Stability of Holographic Superconductors

TL;DR

This paper analyzes the stability of holographic superconductors by classifying perturbations, proving vector sector stability, and demonstrating scalar sector stability through an effective mass analysis near critical points.

Contribution

It provides a detailed stability analysis of holographic superconductors, including a Hamiltonian proof for vector stability and a scalar sector mechanism based on effective mass considerations.

Findings

Vector perturbations are stable via a positive definite Hamiltonian.

Scalar sector stability is linked to the effective mass of charged scalar fields.

Superconducting phase remains stable near and away from the critical point.

Abstract

We study the dynamical stability of holographic superconductors. We first classify perturbations around black hole background solutions into vector and scalar sectors by means of a 2-dimensional rotational symmetry. We prove the stability of the vector sector by explicitly constructing the positive definite Hamiltonian. To reveal a mechanism for the stabilization of a superconducting phase, we construct a quadratic action for the scalar sector. From the action, we see the stability of black holes near a critical point is determined by the equation of motion for a charged scalar field. We show the effective mass of the charged scalar field in hairy black holes is always above the Breitenlohner-Freedman bound near the critical point due to the backreaction of a gauge field. It implies the stability of the superconducting phase. We also argue that the stability continues away from the critical point.