Low-temperature behavior of the Abelian Higgs model in anti-de Sitter space

TL;DR

This work analyzes the fully back-reacted Abelian Higgs model in $AdS_4$, revealing a second-order finite-temperature transition to superconducting black holes and, for sufficiently large $qL$, a domain-wall–like IR region with a nontrivial index of refraction. By deriving and solving the coupled gravity–matter equations, it shows superconducting solutions are thermodynamically preferred below $T_c$ and that the IR geometry can exhibit $SO(2,1)$ symmetry in certain regimes. Numerical results map out thermodynamics, IR refractive structure, and AC conductivity, uncovering a robust domain-wall-like IR behavior and transport features with a persistent delta-function contribution due to translation invariance. These findings advance the holographic understanding of superconductivity with backreaction and finite-temperature domain-wall dynamics in AdS/CFT.

Abstract

We explore the low-temperature behavior of the Abelian Higgs model in AdS_4, away from the probe limit in which back-reaction of matter fields on the metric can be neglected. Over a significant range of charges for the complex scalar, we observe a second order phase transition at finite temperature. The symmetry-breaking states are superconducting black holes. At least when the charge of the scalar is not too small, we observe at low temperatures the emergence of a domain wall structure characterized by a definite index of refraction. We also compute the conductivity as a function of frequency.

Figures (3)

• Figure 1: (Color online) Thermodynamic quantities as functions of $qL$ and temperature. In (A)-(D), the curves for each value of $qL$ have different lengths because they terminate either where numerical errors began to cast doubt on the validity of the coldest solutions, or where we simply ran out of CPU time. All logarithms are natural logs.
• Figure 2: (Color online) (A) The inverse index of refraction, $1/n_{\rm IR}$, as a function of $qL$. (B) The critical temperature $T_c$ and the inflection point temperature $T_*$ compared to the rescaled charge density $\hat{\rho}$ as functions of $qL$. (C) For $qL=0.7$ and the coldest temperature we could reach, there is an inflection point in $h(r)$ at the location of the black dot. In order to see a distinctive double shelf develop around this inflection point, we had to cool the black hole quite a bit more than we did for figure \ref{['SCALINGPLOTS']}. (D) For $qL=1.4$, there is a distinctive double-shelf structure in $h(r)$ at a less extreme temperature than for $qL=0.7$. The black dot is again at an inflection point of $h(r)$.
• Figure 3: (Color online) (A), (B) The real and imaginary parts of the conductivity for $qL=0.7$ at various values of $T/T_c$. (C), (D) The real and imaginary parts of the conductivity for $qL=1.4$ at various values of $T/T_c$. (E), (F) The real and imaginary parts of the conductivity for $qL=2$ at various values of $T/T_c$.