In the probe limit, we numerically build a holographic $p$-wave superfluid model in the four-dimensional Lifshitz black hole coupled to a Maxwell-complex vector field. We observe the rich phase structure and find that the Lifshitz dynamical exponent
$z$ contributes evidently to the effective mass of the matter field and dimension of the gravitational background. Concretely, we obtain the Cave of Winds appeared only in the five-dimensional anti-de Sitter~(AdS) spacetime, and the increasing $z$ hinders not only the condensate but also the appearance of the first-order phase transition. Furthermore, our results agree with the Ginzburg-Landau results near the critical temperature. In addition, the previous AdS superfluid model is generalized to the Lifshitz spacetime.
In the probe limit, we numerically construct a holographic p-wave superfluid model in the 4D and 5D AdS black holes coupled to a Maxwell-complex vector field. We find that, for the condensate with the fixed superfluid velocity, the results are simi
lar to the s-wave cases in both 4D and 5D spacetimes. In particular, The Cave of Winds and the phase transition always being the second order take place in the 5D case. Moreover, we find the second-first order translating point $frac{S_y}{mu}$ increases with the mass squared. Furthermore, for the supercurrent with the fixed temperature, the results agree with the GL prediction near the critical temperature. In addition, this complex vector superfluid model is still a generalization of the SU(2) superfluid model, and also provides a holographic realization of the $He_3$ superfluid system.
