Applicability of holography in thermodynamic equilibrium

Abstract

For a strongly coupled system that has a gravity dual description, we show that the standard holographic dictionary yields a non-negative 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.