# Non-linear effects on the holographic free energy and thermodynamic geometry

## TL;DR

This paper analytically explores how non-linear Born-Infeld electrodynamics influences the free energy and thermodynamic geometry of 2+1-dimensional holographic superconductors, revealing subtle effects of non-linearity on critical temperature and charge density.

## Contribution

It introduces an analytical approach to study non-linearity effects in holographic superconductors using thermodynamic geometry and compares two methods for determining critical temperature.

## Key findings

- Non-linearity affects the relation between critical temperature and charge density.
- Thermodynamic geometry captures effects of non-linearity.
- Results slightly differ from numerical data, indicating analytical approximations.

## Abstract

We have analytically investigated the effects of non-linearity on the free energy and thermodynamic geometry of holographic superconductors in $2+1 -$dimensions. The non-linear effect is introduced by considering the coupling of the massive charged scalar field with Born-Infeld electrodynamics. We then calculate the relation between critical temperature and charge density from two different methods, namely, the matching method and the divergence of the scalar curvature which is obtained by investigating the thermodynamic geometry of the model. The two results are slightly different from numerical values but the effects of non-linearity gets captured in our analysis.