Holographic model of superfluidity

We study a holographic model of a relativistic quantum system with a global $U(1)$ symmetry, at nonzero temperature and density. When the temperature falls below a critical value, we find a second-order superfluid phase transition with mean-field critical exponents. In the symmetry-broken phase, we determine the speed of second sound as a function of temperature. As the velocity of the superfluid component relative to the normal component increases, the superfluid transition goes through a tricritical point and becomes first order.

Figure 1

The phase diagrams for the theory with a scalar with (a) conformal dimension one and (b) conformal dimension two. The solid blue lines indicate a second-order phase transition while the solid red lines (in between the dashed lines) indicate a first-order phase transition. The dashed blue lines are spinodal curves, while the red dot indicates the tricritical point.

Figure 2

The condensate as a function of temperature for the two operators: (a) $O_1$ and (b) $O_2$. The curves in the plots, from right to left, are for $|\xi /\mu |=0$, $1/4$, $1/3$, $2/5$, and $1/2$.

Figure 3

The difference in free energy $\Delta {\Omega }_1$ between the phase with a scalar condensate and without one as a function of $T/|\mu |$: (a) $\xi =0$ and (b) $\xi /|\mu |=4/7$.

Figure 4

The speed of second sound as a function of $T/|\mu |$, computed by evaluating thermodynamic derivatives in Eq. (18): (a) $O_1$ scalar, (b) $O_2$ scalar. The speed of second sound vanishes as $T\to T_c$ and appears to approach a constant value as $T\to 0$.