Thermodynamic geometry and complexity of black holes in theories with broken translational invariance

TL;DR

This paper explores the connection between thermodynamic geometry and holographic complexity in black holes with broken translational symmetry, revealing that thermodynamic curvature singularities align with zero complexity points and proposing action complexity as an order parameter.

Contribution

It demonstrates the correspondence between thermodynamic curvature singularities and complexity zeros in black holes with broken translational invariance, introducing action complexity as an order parameter.

Findings

Thermodynamic curvature diverges at minimal mass and temperature.

Complexity of volume and action vanish at these minimal points.

Action complexity can serve as an order parameter for black hole thermodynamics.

Abstract

The relationship between thermodynamics and the Lloyd bound on the holographic complexity for a black hole has been of interest. We consider $D$ dimensional anti-de Sitter black holes with hyperbolic geometry as well as black holes with momentum relaxation that have a minimum for temperature and mass. We show that the singular points of the thermodynamic curvature of the black holes, as thermodynamic systems, correspond to the zero points of the action and volume complexity at the Lloyd bound. For such black holes with a single horizon, the complexity of volume and the complexity of action at minimum mass and minimum temperature are zero, respectively. We show that the thermodynamic curvature diverges at these minimal values. Because of the behaviour of action complexity and thermodynamic curvature at minimum temperature, we propose the action complexity as an order parameter of the black holes as thermodynamic systems. Also, we derive the critical exponent related to the thermodynamic curvature in different dimensions.