Black holes dual to helical current phases

TL;DR

This paper analyzes $d=4$ CFTs at finite $T$ and chemical potential using $D=5$ Einstein-Maxwell theory with a Chern-Simons term. For sufficiently large CS coupling $\gamma$ the authors go beyond the homogeneous AdS-RN phase and construct fully back-reacted, electrically charged AdS$_5$ black holes with Bianchi VII$_0$ symmetry that are dual to a low-$T$ spatially modulated helical current phase; the transition is second order with mean-field behavior, and the pitch of the helix grows as $T$ decreases while the entropy density vanishes at $T\to0$. The study reveals a two-parameter moduli space of solutions labeled by $T$ and the wave-number $k$, with a thermodynamically preferred line that minimizes the free energy and features $c_h=0$ along it. These results provide a concrete holographic realization of a helical current phase and lay groundwork for future transport and hydrodynamic analyses of spatially modulated holographic matter.

Abstract

We consider the class of d=4 CFTs at finite temperature and chemical potential that are holographically described within D=5 Einstein-Maxwell theory with a Chern-Simons term. The high temperature phase, which is spatially homogeneous and isotropic, is dual to the AdS-Reissner-Nordstrom black brane solution. For sufficiently large Chern-Simons coupling, we construct new electrically charged AdS black hole solutions that are dual to the low temperature, spatially modulated phase. In this phase the current, associated with the abelian global symmetry, spontaneously acquires a helical order. The new black holes are stationary and also have Bianchi VII$_0$ symmetry.

Figures (4)

• Figure 1: The curve denotes the critical temperature at which the AdS-RN black brane becomes unstable and also where the new branches of helical black holes, given in figure \ref{['fig:1']}, appear. The plot is for $\gamma=1.7$ and $\mu=1$.
• Figure 2: The two-parameter family of helical black holes, labelled by temperature $T$ and wave-number $k$, and their free-energy $w$. The red line denotes the thermodynamically preferred locus, which minimises $w$ over the moduli space of solutions at fixed $T$ labelled by $k$. The plot is for $\gamma=1.7$ and $\mu=1$.
• Figure 3: Plot of $c_h$ versus $k$ for the one parameter family of helical black hole solutions given in figure \ref{['fig:1']} for the representative temperature $T\approx 0.0535$. In particular $c_h=0$ on the thermodynamically preferred red line of solutions given in \ref{['fig:1']}. The plot is for $\gamma=1.7$ and $\mu=1$.
• Figure 4: The red lines plot various physical quantities against temperature $T$ for the thermodynamically preferred helical black hole solutions on the red line in figure \ref{['fig:1']}. The blue line refers to the AdS-RN black hole solution. $w$ is the free energy and $k$ is the wave-number of the helical order. $c_Q$ and $c_\alpha$ fix the spatially modulated momentum and stress/strain in the $(x_2,x_3)$ plane, respectively. $q$ and $c_b$ determine the size of the charge and the spatially modulated current, respectively, and $s$ is the entropy density. The plots are for $\gamma=1.7$ and $\mu=1$.