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arXiv:2401.03172 · Journal of Physics A Mathematical and Theoretical

Root patterns and exact surface energy of the spin-1 Heisenberg model with generic open boundaries

Jiasheng Dong, Pengcheng Lu, Junpeng Cao, Wen‐Li Yang, Ian Marquette, Yao-Zhong Zhang

Published Jul 3, 2024• 2 citations
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Abstract

Abstract We investigate the thermodynamic limit and exact surface energy of the isotropic spin-1 Heisenberg chain with integrable generic open boundary conditions by a novel Bethe ansatz method. We obtain the homogeneous (or two-term) Bethe ansatz like equations for the zero roots of the transfer matrix. Based on the patterns of the zero roots, we analytical calculate the densities of zero roots and the surface energies of the model in all regimes of the boundary parameters.

DOI
10.1088/1751-8121/ad5ee3

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Abstract We investigate the thermodynamic limit and exact surface energy of the isotropic spin-1 Heisenberg chain with integrable generic open boundary conditions by a novel Bethe ansatz method. We obtain the homogeneous (or two-term) Bethe ansatz like equations for the zero roots of the transfer matrix. Based on the patterns of the zero roots, we analytical calculate the densities of zero roots and the surface energies of the model in all regimes of the boundary parameters.
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We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.
Exact physical quantities of the D2(2) {D}_2^{(2)} D2(2)​ spin chain model with generic open boundary conditions
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The thermodynamic limit and boundary energy of the isotropic spin-1 Heisenberg chain with non-diagonal boundary fields are studied. The finite size scaling properties of the inhomogeneous term in the T-Q relation at the ground state are calculated by the density matrix renormalization group. Based on our findings, the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations of a related model with parallel boundary fields. These results can be generalized to the SU(2) symmetric high spin Heisenberg model directly.
The string hypothesis of Bethe roots is a cornerstone in the thermodynamic analysis of quantum integrable systems, since it connects root configurations with physical quantities such as the ground-state energy, surface energy and excitation spectra. For integrable models with U(1) U ( 1 ) symmetry, this connection is well established. When the U(1) U ( 1 ) symmetry is broken by generic non-diagonal boundary fields, however, the off-diagonal Bethe Ansatz leads to an inhomogeneous T\text{--}Q T – Q relation whose Bethe roots have highly nontrivial distributions. This raises two fundamental questions: whether the zero roots and the ODBA Bethe roots still possess regular and classifiable structures in the large-size limit, and whether such structures can be used to extract physical quantities. In this work, we address these two questions for the isotropic Heisenberg spin chain with non-diagonal open boundaries. By combining tensor-network algorithms with Bethe-Ansatz techniques, we determine the zero-root and Bethe-root configurations associated with the \Lambda\text{--}\theta Λ – θ relation and the inhomogeneous Bethe Ansatz equations for large system sizes, up to N\simeq 60 N ≃ 60 and 100 100 . We find that, despite the absence of U(1) U ( 1 ) symmetry, the roots exhibit well-organized patterns. The zero roots form bulk strings, boundary strings, additional roots, and a helical half-spinon, while the ODBA Bethe roots split into four geometric classes: regular roots, line roots, arc roots and paired-line roots. We further show that these root structures are not merely numerical patterns, but contain direct thermodynamic information. In particular, the geometric structure of the ODBA Bethe roots allows us to derive analytic expressions for the surface energy and elementary excitation energies. This demonstrates that the full ODBA Bethe-root configuration can serve as a thermodynamically meaningful object in a U(1) U ( 1 ) -symmetry-broken open spin chain. A comparison with the t\text{--}W t – W scheme shows that the zero-root description has a simpler topology, while the ODBA Bethe roots provide a direct route to physical quantities. Our results therefore provide a framework for studying root structures and thermodynamic properties in quantum integrable models without U(1) U ( 1 ) symmetry.
A bstract The Izergin-Korepin model is an integrable model with the simplest twisted quantum affine algebra U q ( A2(2) {A}_2^{(2)} A2(2)​ A 2 2 ) symmetry. Applying the t - W method, we derive the homogeneous zeroes Bethe ansatz equations and the corresponding zeroes patterns of the Izergin-Korepin model with generic integrable boundaries. Based on these results, we analytically compute the surface energies and boundary excitations in different regimes of boundary parameters of the model. It is shown that in some regimes, correlation effect appears between two boundary fields.