# Relaxation of the order-parameter statistics in the Ising quantum chain

## TL;DR

This paper analyzes the out-of-equilibrium distribution of the local order parameter in the transverse field Ising chain after a quench, revealing phase-dependent memory effects and a connection to classical models.

## Contribution

It provides an analytical description of the late-time stationary distribution of the order parameter, linking it to a classical partition function and ground-state properties.

## Key findings

- Stationary distribution depends on the phase of the post-quench Hamiltonian.
- Deep in the broken-symmetry phase, the distribution relates to the ground state.
- The distribution retains memory of initial long-range order in certain phases.

## Abstract

We study the out-of-equilibrium probability distribution function of the local order parameter in the transverse field Ising quantum chain. Starting from a fully polarised state, the relaxation of the ferromagnetic order is analysed: we obtain a full analytical description of the late-time stationary distribution by means of a remarkable relation to the partition function of a 3-states classical model. Accordingly, depending on the phase whereto the post-quench Hamiltonian belongs, the probability distribution may locally retain memories of the initial long-range order. When quenching deep in the broken-symmetry phase, we show that the stationary order-parameter statistics is indeed related to that of the ground state. We highlight this connection by inspecting the ground-state equilibrium properties, where we propose an effective description based on the block-diagonal approximation of the $n$-point spin correlation functions.