# Nonanalyticity of circuit complexity across topological phase transitions

## TL;DR

This paper demonstrates that circuit complexity can detect topological phase transitions through nonanalytic behavior, both in and out of equilibrium, in one- and two-dimensional models, with implications for quantum simulation.

## Contribution

It introduces a specific measure of circuit complexity that reveals topological phase transitions via nonanalyticity, extending understanding of complexity in quantum phase transitions.

## Key findings

- Nonanalyticity in circuit complexity signals topological phase transitions.
- Quench dynamics show linear or quadratic growth of complexity.
- Dimensional increase weakens but does not eliminate nonanalyticity.

## Abstract

The presence of nonanalyticity in observables is a manifestation of phase transitions. Through the study of two paradigmatic topological models in one and two dimensions, in this work we show that the circuit complexity based on our specific quantification can reveal the occurrence of topological phase transitions, both in and out of equilibrium, by the presence of nonanalyticity. By quenching the system out of equilibrium, we find that the circuit complexity grows linearly or quadratically in the short-time regime if the quench is finished instantaneously or in a finite time, respectively. Notably, we find that for both the sudden quench and the finite-time quench, a topological phase transition in the pre-quench Hamiltonian will be manifested by the presence of nonanalyticity in the first-order or second-order derivative of circuit complexity with respect to time in the short-time regime, and a topological phase transition in the post-quench Hamiltonian will be manifested by the presence of nonanalyticity in the steady value of circuit complexity in the long-time regime. We also show that the increase of dimension does not remove, but only weakens the nonanalyticity of circuit complexity. Our findings can be tested in quantum simulators and cold-atom systems.