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Colorful horizons with charge in anti-de Sitter space

Steven S. Gubser

TL;DR

This work addresses spontaneous breaking of a $U(1)$ gauge symmetry near the horizon of a charged AdS$_4$ black hole by a non‑abelian condensate of $SU(2)$ gauge fields. Using a symmetry‑based Lagrangian and a horizon‑regular, asymptotically AdS ansatz for the gauge field, the author solves the coupled Einstein–Yang–Mills equations to map the phase structure, reading thermodynamic quantities from horizon and boundary data. The main finding is a second‑order phase transition from the RNAdS phase to a phase with a normalizable non‑abelian condensate $w$, with universal scaling behavior emerging at large gauge coupling $gL$ (negligible back‑reaction) and characteristic scaling forms for the charge fraction outside the horizon, the order parameter, and the free‑energy difference. The results illuminate holographic realizations of superconductivity‑like phenomena in strongly coupled gauge theories and suggest avenues for extending to $p$‑wave or $d$‑wave analogs, while noting cautions about the large‑$N$ limit and its relation to real materials.

Abstract

An abelian gauge symmetry can be spontaneously broken near a black hole horizon in anti-de Sitter space using a condensate of non-abelian gauge fields. There is a second order phase transition between Reissner-Nordstrom-anti-de Sitter solutions, which are preferred at high temperatures, and symmetry breaking solutions, which are preferred at low temperatures.

Colorful horizons with charge in anti-de Sitter space

TL;DR

This work addresses spontaneous breaking of a gauge symmetry near the horizon of a charged AdS black hole by a non‑abelian condensate of gauge fields. Using a symmetry‑based Lagrangian and a horizon‑regular, asymptotically AdS ansatz for the gauge field, the author solves the coupled Einstein–Yang–Mills equations to map the phase structure, reading thermodynamic quantities from horizon and boundary data. The main finding is a second‑order phase transition from the RNAdS phase to a phase with a normalizable non‑abelian condensate , with universal scaling behavior emerging at large gauge coupling (negligible back‑reaction) and characteristic scaling forms for the charge fraction outside the horizon, the order parameter, and the free‑energy difference. The results illuminate holographic realizations of superconductivity‑like phenomena in strongly coupled gauge theories and suggest avenues for extending to ‑wave or ‑wave analogs, while noting cautions about the large‑ limit and its relation to real materials.

Abstract

An abelian gauge symmetry can be spontaneously broken near a black hole horizon in anti-de Sitter space using a condensate of non-abelian gauge fields. There is a second order phase transition between Reissner-Nordstrom-anti-de Sitter solutions, which are preferred at high temperatures, and symmetry breaking solutions, which are preferred at low temperatures.

Paper Structure

This paper contains 5 sections, 31 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Symmetry breaking solutions
  3. Summary of results
  4. Static perturbations
  5. Strong coupling limit

Figures (3)

  • Figure 1: (A) The phase diagram. Each point corresponds to a numerically constructed black hole solution. (B) $\hat{J}/\hat{\rho}$ as a function of $T/T_c$. (C) $q$ as a function of $T/T_c$. (D) $\Delta\hat{f}/\hat{\rho}^{3/2}$ as a function of $T/T_c$.
  • Figure 2: Left: Temperatures where static linear perturbations of the RNAdS solution arise. The curve labeled "no back-reaction" is given analytically by (\ref{['FoundTc']}). It comes from neglecting the back-reaction of $\Phi$ as well as of $w$ on the metric. As discussed in section \ref{['STRONG']}, neglecting back-reaction is a good approximation when $gL$ is large. Right: Static normalizable perturbations, scaled so that $w=1$ at the horizon. The solution that is everywhere positive corresponds to the dot at $gL=6$ on the $T=T_c$ curve in the left-hand plot, and the other solutions correspond to the dots on the other curves.
  • Figure 3: Plots of $\hat{J}/\hat{\rho}$ and $q$ as a function of $t = 1-T/T_c$ for several values of $gL$. The $gL=\infty$ curve is obtained from solutions to (\ref{['YMscaled']}), where back-reaction of the gauge fields on the metric is neglected.