Non-ergodic states induced by impurity levels in quantum spin chains

TL;DR

This paper investigates how impurity-induced localized states in a quantum spin chain can lead to non-ergodic behavior, with exact solutions showing the conditions for relaxation or localization effects.

Contribution

It provides an exact analytic study of a boundary impurity in a quantum XY spin chain, identifying conditions for ergodic and non-ergodic dynamics based on localized states.

Findings

Localized states split off from the continuum for certain impurity parameters

Ergodic behavior occurs when no localized states are present

Localization phenomena can prevent the system from relaxing to equilibrium

Abstract

The semi-infinite XY spin chain with an impurity at the boundary has been chosen as a prototype of interacting many-body systems to test for non-ergodic behavior. The model is exactly solvable in analytic way in the thermodynamic limit, where energy eigenstates and the spectrum are obtained in closed form. In addition of a continuous band, localized states may split off from the continuum, for some values of the impurity parameters. In the next step, after the preparation of an arbitrary non-equilibrium state, we observe the time evolution of the site magnetization. Relaxation properties are described by the long-time behavior, which is estimated using the stationary phase method. Absence of localized states defines an ergodic region in parameter space, where the system relaxes to a homogeneous magnetization. Out of this region, impurity levels split from the band, and localization phenomena may lead to non-ergodicity.