# Topological invariants of the Ryu-Takayanagi ($RT$) surface used to observe holographic superconductor phase transition

## TL;DR

This paper uses topological invariants of the Ryu-Takayanagi surface in Lifshitz black hole backgrounds to identify and analyze phase transitions in holographic superconductors, providing clearer insights than traditional methods.

## Contribution

It introduces topological invariants of the RT surface as tools to detect phase transitions and their order in holographic superconductor models, surpassing entanglement entropy and complexity measures.

## Key findings

- Topological invariants identify phase transition points and their order.
- Discontinuity slopes at critical points indicate second-order transitions.
- Backreaction parameter influences the nature of the phase transition.

## Abstract

We study the phase transitions in the metal/superconductor system using topological invariants of the Ryu-Takayanagi ($RT$) surface and the volume enclosed by the $RT$ surface in the Lifshitz black hole background. It is shown that these topological invariant quantities identify not only the phase transition but also its order. According to these findings a discontinuity slope is observed at the critical points for these invariant quantities that correspond to the second order of phase transition. These topological invariants provide a clearer illustration of the superconductor phase transition than do the holographic entanglement entropy and the holographic complexity. Also, the backreaction parameter, $k$, is found to have an important role in distinguishing the critical points. The reducing values of the parameter $k$ means that the backreaction of the matter fields are negligible. A continuous slope is observed around the critical points which is characteristic of the probe limit. In addition, exploring the nonlinear electrodynamic, the effects of the nonlinear parameter, $\beta$, is investigated. Finally the properties of conductivity are numerically explored in our model.