Long time behaviour of a local perturbation in the isotropic XY chain under periodic forcing

TL;DR

This paper investigates the long-term behavior of a quantum spin chain under periodic magnetic forcing, demonstrating the existence of a synchronized periodic state using KAM techniques under certain frequency conditions.

Contribution

It introduces a novel application of KAM-type renormalization to analyze the asymptotic periodicity of the isotropic XY chain with periodic forcing.

Findings

Existence of a periodic solution under diophantine-like frequency conditions

Asymptotic state of the chain is a periodic function synchronized with the forcing

Applicable to low-frequency forcing scenarios

Abstract

We study the isotropic XY quantum spin chain with a time-periodic transverse magnetic field acting on a single site. The asymptotic problem can be mapped into a highly resonant Floquet-Schr\"odinger equation, for which, under a diophantine-like assumption on the frequency, we show the existence of a periodic solution. The proof is based on a KAM-type renormalisation. This in turn implies the state of the quantum spin chain to be asymptotically a periodic function synchronised with the forcing also at low frequencies.