# On the Complexity of a $2+1$--dimensional Holographic Superconductor

## TL;DR

This paper investigates the subregion complexity and entanglement entropy of a 2+1-dimensional holographic superconductor with backreaction, revealing phase transition behaviors and the growth of complexity with strip width.

## Contribution

It provides the first detailed analysis of subregion complexity in a fully backreacted holographic superconductor, including phase transition effects and multivalued complexity behavior.

## Key findings

- Complexity grows linearly with strip width.
- Phase transition occurs at the same critical temperature as free energy analysis.
- Multivalued complexity (

## Abstract

We present the results of our computation of the subregion complexity and also compare it with the entanglement entropy of a $2+1$--dimensional holographic superconductor which has a fully backreacted gravity dual with a stable ground sate. We follow the "complexity equals volume" or the CV conjecture. We find that there is only a single divergence for a strip entangling surface and the complexity grows linearly with the large strip width. During the normal phase the complexity increases with decreasing temperature, but during the superconducting phase it behaves differently depending on the order of phase transition. We also show that the universal term is finite and the phase transition occurs at the same critical temperature as obtained previously from the free energy computation of the system. In one case, we observe multivaluedness in the complexity in the form of an "S" curve.