# Loschmidt Echo in Many-Body Localized Phase

## TL;DR

This paper investigates the behavior of the Loschmidt echo in the many-body localized phase, revealing slow, power-law decay of fluctuations and its relation to operator spreading, distinguishing it from ergodic and Anderson insulator phases.

## Contribution

It demonstrates that the Loschmidt echo exhibits power-law decay in the MBL phase and links this behavior to operator spreading and local integrals of motion, providing new insights into non-ergodic dynamics.

## Key findings

- Loschmidt echo fluctuations decay as a power law in MBL phase
- Spin-echo correlation saturates to a finite value in MBL
- Loschmidt echo probes operator spreading and local integrals of motion

## Abstract

The Loschmidt echo, defined as the overlap between quantum wave function evolved with different Hamiltonians, quantifies the sensitivity of quantum dynamics to perturbations and is often used as a probe of quantum chaos. In this work we consider the behavior of the Loschmidt echo in the many body localized phase, which is characterized by emergent local integrals of motion, and provides a generic example of non-ergodic dynamics. We demonstrate that the fluctuations of the Loschmidt echo decay as a power law in time in the many-body localized phase, in contrast to the exponential decay in few-body ergodic systems. We consider the spin-echo generalization of the Loschmidt echo, and argue that the corresponding correlation function saturates to a finite value in localized systems. Slow, power-law decay of fluctuations of such spin-echo-type overlap is related to the operator spreading and is present only in the many-body localized phase, but not in a non-interacting Anderson insulator. While most of the previously considered probes of dephasing dynamics could be understood by approximating physical spin operators with local integrals of motion, the Loschmidt echo and its generalizations crucially depend on the full expansion of the physical operators via local integrals of motion operators, as well as operators which flip local integrals of motion. Hence, these probes allow to get insights into the relation between physical operators and local integrals of motion, and access the operator spreading in the many-body localized phase.