Holographic $p$-wave superfluid

In the probe limit, we numerically construct a holographic $p$-wave superfluid model in the four-dimensional (4D) and five-dimensional (5D) anti-de Sitter black holes coupled to a Maxwell-complex vector field. We find that, for the condensate with the fixed superfluid velocity, the results are similar to the $s$-wave cases in both 4D and 5D spacetimes. In particular, the Cave of Winds and the phase transition, always being of second order, take place in the 5D case. Moreover, we find that the translating superfluid velocity from second order to first order $\frac{S_y}{\mu }$ increases with the mass squared. Furthermore, for the supercurrent with fixed temperature, the results agree with the Ginzburg-Landau prediction near the critical temperature. In addition, this complex vector superfluid model is still a generalization of the SU(2) superfluid model, and it also provides a holographic realization of the $H{e}_3$ superfluid system.

Figure 1

The condensate versus the reduced temperature with $m^2=-\frac{1}{4}$ (a), $\frac{3}{4}$ (b), and 2 (c), where all curves respectively correspond to $\frac{S_y}{\mu }=0,0.5$, and 0.65 from right to left. Panel (d) illustrates the grand potential in the case of $m^2=\frac{3}{4}$ corresponding to $\frac{S_y}{\mu }=0.5$ (red dashed line), 0.65 (black solid line), and the normal phase (magenta dotted line), respectively.

Figure 2

The translating superfluid velocity $\frac{S_y}{\mu }$ from the second to the first order as a function of the mass squared of the vector field ${\rho }_{\mu }$.

Figure 3

The condensate (a) and grand potential (b) versus the reduced temperature with $m^2=0$ in the 5D black hole. The curves in (a) correspond to $\frac{S_y}{\mu }=0$, 0.42, 0.75, and 0.85 from right to left, while the curves in (b) to $\frac{S_y}{\mu }=0.75$ (red dashed), 0.85 (black solid), and the normal phase (magenta dotted), respectively.

Figure 4

The condensates (a), (b) and the grand potential (c) as a function of the reduced temperature with $m^2=\frac{5}{4}$ in the 5D black hole. The curves in (a) correspond to $\frac{S_y}{\mu }=0$, 0.5, 0.7, and 0.86 from right to left, and the curve in (b) to $\frac{S_y}{\mu }=0.86$, while ones in (c) to $\frac{S_y}{\mu }=0.86$ (black solid) and the normal phase (magenta dotted), respectively.

Figure 5

The condensate (a) and the grand potential (b) versus the reduced temperature with $m^2=3$ in the 5D black hole. The curves in (a) correspond, respectively, to $\frac{S_y}{\mu }=0$, 0.5, 0.7, and 0.86 from right to left, while the ones in (b) correspond to $\frac{S_y}{\mu }=0.86$ (black solid) and the normal phase (magenta dotted).

Figure 6

The supercurrent versus superfluid velocity for $m^2=-\frac{1}{4}$ (a), 0 (b), $\frac{3}{4}$ (c), and 2 (d) in the 4D case. The curves in all panels correspond to $\frac{T}{T_0}=0.2,0.45,0.7$, and 0.98 from right to left, respectively.

Figure 7

The ratio $\alpha ≔{(⟨O_x{⟩}_c/⟨O_x{⟩}_{\infty })}^2$ versus the reduced temperature. The curves from top to bottom correspond to the mass squared $m^2=-\frac{1}{4},0,\frac{3}{4}$, and 2 in the 4D spacetime (a), and to $m^2=-1,0,\frac{5}{4}$, and 3 in the 5D case (b), respectively.

Figure 8

The supercurrent versus superfluid velocity for $m^2=-1$ (a), $\frac{5}{4}$ (b), 3 (c), and $\frac{5}{4}$ (d) in the 5D spacetime. The curves in (a), (b), and (c), respectively, correspond to $\frac{T}{T_0}=0.2,0.45,0.7$, and 0.98 from right to left, while the curve in (d) corresponds to $\frac{T}{T_0}=0.2$. Note that the case of $\frac{T}{T_c}=0.98$ corresponds to the lowest purple dotted line.