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Off-shell phase diagram of BPS black holes in AdS$_5$

Debabrata Sahu, Chandrasekhar Bhamidipati

Abstract

We construct the off-shell free energy of supersymmetric black holes in AdS$_5$, and study the phase diagram in various limiting cases, with particular emphasis on BPS thermodynamics. The changes to the free energy following from the four-derivative corrections to five-dimensional minimal gauged supergravity action are computed, and the modifications to the phase diagram are studied. Starting from Landau's theory, an exact method is systematically developed to construct the off shell BPS free energy, which in certain limiting cases, can be rearranged in terms of an effective energy and entropy of the system, with the later being conjugate to an effective BPS temperature. The off-shell BPS phase diagram shows features which resemble the phases of general AdS Schwarzschild black holes, with some nuances in the asymptotic structure, modified by four-derivative corrections. Using AdS/CFT, phenomenological effective potentials in the boundary gauge theory are proposed, dual to both general black holes and their BPS counterparts. The saddle points of the effective potential capture the various locally stable and unstable phases of the gauge theory at finite temperature and chemical potential.

Off-shell phase diagram of BPS black holes in AdS$_5$

Abstract

We construct the off-shell free energy of supersymmetric black holes in AdS, and study the phase diagram in various limiting cases, with particular emphasis on BPS thermodynamics. The changes to the free energy following from the four-derivative corrections to five-dimensional minimal gauged supergravity action are computed, and the modifications to the phase diagram are studied. Starting from Landau's theory, an exact method is systematically developed to construct the off shell BPS free energy, which in certain limiting cases, can be rearranged in terms of an effective energy and entropy of the system, with the later being conjugate to an effective BPS temperature. The off-shell BPS phase diagram shows features which resemble the phases of general AdS Schwarzschild black holes, with some nuances in the asymptotic structure, modified by four-derivative corrections. Using AdS/CFT, phenomenological effective potentials in the boundary gauge theory are proposed, dual to both general black holes and their BPS counterparts. The saddle points of the effective potential capture the various locally stable and unstable phases of the gauge theory at finite temperature and chemical potential.

Paper Structure

This paper contains 26 sections, 120 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Off-shell Free energy of AdS$_5$ black holes
  3. General AdS5 black hole thermodynamics
  4. Off-shell free energy in the grand canonical ensemble
  5. Nonrotating ($\Omega = 0$) black holes, subcritical electric potential : $0 < {\Phi} < \sqrt{3}$
  6. Subcritical electric potential : $0 < {\Phi} < \sqrt{3}$ with $\Omega \neq 0$
  7. Vanishing electric potential limit: $\Phi = 0$
  8. Critical electric potential $\Phi = \Phi^* = \sqrt{3}$
  9. Maximal rotational velocity $\Omega = 1$
  10. Phase diagram of AdS$_5$ black holes with four-derivative corrections
  11. Off-shell free energy of BPS black holes
  12. On-shell free energy
  13. Off-shell BPS free energy from Landau's phenomenological approach
  14. Exact Method for off-shell BPS free energy
  15. Two derivative case
  16. ...and 11 more sections

Figures (19)

  • Figure 1: (a) For subcritical electric potential ${\Phi} = \frac{\sqrt{3}}{2}, {\Omega} = 0$ : Behaviour of BW free energy as a function of the order parameter $r_+$ at different temperatures ${T}$. Temperature of the curves increases from top to bottom. Hawking-Page transition happens at the temperature ${T}_{HP} = 0.413$ (solid red curve). (b) Behaviour of BW free energy as a function of the order parameter $r_+$ for $\Omega = 0$: The yellow, red, blue curves represent ${\Phi } = \frac{\sqrt{3}}{4}, \frac{\sqrt{3}}{2}, \frac{3\sqrt{3}}{4}$ with respective Hawking-Page temperatures ${T}_{HP} = 0.462, 0.413, 0.315$.
  • Figure 2: Free energy as a function of the order parameter $r_+$ for ${\Phi} = \frac{\sqrt{3}}{2}$: The yellow, red, blue curves represent ${\Omega} = 0.5, 0.8, 0,9$ with respective Hawking-Page temperatures ${T}_{HP} = 0.372, 0.291, 0.242$ .
  • Figure 3: Free energy as a function of the order parameter $r_+$ for fixed ${\Phi} = \frac{\sqrt{3}}{2}$ and varying $\Omega$. The yellow, red, blue curve represents ${\Omega} = 0.5, 0.6, 0.7$ with common temperature $T = 0.39 >T_{\rm HP}$ of all three curves, for (a) relatively large range of $r_+$ (large black holes). (b) small range of $r_+$ (small black holes).
  • Figure 4: BW free energy as a function of the order parameter $r_+$ for ${\Phi} = 0$: The yellow, red, blue curves represent ${\Omega} = 0.5, 0.8, 0,9$ with respective Hawking-Page temperatures ${T}_{HP} = 0.434, 0.350, 0.297$.
  • Figure 5: (a) BW free energy as a function of the order parameter $r_+$ for ${\Phi} = \Phi^* = \sqrt{3}$ and $\Omega = 0.5$ : The yellow, red, blue curves represents temperatures ${T} = 0.25, 0.3, 0.35$ respectively. (b) BW free energy as a function of the order parameter $r_+$ for ${\Phi} = \Phi^* = \sqrt{3}$ and common temperature $T = 0.3$ : The yellow, red, blue curves represent $\Omega = 0.5, 0.7,0.8$ respectively.
  • ...and 14 more figures