Superfluid black branes in AdS_4\times S^7

TL;DR

This work studies finite-temperature phases of the $d=3$ $N=8$ SCFT holographically via $AdS_4\times S^7$, constructing back-react black branes in $D=11$ supergravity that capture the highest-temperature superfluid instability for a diagonal $U(1)_R$. It demonstrates that the existence and thermodynamic relevance of these holographic superconductors depend crucially on which $SO(8)$ truncation is used, revealing a spectrum of back-reacted solutions including Gubser–Mitra branches and domain-wall flows to IR $AdS_4$ vacua in an alternative quantisation. In particular, the $SU(3)\times U(1)$ infrared fixed point can be reached by zero-temperature domain walls, yielding an emergent superconformal symmetry in the IR and providing top-down realizations of holographic superconductivity with nontrivial zero-temperature ground states. The results underscore the need to consider full or broader truncations (and possibly full $SO(8)$ or $D=11$ embeddings) to map the complete phase diagram and ground states of the dual CFT.

Abstract

We consider the d=3 N=8 SCFT dual to AdS_4\times S^7 when held at finite temperature and chemical potential with respect to a diagonal U(1)_R\subset SO(8) global symmetry and construct black brane solutions of D=11 supergravity that are associated with the superfluid instability with the highest known critical temperature. We construct a rich array of solutions using different sub-truncations of SO(8) gauged supergravity finding results that strongly depend on the truncation used. Our constructions include black brane solutions associated with the Gubser-Mitra instability which preserve the U(1)_R symmetry, and these, in turn, can have further superfluid instabilities. In addition, we also construct superfluid black branes that at zero temperature are domain walls that interpolate between the SO(8) AdS_4 vacuum in the UV, in an alternative quantisation, and the supersymmetric SU(3)\times U(1) AdS_4 vacuum in the IR.

Figures (4)

• Figure 1: Black brane solutions in the 4 equal charged scalars truncation \ref{['4lag']} with $\mu_1=1$. The plots show $\varphi_{(1)}\sim <{\cal O}_\varphi>$ and the free energy $w$ versus temperature $T$ for the superfluid black brane solutions (red line) emanating from the AdS-RN black brane solutions (blue line) at $T_c\approx 0.174$.
• Figure 2: Various black brane solutions in the 2+2 equal charged scalars truncation \ref{['2+2lag']} with $\mu_1=1$, $\bar{\mu}_0=0$: additional solutions are found using the ${\mathbb{Z}}_2$ symmetry \ref{['zed2']}. The red branches are superfluid black branes with $\gamma_1\ne 0$ and $\gamma_2=0$. The solid red branch is the superfluid black brane associated with an instability of the AdS-RN black brane. The green line is a branch of superfluid black branes that in addition has $\gamma_{2}\ne 0$. The blue dotted and dashed lines are two branches of black branes with $\gamma_1=\gamma_2=0$ associated with the Gubser-Mitra instability of the AdS-RN black brane. The solid blue line in the bottom figure is the AdS-RN black brane, but in the top figure it also represents the back reacted Gubser-Mitra branes. The solid dots on lines that meet indicate the points at which a new branch of black branes is appearing. The dashed circles indicate the thermodynamically preferred phase transitions within this truncation: as one lowers $T$ the system moves from the AdS-RN branch to the solid red superfluid branch, then, discontinuously, to the green branch.
• Figure 3: Various black brane solutions in the $SU(3)$ truncation \ref{['su3lag']} with $\mu_1=1$, $\mu_0=0$. The solid red branch is the superfluid black brane associated with an instability of the AdS-RN black brane and has $\zeta_1\ne 0$ and $\zeta_2= 0$. It branches into the solid green branch which also has $\zeta_2\ne 0$. The blue dotted and dashed lines are two branches of black branes with $\zeta_1=\zeta_2=0$ associated with the Gubser-Mitra instability of the AdS-RN black brane. The solid blue line in each of the bottom figures represents the AdS-RN black branes, but in the top figure it also represents the back reacted Gubser-Mitra branes. The solid black line are the superfluid black branes with $\zeta_1=0$ and $\zeta_2\ne 0$ found in Gauntlett:2009dn. Additional black branes are also shown. The solid dots indicate the points at which a new branch of black branes is appearing on lines that meet. The dashed circles indicate the thermodynamically preferred phase transitions within this truncation: as one lowers $T$ the system moves from the AdS-RN branch to the solid red superfluid branch, then, discontinuously, to the dotted Gubser-Mitra branch.
• Figure 4: Black brane solutions of the $SU(3)$ invariant truncation \ref{['su3lag']} all of which have $\zeta_1=0$. The two chemical potentials are $\mu_{0}\approx0.98, \mu_{1}=1$. We are using an alternative quantisation in which $\Delta({\cal O}_{\zeta_2})=1$ and $\Delta({\cal O}_{\lambda})=2$ and the deformation parameter $\lambda_{(1)}$ is given by $\lambda_{(1)}\approx0.38$. The blue branch is the unbroken phase black branes and the red branch is the superfluid black branes. The solid dots at zero temperature are the relevant quantities for the charged domain walls that interpolate between the $SO(8)$$AdS_4$ and the $SU(3)\times U(1)$$AdS_4$ vacua.