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

TL;DR

Using 5D Einstein–Maxwell theory with a Chern–Simons term, the paper constructs asymptotically AdS_5 charged magnetic branes dual to 4D gauge theories under a uniform magnetic field and electric density. The low-temperature behavior depends sensitively on the CS coupling k: for k>1 a small B dramatically reduces the extremal entropy and the near-horizon geometry destabilizes, while for k≤1 the extremal entropy remains finite with mild horizon singularities; at k=1 there are exact warped AdS_3×R^2 near-horizon states. A perturbative B^2 analysis and extensive numerical studies map the phase diagram, and the authors propose an AdS/CFT version of Nernst's theorem to interpret these results. The work has implications for extremal black branes, quantum criticality, and holographic condensed matter applications.

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 ρ. 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.

Figures (7)

• Figure 1: Illustration of fine tuning required to maintain nonzero extremal entropy. Small nonzero couplings $\lambda_i$ lead to no appreciable effect at high temperature, but cause the low temperature entropy to flow to zero.
• Figure 2: Schematic illustration of flows in parameter space for three ranges of $k$. Blue lines are flows at various fixed values of $B^3/\rho^2$. Arrows indicate direction of decreasing temperature. Red lines indicate the boundary of allowed $(b,q)$ values where nonsingular solutions are possible. The near-horizon geometries at the end points of the flows are indicated in the $k<1$ and $k=1$ cases. For $k>1$ there exists an AdS$_3\times R^2$ solution at $b=\sqrt{3}$, indicated by the dot, but the flows do not reach this point. In the $k<1$ and $k>1$ cases, near the endpoint of the flow $B^3/\rho^2$ becomes a very sensitive function of $(b,q)$, depending on the precise direction of approach.
• Figure 3: Plot of the entropy versus temperature at fixed $B^3/\rho^2=0$ and $B^3/\rho^2 = .15 \pm .002$, for $k=2/\sqrt{3}$ (supersymmetric value). The numerical results show that a small $B$ field causes a large drop in the entropy at low temperature.
• Figure 4: Plot of the entropy versus temperature at fixed $B^3/\rho^2=0$ and $B^3/\rho^2 =10.5 \pm .6$, for $k=0$. The entropy appears to go to a finite value (but see the text for comments on the strict zero temperature limit).
• Figure 5: The function $\hat{\tau} (\lambda )$.