# $Q$-$\Phi$ criticality and microstructure of charged AdS black holes in $f(R)$ gravity

## TL;DR

This paper explores the phase transitions and microscopic properties of charged AdS black holes in $f(R)$ gravity, revealing Van der Waals-like critical behaviour and a Fermi gas analogy through thermodynamic geometry analysis.

## Contribution

It introduces new thermodynamic parameters and investigates the impact of $f(R)$ gravity modifications on black hole phase structures and microscopic properties.

## Key findings

- Black holes undergo first-order small-large phase transitions.
- Critical behaviour resembles Van der Waals liquid-vapor system.
- Black hole microstructure shows Fermi gas properties.

## Abstract

The phase transition and critical behaviours of charged AdS black holes in $f(R)$ gravity with a conformally invariant Maxwell (CIM) source and constant curvature are further investigated. As a highlight, this research is carried out by employing new state parameters $(T,Q,\Phi)$ and contributes to deeper understanding the thermodynamics and phase structure of black holes. Our analyses manifest that the charged $f(R)$-CIM AdS black hole undergoes a first order small-large black hole phase transition, and the critical behaviours qualitatively behave like a Van der Waals liquid-vapor system. However, differing from the case in Einstein's gravity, phase structures of the black holes in $f(R)$ theory exhibit an interesting dependence on gravity modification parameters. Moreover, we adopt the thermodynamic geometry to probe the black hole microscopic properties. The results show that, on the one hand, both the Ruppeiner curvature and heat capacity diverge exactly at the critical point, on the other hand, the $f(R)$-CIM AdS black hole possesses the property as ideal Fermi gases. Of special interest, we discover a microscopic similarity between the black holes and a Van der Waals liquid-vapor system.