Holographic superfluid with gauge-axion coupling

We have constructed a holographic superfluid with gauge-axion coupling. Depending on whether the coupling is positive or negative, the system displays metallic or insulating behavior in its normal state. A significant feature of the system is the appearance of a mid-IR peak in the alternating current (AC) conductivity in a certain range of parameters. This peak arises due to competition between explicit symmetry breaking (ESB) and spontaneous symmetry breaking (SSB), which results in the presence of a pseudo-Goldstone mode. Moreover, a dip in low-frequency AC conductivity is observed, stemming from the excitation of the SSB Goldstone mode. In the superfluid phase, the effect of gauge-axion coupling on the condensation or superfluid energy gap is only amplified in the presence of strong momentum dissipation. Notably, for the case with negative gauge-axion coupling, a hard-gap-like behavior at low frequency and a pronounced peak at intermediate frequency are observed, indicating that the evolution of the superfluid component is distinct from that of positive coupling.