On the applicability of holography in thermodynamic equilibrium

TL;DR

This paper investigates the application of holography to thermodynamic equilibrium systems, revealing conditions for susceptibility positivity, violations of the trace energy condition, and proposing a candidate c-function, while addressing a related paradox.

Contribution

It demonstrates that holographic susceptibility is nonnegative in equilibrium, identifies violations of the trace energy condition, and introduces a monotonic c-function candidate, advancing understanding of holographic thermodynamics.

Findings

Holographic susceptibility is nonnegative in equilibrium.

Violations of the trace energy condition occur in certain cases.

A monotonic c-function candidate is proposed.

Abstract

For a strongly coupled system that has a gravity dual description, we show that the standard holographic dictionary yields a nonnegative susceptibility when the system is in thermodynamic equilibrium and the correlation function is absolutely integrable. When the system has no spontaneous condensation or has a spontaneous $\mathbb{Z}_2$-symmetry breaking, we find that the ``trace energy condition" is violated in many cases. There is a normalized grand potential density that is monotonic as accessing to lower scales, providing a candidate $c$-function characterizing the number of effective degrees of freedom. Finally, we discuss a ``paradox'' raising by the negative susceptibility in holography and its resolution.