A gravity derivation of the Tisza-Landau Model in AdS/CFT

TL;DR

This work derives fully backreacted gravity solutions dual to boundary superfluids with finite supercurrent in AdS/CFT and shows that the non-dissipative fluid dynamics are governed by a relativistic two-fluid (Tisza-Landau) model, including the Josephson constraint. By solving the Einstein–Maxwell–scalar system and performing holographic renormalization, the authors obtain boundary stress-energy and current that match a Landau-type constitutive framework, with a detailed thermodynamic structure incorporating a Goldstone chemical potential $\mu_s$ and supercurrent data. The paper reveals that, at strong backreaction and low charge $q$, the superfluid phase transition is second order for all admissible superfluid fractions $\zeta$, while larger $q$ can produce first-order transitions above a critical $\zeta_c$, as shown by free-energy landscapes and condensate behavior. Numerically, horizon regularity enforces consistency relations between asymptotic data, and the results illuminate how backreaction shapes transport and phase structure in strongly coupled holographic superfluids; the approach provides a controlled framework to study non-dissipative relativistic two-fluid hydrodynamics in AdS/CFT and sets the stage for extensions to higher dimensions and dissipative order.

Abstract

We derive the fully backreacted bulk solution dual to a boundary superfluid with finite supercurrent density in AdS/CFT. The non-linear boundary hydrodynamical description of this solution is shown to be governed by a relativistic version of the Tisza-Landau two-fluid model to non-dissipative order. As previously noted, the phase transition can be both first order and second order, but in the strongly-backreacted regime at low charge q we find that the transition remains second order for all allowed fractions of superfluid density.

Figures (6)

• Figure 1: (colour online) The condensate $\langle {\cal O}_\psi\rangle$ as a function of reduced temperature. For larger values of the charge $q$ the condensate becomes multi-valued beyond a critical fraction of superfluid $\zeta_c$ indicating that the phase transition is now first order. The normalisation of the condensate is chosen for ease of comparison with the probe calculations of Herzog:2008he.
• Figure 2: (colour online) The quantity $\rho_s/\rho$ as a function of reduced temperature. This fraction approaches unity for $q > 1$, but for $q=1$ there always remains a normal component, even at very low temperatures. Some of the curves are multi-valued as a consequence of the first-order phase transition. In those cases only the upper branch below the critical temperature $t_c(\zeta)$ is physical.
• Figure 3: (colour online) The free energy at fixed $\zeta$ as a function of reduced temperature for $q=1,2$. The unbroken branch is shown as a dashed line. For the lowest $q$ the broken branch joins the unbroken branch smoothly for all allowed values of $\zeta$. For $q=2$, we see the swallow-tail behaviour characteristic of first-order transitions for fractions above a critical value $\zeta_c$. The behaviour for $q>2$ is very similar and we do not reproduce these plots here.
• Figure 4: (colour online) Figure demonstrating to numerical accuracy (in this case the numerical error is of the order of $10^{-4}$) the constraints among asymptotic data following from horizon regularity. Both constraints are illustrated based on the data points we used to generate the plots at $q=8$ and $\zeta=\frac{2}{5}$.
• Figure 5: (colour online) Temperature, at which the marginal mode occurs as a function of superfluid fraction $\zeta$. The curves for ascending $q=1,2,8,16,32,100$ are shown from bottom to top. The vertical dashed lines indicate fractions of the condensate of $\zeta=1/4, 1/3, 2/5, 1/2$ and the horizontal dashed lines are visual aids in order to identify the critical temperatures for the $q=100$ curve.