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Condensate flow in holographic models in the presence of dark matter

Holographic model of a three-dimensional current carrying superconductor or superfluid with {\it dark matter} sector described by the additional $U(1)$-gauge field coupled to the ordinary Maxwell one, has been studied in the probe limit. We investigated analytically by the Sturm-Liouville variational method, the holographic s-wave and p-wave models in the background of the AdS soliton as well as five-dimensional AdS black hole spacetimes. The two models of p-wave superfluids were considered, the so called $SU(2)$ and the Maxwell-vector. Special attention has been paid to the dependence of the critical chemical potential and critical transition temperature on the velocity of the condensate and {\it dark matter} parameters. The current $J$ in holographic three-dimensional superconductor studied here, shows the linear dependence on $T_c-T$ for both s and p-wave symmetry. This is in a significant contrast with the previously obtained results for two-dimensional superconductors, which reveal the $(T-T_c)^{3/2}$ temperature dependence. The coupling constant $α$, as well as, chemical potential $μ_D$ and the velocity $S_D$ of the {\it dark matter}, affect the critical chemical potential of the p-wave holographic $SU(2)$ system. On the other hand, $α$, {\it dark matter} velocity $S_D$ and density $ρ_D$ determine the actual value of the transition temperature of the same superconductor/superfluid set up. However, the {\it dark matter} does not affect the value of the current.