Bethe ansatz description of edge-localization in the open-boundary XXZ spin chain

TL;DR

This paper uses Bethe ansatz to analytically describe edge-localization phenomena in the open-boundary XXZ spin chain for anisotropies greater than one, revealing how eigenstates localize at edges and how this behavior changes with anisotropy and chain size.

Contribution

It provides a Bethe ansatz framework for understanding edge-locking in the XXZ chain across the entire >1 region, including analytic expansions and the behavior at exceptional points.

Findings

Edge-locking associated with imaginary Bethe solutions at large .

Eigenstates can have particles locked at edges or delocalized, depending on anisotropy.

Edge-locking becomes stable for all >1 in the large chain limit.

Abstract

At large values of the anisotropy \Delta, the open-boundary Heisenberg spin-1/2 chain has eigenstates displaying localization at the edges. We present a Bethe ansatz description of this `edge-locking' phenomenon in the entire \Delta>1 region. We focus on the simplest spin sectors, namely the highly polarized sectors with only one or two overturned spins, i.e., one-particle and two-particle sectors. Edge-locking is associated with pure imaginary solutions of the Bethe equations, which are not commonly encountered in periodic chains. In the one-particle case, at large anisotropies there are two eigenstates with imaginary Bethe momenta, related to localization at the two edges. For any finite chain size, one of the two solutions become real as the anisotropy is lowered below a certain value. For two particles, a richer scenario is observed, with eigenstates having the possibility of both particles locked on the same or different edge, one locked and the other free, and both free either as single magnons or as bound composites corresponding to `string' solutions. For finite chains, some of the edge-locked spins get delocalized at certain values of the anisotropy (`exceptional points'), corresponding to imaginary solutions becoming real. We characterize these phenomena thoroughly by providing analytic expansions of the Bethe momenta for large chains, large anisotropy, and near the exceptional points. In the large-chain limit all the exceptional points coalesce at the isotropic point (\Delta=1) and edge-locking becomes stable in the whole \Delta>1 region.