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

Abstract

At large values of the anisotropy Δ, the open-boundary Heisenberg spin- chain has eigenstates displaying localization at the edges. We present a Bethe ansatz description of this ‘edge-locking’ phenomenon in the entire Δ > 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 Δ 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 Δ 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, or 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 Δ (‘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 Δ, and near the exceptional points. In the large-chain limit all the exceptional points coalesce at the isotropic point (Δ = 1) and edge-locking becomes stable in the whole Δ > 1 region.