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

Vincenzo Alba, Kush Saha, Masudul Haque

Published Oct 1, 2013• 17 citations
OpenAlex

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.

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

Vincenzo Alba, Kush Saha, Masudul Haque

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.

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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.
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.
We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.
We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω=±1 with Ω the eigenvalue of T. We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω, ϒ)=(±1, ∓1) where ϒ is the eigenvalue of V. Therefore, the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω, ϒ)=(±1, ∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.
For an anisotropic Heisenberg (XXZ\mathit{XXZ}XXZ) spin chain, we show that an open boundary induces a series of approximately self-similar features at different energy scales, high up in the eigenvalue spectrum. We present a nonequilibrium phenomenon related to this fractal structure, involving states in which a connected block near the edge is polarized oppositely to the rest of the chain. We show that such oppositely polarized blocks can be ``locked'' to the edge of the spin chain and that there is a hierarchy of edge-locking effects at various orders of the anisotropy. The phenomenon enables dramatic control of quantum-state transmission and magnetization control.