peak disappears. In the superconducting phase, this gauge-axion coupling also plays a key role leading to a more evident gap at the low frequency conductivity. In addition, we also study the combined effects of the strength of broken translations and various couplings among the gauge field, axion fields and the complex scalar field.