# Phase transition in complex-time Loschmidt echo of short and long range spin chain

## TL;DR

This paper investigates phase transitions in the complex-time Loschmidt echo of XX spin chains, revealing third order phase transitions in models with various interaction ranges using random matrix and Toeplitz determinant techniques.

## Contribution

It introduces a random matrix formulation for the Loschmidt echo in generalized XX spin chains and characterizes phase transitions for long-range interactions.

## Key findings

- Identifies third order phase transitions in complex-time Loschmidt echo.
- Derives exact Loschmidt echo expressions for long-range models.
- Analyzes Stokes phenomena related to Fisher-Hartwig singularities.

## Abstract

We explain and exploit the random matrix formulation of the Loschmidt echo for the XX spin chain, valid for multiple domain wall initial states and also for a XX spin chain generalized with additional interactions to more neighbours. For models with interactions decaying as $e^{-\alpha \left\vert l-j\right\vert }/\left\vert l-j\right\vert ^{p+1}$, with $p$ integer or natural number and $\alpha \geq 0$, we show that there are third order phase transitions in a double scaling limit of the complex-time Loschmidt echo amplitudes. For the long-range version of the chain, we use an exact result for Toeplitz determinants with a pure Fisher-Hartwig singularity, to obtain exactly the Loschmidt echo for complex times and discuss the associated Stokes phenomena. We also study the case of a finite chain for one of the generalized XX models.