Fourth sound of holographic superfluids

TL;DR

The paper analyzes fourth sound in holographic superfluids realized by a charged scalar coupled to a gauge field in AdS${}_4$ within the probe limit. It shows that for condensates with large scaling dimension $\Delta$, fourth sound tends to the conformal first-sound value $v_4^2\to 1/(d-1)$ as $T\to 0$, while for small $\Delta$ the low-temperature behavior is non-conformal due to anomalous scaling of the charge density and order parameter. By introducing a scalar potential with a finite infrared minimum, conformal invariance is restored at low temperatures, yielding $v_4^2\to 1/2$ for $\Delta=1,2$ in the probe limit and revealing a link between $\langle O_{\Delta} \rangle/\mu^{\Delta}$ and the conformal limit. The results connect holographic two-fluid hydrodynamics with He-like fourth-sound phenomenology and suggest directions for including backreaction to study second sound more completely.

Abstract

We compute fourth sound for superfluids dual to a charged scalar and a gauge field in an AdS_4 background. For holographic superfluids with condensates that have a large scaling dimension (greater than approximately two), we find that fourth sound approaches first sound at low temperatures. For condensates that a have a small scaling dimension it exhibits non-conformal behavior at low temperatures which may be tied to the non-conformal behavior of the order parameter of the superfluid. We show that by introducing an appropriate scalar potential, conformal invariance can be enforced at low temperatures.

Figures (6)

• Figure 1: The critical temperature $T_c$, where $\psi$ develops a zero mode, as a function of the dimension of the condensate $\Delta$ for $1/2< \Delta < 5$.
• Figure 2: (Color online) Fourth sound versus temperature for condensates of dimension $2/3 \leq \Delta \leq 2$ in the abelian Higgs model. The temperature is measured relative to the critical temperature $T_c$. The curves are color coded according to the values of $\Delta$.
• Figure 3: Values of $a_{\Delta}$ and $n_{\Delta}$, obtained by fitting the total charge density $\rho = a_{\Delta} \mu^2 \left(\frac{\mu}{T}\right)^{n_{\Delta}}$ to the numerics. The fit was carried out for data points for which $(\rho-{\rho_{\rm s}})/\rho < 10^{-3}$ where ${\rho_{\rm s}}$ is the charge density of the superfluid phase. $\Delta$ is the dimension of the condensate.
• Figure 4: (Color online) The dimensionless ratio $\langle O_{\Delta} \rangle^{1/\Delta}\mu/\rho$ versus temperature. At low temperatures this ratio approaches a constant, which implies via \ref{['E:rhobehavior']} that $\langle O_{\Delta} \rangle/\mu^{\Delta}$ diverges at low temperatures for small $\Delta$. The curves are color coded according to the value of $\Delta$.
• Figure 5: (Color online) The asymptotic value of $\langle O_{\Delta} \rangle^{1/\Delta}\mu/\rho$ at low temperatures, as a function of $\Delta$, the dimension of the scalar condensate. The blue curve shows the numerical result, displayed in figure \ref{['F:Condensate']}, and the dashed black curve shows the analytic approximation \ref{['E:lowTasymptotics']}, valid for large $\langle O_{\Delta} \rangle/\mu^{\Delta}$.