Charged Magnetic Brane Solutions in AdS_5 and the fate of the third law of thermodynamics

TL;DR

This paper constructs AdS_5 solutions with magnetic fields and electric charge, revealing instability of near-horizon geometries and proposing a new formulation of the third law of thermodynamics for black holes in AdS/CFT.

Contribution

It introduces new charged magnetic brane solutions in AdS_5, analyzes their thermodynamic properties, and connects these findings to the third law of thermodynamics in the holographic context.

Findings

Small magnetic fields drastically reduce entropy at low temperatures.

Purely electrically charged branes with AdS_2 imes R^3 geometry are unstable under magnetic perturbations.

Exact warped AdS_3 black hole solutions are found for specific Chern-Simons couplings.

Abstract

We construct asymptotically AdS_5 solutions to 5-dimensional Einstein-Maxwell theory with Chern-Simons term which are dual to 4-dimensional gauge theories, including N=4 SYM theory, in the presence of a constant background magnetic field B and a uniform electric charge density \rho. For the solutions corresponding to supersymmetric gauge theories, we find numerically that a small magnetic field causes a drastic decrease in the entropy at low temperatures. The near-horizon AdS_2 \times R^3 geometry of the purely electrically charged brane thus appears to be unstable under the addition of a small magnetic field. Based on this observation, we propose a formulation of the third law of thermodynamics (or Nernst theorem) that can be applied to black holes in the AdS/CFT context. We also find interesting behavior for smaller, non-supersymmetric, values of the Chern-Simons coupling k. For k=1 we exhibit exact solutions corresponding to warped AdS_3 black holes, and show that these can be connected to asymptotically AdS_5 spacetime. For k\leq 1 the entropy appears to go to a finite value at extremality, but the solutions still exhibit a mild singularity at strictly zero temperature. In addition to our numerics, we carry out a complete perturbative analysis valid to order B^2, and find that this corroborates our numerical results insofar as they overlap.