Thermodynamic curvature of the Schwarzschild-AdS black hole and Bose condensation

Sandip Mahish,Aritra Ghosh,Chandrasekhar Bhamidipati

doi:10.1016/j.physletb.2020.135958

Sandip Mahish, Aritra Ghosh + Show 1 more

Open Access

https://doi.org/10.1016/j.physletb.2020.135958

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Abstract

In the AdS/CFT correspondence, a dynamical cosmological constant Λ in the bulk corresponds to varying the number of colors N in the boundary gauge theory with a chemical potential μ as its thermodynamic conjugate. In this work, within the context of Schwarzschild black holes in AdS5×S5 and its dual finite temperature N=4 superconformal Yang-Mills theory at large N, we investigate thermodynamic geometry through the behavior of the Ruppeiner scalar R. The sign of R is an empirical indicator of the nature of microscopic interactions and is found to be negative for the large black hole branch implying that its thermodynamic characteristics bear qualitative similarities with that of an attraction dominated system, such as an ideal gas of bosons. We find that as the system's fugacity approaches unity, R takes increasingly negative values signifying long range correlations and strong quantum fluctuations signaling the onset of Bose condensation. On the other hand, R for the small black hole branch is negative at low temperatures and positive at high temperatures with a second order critical point which roughly separates the two regions.

Highlights

Thermodynamics is built on a macroscopic framework without per se any reference to the microscopic physics, it is robust enough to yield deep insights into a wide range of phenomena which principally originate at the microscopic level
We studied the thermodynamic geometry of the Schwarzschild-AdS black hole in AdS5 × S5 in the context of the AdS/conformal field theory (CFT) correspondence where a dynamical cosmological constant in the bulk corresponds to varying number of colors in the dual N = 4 superconformal Yang-Mills theory at finite temperature and large N
We learnt from the analysis of the thermodynamic curvature or Ruppeiner scalar R that the supposed behavior of the large black hole branch greatly mimics that of an ideal Bose gas showing the characteristics of Bose condensation

Summary

INTRODUCTION

Thermodynamics is built on a macroscopic framework without per se any reference to the microscopic physics, it is robust enough to yield deep insights into a wide range of phenomena which principally originate at the microscopic level. Since a dynamical cosmological constant leads to the introduction of a pressure in bulk, it is interesting to ask whether the same interpretation is applicable to the boundary CFT as well Examining this interpretation in the case of AdS5 × S5 and the finite temperature N = 4 superconformal Yang-Mills theory at large N reveals that it is more suitable to view a dynamical Λ as being associated to varying number of colors [9, 13,14,15,16,17] of the dual CFT. The temperature at which it approaches zero is close to the point where the black hole undergoes the Hawking-Page (HP) transition and it was proposed to correspond to Bose condensation This was improved further in [28], where it was shown that μ = 0 might be reached in other systems involving charged, rotating and higher curvature AdS black holes, exactly at or above the HP transition temperature if one makes use of densities instead of absolute thermodynamic quantities. We infer empirically from the sign of the Ruppeiner scalar that the small black hole branch is attraction dominated at low temperatures and repulsion dominated at high temperatures with a second order critical point approximately separating the two regions

THERMODYNAMICS WITH CHEMICAL POTENTIAL

Chemical potential

Behavior of the specific heats

THERMODYNAMIC GEOMETRY

DISCUSSION