Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

TL;DR

This paper derives exact formulas for the return probability after a domain wall quench in the XXZ spin chain, revealing complex decay behaviors and connections to statistical models, with implications for spin transport properties.

Contribution

It establishes exact Fredholm determinant formulas for return probabilities in the XXZ chain after a domain wall quench, linking to the six vertex model and analyzing decay behaviors.

Findings

Decay is Gaussian at roots of unity, exponential otherwise for ||<1

Front position moves as t(t)=t(1-^2)

At ||=1, return probability decays as e^{-\u03b6(3/2)\,rac{ t}{}}t^{1/2}

Abstract

We study the return probability and its imaginary ($\tau$) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy $|\Delta|< 1$. We establish exact Fredholm determinant formulas for those, by exploiting a connection to the six vertex model with domain wall boundary conditions. In imaginary time, we find the expected scaling for a partition function of a statistical mechanical model of area proportional to $\tau^2$, which reflects the fact that the model exhibits the limit shape phenomenon. In real time, we observe that in the region $|\Delta|<1$ the decay for large times $t$ is nowhere continuous as a function of anisotropy: it is either gaussian at root of unity or exponential otherwise. As an aside, we also determine that the front moves as $x_{\rm f}(t)=t\sqrt{1-\Delta^2}$, by analytic continuation of known arctic curves in the six vertex model. Exactly at $|\Delta|=1$, we find the return probability decays as $e^{-\zeta(3/2) \sqrt{t/\pi}}t^{1/2}O(1)$. It is argued that this result provides an upper bound on spin transport. In particular, it suggests that transport should be diffusive at the isotropic point for this quench.