We study a charged scalar field in a bulk $3+1$-dimensional anti–de Sitter (AdS) spacetime with a planar black hole background metric. Through the $\mathrm{AdS}/\mathrm{CFT}$ 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 $\mathrm{AdS}/\mathrm{CFT}$ 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 quasistatic evolution at least, it may be reasonable to think of this link as an extension of the $\mathrm{AdS}/\mathrm{CFT}$ correspondence.

• Received 12 March 2019

DOI:https://doi.org/10.1103/PhysRevD.100.046004

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Published by the American Physical Society