Non-equilibrium quantum relaxation across a localization-delocalization transition

TL;DR

This paper investigates the non-equilibrium relaxation dynamics of a one-dimensional quasi-periodic quantum system across a localization-delocalization transition, revealing distinct entanglement and order parameter behaviors in different phases.

Contribution

It provides a detailed analysis of quench and adiabatic dynamics in the 1D $XX$-model with a quasi-periodic potential, highlighting new scaling laws and dynamical signatures of the transition.

Findings

Entanglement entropy grows linearly in delocalized phase

Entropy saturates in localized phase

Defect density scales with the rate of adiabatic change

Abstract

We consider the one-dimensional $XX$-model in a quasi-periodic transverse-field described by the Harper potential, which is equivalent to a tight-binding model of spinless fermions with a quasi-periodic chemical potential. For weak transverse field (chemical potential), $h<h_c$, the excitations (fermions) are delocalized, but become localized for $h>h_c$. We study the non-equilibrium relaxation of the system by applying two protocols: a sudden change of $h$ (quench dynamics) and a slow change of $h$ in time (adiabatic dynamics). For a quench into the delocalized (localized) phase, the entanglement entropy grows linearly (saturates) and the order parameter decreases exponentially (has a finite limiting value). For a critical quench the entropy increases algebraically with time, whereas the order parameter decreases with a stretched-exponential. The density of defects after an adiabatic field change through the critical point is shown to scale with a power of the rate of field change and a scaling relation for the exponent is derived.