Bound states and entanglement in the excited states of quantum spin chains

TL;DR

This paper studies the entanglement properties of excited states in the spin-1/2 Heisenberg chain using Bethe ansatz, revealing bounds, band structures, and effects of bound states on entanglement entropy.

Contribution

It provides a detailed analysis of entanglement in excited states, including bounds, band organization, and the impact of bound states, using exact Bethe ansatz solutions.

Findings

Entanglement entropy exhibits an intermediate scaling between logarithmic and extensive.

Bound states significantly reduce entanglement entropy, acting as single particles.

Highly-entangled states at finite density show extensive entanglement proportional to chain length.

Abstract

We investigate entanglement properties of the excited states of the spin-1/2 Heisenberg (XXX) chain with isotropic antiferromagnetic interactions, by exploiting the Bethe ansatz solution of the model. We consider eigenstates obtained from both real and complex solutions ("strings") of the Bethe equations. Physically, the former are states of interacting magnons, whereas the latter contain bound states of groups of particles. We first focus on the situation with few particles in the chain. Using exact results and semiclassical arguments, we derive an upper bound S_MAX for the entanglement entropy. This exhibits an intermediate behavior between logarithmic and extensive, and it is saturated for highly-entangled states. As a function of the eigenstate energy, the entanglement entropy is organized in bands. Their number depends on the number of blocks of contiguous Bethe-Takahashi quantum numbers. In presence of bound states a significant reduction in the entanglement entropy occurs, reflecting that a group of bound particles behaves effectively as a single particle. Interestingly, the associated entanglement spectrum shows edge-related levels. At finite particle density, the semiclassical bound S_MAX becomes inaccurate. For highly-entangled states S_A\propto L_c, with L_c the chord length, signaling the crossover to extensive entanglement. Finally, we consider eigenstates containing a single pair of bound particles. No significant entanglement reduction occurs, in contrast with the few-particle case.