Abstract
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 $\alpha$, as well as, chemical potential $\mu_D$ 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, $\alpha$, {\it dark matter} velocity $S_D$ and density $\rho_D$ 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.
Highlights
The aforementioned studies were generalized in many ways, e.g., back-reaction of the order parameter on spacetime metric was considered enabling the second order phase transition to be replaced by the first order one [17], the gravitational background of AdS soliton [24] was proposed to study holographic insulator/superconductor transition at zero temperature [25]–[27]
In the paper we have considered the properties of holographic superfluid with the superflow of the condensate, i.e., the situation when on the gravity side one accepts t-component of the gauge fields and takes into account the spatial one, as well
We have analyzed the s-wave and p-wave current carrying superfluids with dark matter sector which has been described by additional U(1)-gauge field coupled to the ordinary one, in the background of the AdS soliton (T=0) or black hole (T= 0)
Summary
We introduce a set up for zero temperature s-wave holographic superfluid. Α stands for the coupling constant between both gauge fields, m, q is mass and charge of the scalar field ψ, respectively This model was widely used in the probe limit studies, as well as, backreaction effects were taken into account in order to envisage the influence of the dark matter on the properties of holographic s and p-wave superconductors and vortices [57]–[63]. The coupling among condensing ψ and the aforementioned components of gauge field, may cause the black object (black soliton or black hole) to be unstable to forming scalar hair. Let us remark that the form of equation (2.6) is such that the influence of the dark matter sector shows up by appearing α-coupling constant in the relations governing the ordinary Maxwell field. We solve these equations analytically close to the transition point [90] and obtain the critical value of the chemical potential μc, the behavior of the order parameter and Aφ component of the Maxwell field
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