Overlap distributions for quantum quenches in the anisotropic Heisenberg chain

TL;DR

This paper analyzes the overlap distributions in the anisotropic Heisenberg chain after quantum quenches, providing insights into the eigenstate contributions and deriving formulas for dynamic quantities, with a focus on specific initial and final anisotropy states.

Contribution

It offers a detailed analysis of overlap distributions for quenches in the XXZ chain, including explicit formulas and selection rules, enhancing understanding of post-quench dynamics.

Findings

Only a small fraction of eigenstates contribute after a Ne9el to XX quench.

Contributions from eigenstates have identical overlap magnitudes.

Derived concise expressions for Loschmidt echo and correlators.

Abstract

The dynamics after a quantum quench is determined by the weights of the initial state in the eigenspectrum of the final Hamiltonian, i.e., by the distribution of overlaps in the energy spectrum. We present an analysis of such overlap distributions for quenches of the anisotropy parameter in the one-dimensional anisotropic spin-1/2 Heisenberg model (XXZ chain). We provide an overview of the form of the overlap distribution for quenches from various initial anisotropies to various final ones, using numerical exact diagonalization. We show that if the system is prepared in the antiferromagnetic N\'eel state (infinite anisotropy) and released into a non-interacting setup (zero anisotropy, XX point) only a small fraction of the final eigenstates gives contributions to the post-quench dynamics, and that these eigenstates have identical overlap magnitudes. We derive expressions for the overlaps, and present the selection rules that determine the final eigenstates having nonzero overlap. We use these results to derive concise expressions for time-dependent quantities (Loschmidt echo, longitudinal and transverse correlators) after the quench. We use perturbative analyses to understand the overlap distribution for quenches from infinite to small nonzero anisotropies, and for quenches from large to zero anisotropy.