# A Non-Equilibrium Approach To Holographic Superconductors Using Gradient Flow

## TL;DR

This paper investigates holographic superconductors using gradient flow in a 3+1D anti-de Sitter spacetime, revealing how fields transition between normal and superconducting phases while minimizing free energy, and proposing an extension of AdS/CFT correspondence off equilibrium.

## Contribution

It introduces a numerical gradient flow approach to study phase transitions in holographic superconductors and explores the link between off-shell bulk configurations and boundary states.

## Key findings

- Gradient flow moves fields between fixed points minimizing free energy.
- Flow equations connect bulk off-shell states with boundary phases.
- Proposes an extension of AdS/CFT for quasi-static, off-equilibrium configurations.

## Abstract

We study a charged scalar field in a bulk 3+1 dimensional anti-deSitter spacetime with a planar black hole background metric. Through the AdS/CFT correspondence this is equivalent to a strongly coupled field theory in 2+1 dimensions describing a superconductor. We use the gradient flow method and solve the flow equations numerically between two fixed points: a vacuum solution and a hairy black hole solution. We study the corresponding flow on the boundary between a normal metal phase and a superconducting phase. We show how the gradient flow moves fields between two fixed points in a way that minimizes the free energy of the system. At the fixed points of the flow the AdS/CFT correspondence provides an equivalence between the Euclidean on-shell action in the bulk and the free energy of the boundary, but it does not tell us about fields away from equilibrium. However, we can formally link static off-shell configurations in the bulk and in the boundary at the same point along the flow. For quasi-static evolution at least, it may be reasonable to think of this link as an extension of the AdS/CFT correspondance.