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A Simple Holographic Superconductor with Momentum Relaxation

We study a holographic superconductor model with momentum relaxation due to massless scalar fields linear to spatial coordinates($ψ_I = βδ_{Ii} x^i$), where $β$ is the strength of momentum relaxation. In addition to the original superconductor induced by the chemical potential($μ$) at $β=0$, there exists a new type of superconductor induced by $β$ even at $μ=0$. It may imply a new `pairing' mechanism of particles and antiparticles interacting with $β$, which may be interpreted as `impurity'. Two parameters $μ$ and $β$ compete in forming a superconducting phase. As a result, the critical temperature behaves differently depending on $β/μ$. It decreases when $β/μ$ is small and increases when $β/μ$ is large, which is a novel feature compared to other models. After analysing ground states and phase diagrams for various $β/μ$, we study optical electric($σ$), thermoelectric($α$), and thermal($\barκ$) conductivities. When the system undergoes a phase transition from a normal to a superconducting phase, $1/ω$ pole appears in the imaginary part of the electric conductivity, implying infinite DC conductivity. If $β/μ<1$, at small $ω$, a two-fluid model with an imaginary $1/ω$ pole and the Drude peak works for $σ$, $α$, and $\barκ$, but if $β/μ>1$ a non-Drude peak replaces the Drude peak. It is consistent with the coherent/incoherent metal transition in its metal phase. The Ferrell-Glover-Tinkham (FGT) sum rule is satisfied for all cases even when $μ=0$.