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arXiv:2606.14140 · Journal of Statistical Mechanics Theory and Experiment

Quantum Entanglement of Bethe States

Yu Hao, Yunfeng Jiang, Bi-Quan Yang, De-liang Zhong

hep-thcond-mat.stat-mechquant-phSubmitted Jun 12, 2026• Announced Jun 15, 2026• 0 citations
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Abstract

We investigate the quantum entanglement of Bethe states across a family of integrable spin chains, including the XXX$_{\frac{1}{2}}$ model, its higher-spin generalizations (XXX$_s$), and the non-compact $SL(2,\mathbb{R})$ chain. For on-shell eigenstates, we perform a comprehensive scan of the bipartite entanglement entropy across the entire spectrum of finite chains with periodic boundary conditions, and identify the Bethe solutions that minimize and maximize the entanglement. These extremal solutions follow systematic, spin-dependent patterns in the Bethe quantum numbers. In the XXX$_{\frac{1}{2}}$ spin chain, for the antiferromagnetic chain, the state with minimal entropy always coincides with the lowest-energy state (the ground state) within a given fixed-magnon sector. For the higher-spin XXX$_s$ model, however, the lowest-entropy state is not always identical to the ground state, and can even be the state of highest energy. By contrast, the Bethe roots that maximize entropy exhibit considerably more intricate structure. Our analysis further reveals how special Bethe root configurations, such as singular and strange solutions, affect entanglement, and it uncovers characteristic entanglement features in the non-compact $SL(2,\mathbb{R})$ chain that are absent from compact spin chains. For off-shell Bethe states, we develop an optimization algorithm that extremizes the entanglement entropy over rapidity distributions, enabling us to explore the maximum entanglement achievable by a Bethe state without imposing the Bethe ansatz equations.

Comments
84 pages, 16 figures

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We investigate the entanglement properties of the excited states of the spin- 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 behaviour 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 the 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 a finite particle density, the semiclassical bound S MAX becomes inaccurate. For highly-entangled states S A ∝ L c , with L c the chord length, signalling 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.
We investigate Bethe Ansatz equations for the one-dimensional spin-12\frac{1}{2}21​ Heisenberg XXX chain with a special interest in a finite system. Solutions for the two-particle sector are obtained. The ground state in antiferromagnetic case has been analytically studied through the logarithmic form of Bethe Ansatz equations.
The spin-1/21/21/2 Heisenberg antiferromagnetic chain is the canonical example of an integrable quantum many-body model. Despite its exact solvability, explicit finite-size solutions are typically only accessible via numerical evaluation of the Bethe ansatz equations. Here, we analyse the algebraic structure of the exact, symbolic ground states for chains up to twelve sites using the Bethe ansatz. We find that the algebraic degree of the ground state magnon phases and Bethe roots scale with respect to system size at a different rate to the ground-state energy and wavefunction coefficients. The ground-state Bethe roots and energy are found to become Galois unsolvable for ten or more sites, thus demonstrating the lack of a symbolic analytical solution in a quantum integrable model due to algebraic complexity.
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Abstract Every solution of the Bethe ansatz equations (BAEs) is characterized by a set of quantum numbers called the Bethe quantum numbers, which are fundamental for evaluating each Bethe root numerically. We rigorously derive the Bethe quantum numbers for the real solutions of the spin-1/2 massive XXZ spin chain in the two down-spin sector, assuming the existence of solutions to some form of BAE. In the sector the quantum numbers J 1 and J 2 were derived for complex solutions, but not for real solutions. We show the exact results in the sector as follows. (i) When two Bethe quantum numbers are different, i.e. for J 1 ≠ J 2 , we introduce a graphical method, which we call a contour method, for deriving the solution of BAE to a given set of Bethe quantum numbers. By the method, we can readily show the existence and the uniqueness of the solution. (ii) When two Bethe quantum numbers are equal, i.e. for J 1 = J 2 , we derive the criteria for the collapse of two-strings and the emergence of an extra two-string by an analytic method. (iii) We obtain the number of real solutions, which depends on the site number N and the XXZ anisotropy parameter ζ. (iv) We derive all infinite-valued solutions of BAE for the XXX spin chain in the two down-spin sector through the XXX limit. (v) We explicitly show the completeness of the Bethe ansatz in terms of the Bethe quantum numbers.
We solve for the spectrum of quantum spin chains based on representations of the Temperley - Lieb algebra associated with the quantum groups for and . We employ a generalization of the coordinate Bethe ansatz developed previously for the deformed biquadratic spin-one chain. As expected, all of these models have equivalent spectra, i.e. they differ only in the degeneracy of their eigenvalues. This is true for finite length and open boundary conditions. For periodic boundary conditions the spectra of the lower-dimensional representations are contained entirely in the higher-dimensional ones. The Bethe states are highest-weight states of the quantum group, except in certain cases where the states have energy zero.