# Dynamical signature of localization-delocalization transition in one-dimensional incommensurate lattice

## TL;DR

This paper studies the dynamical behavior of a one-dimensional incommensurate lattice model during a quench, revealing that zero points in the Loschmidt echo serve as signatures of the localization-delocalization transition.

## Contribution

It provides analytical and numerical analysis of Loschmidt echo behavior across phase transitions in the Aubry-Andr model, linking zero points to phase changes.

## Key findings

- Zero points in Loschmidt echo indicate phase transitions.
- Analytical expressions for Loschmidt echo during specific quenches.
- Loschmidt echo remains positive within same phase, reaches zero across different phases.

## Abstract

We investigate the quench dynamics of a one-dimensional incommensurate lattice described by the Aubry-Andr\'{e} model by a sudden change of the strength of incommensurate potential $\Delta$ and unveil that the dynamical signature of localization-delocalization transition can be characterized by the occurrence of zero points in the Loschmit echo. For the quench process with quenching taking place between two limits of $\Delta=0$ and $\Delta=\infty$, we give analytical expressions of the Loschmidt echo, which indicate the existence of a series of zero points in the Loschmidt echo. For a general quench process, we calculate the Loschmidt echo numerically and analyze its statistical behavior. Our results show that if both the initial and post-quench Hamiltonian are in extended phase or localized phase, Loschmidt echo will always be greater than a positive number; however if they locate in different phases, Loschmidt echo can reach nearby zero at some time intervals.