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arXiv:2606.01918 · Journal of Physics A Mathematical and Theoretical

Exact spectrum of the XX spin chain with constrained non-diagonal boundary fields

Changqing Liu, Xiaotian Xu, Xin Zhang

Published Sep 7, 2026• 0 citations
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Abstract

Abstract We study the exact spectrum of the XX spin chain with constrained non-diagonal boundary fields, which can be analyzed by solving the associated Bethe Ansatz equations. In these equations, the number of Bethe roots has a fixed parity, and all Bethe roots are located at the zeros of a unary function. We investigate the possible positions of the Bethe roots. Based on numerical observations, we analyze the Bethe root configurations for the ground state and the first excited state. Our results show that elementary excitations are characterized by the cooperative change of a pair of Bethe roots. Furthermore, we obtain an analytical expression for the ground state energy in the thermodynamic limit.

DOI
10.1088/1751-8121/aea3ca

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Abstract We study the exact spectrum of the XX spin chain with constrained non-diagonal boundary fields, which can be analyzed by solving the associated Bethe Ansatz equations. In these equations, the number of Bethe roots has a fixed parity, and all Bethe roots are located at the zeros of a unary function. We investigate the possible positions of the Bethe roots. Based on numerical observations, we analyze the Bethe root configurations for the ground state and the first excited state. Our results show that elementary excitations are characterized by the cooperative change of a pair of Bethe roots. Furthermore, we obtain an analytical expression for the ground state energy in the thermodynamic limit.
Recently, the XXX spin chain with arbitrary boundary fields was successfully solved [1] via the off-diagonal Bethe ansatz method [2]. The correctness and the completeness of this solution were numerically verified by Nepomechie for one choice of the parameterizations [3]. In this paper, we discuss further the completeness of another parameterization of the Bethe ansatz equations and its reduction to the parallel boundary field case. The numerical results show that when the relative angle between the boundary fields turns to zero, both the T-Q relations and the Bethe ansatz equations are reduced to the ones obtained by the conventional Bethe ansatz methods. This allows us to establish a one-to-one correspondence between the Bethe roots of the unparallel boundary field case and those of the parallel boundary field case. In the thermodynamic limit, those two sets of Bethe roots tend to the same and the contribution of the relative angle to the energy is in the order of 1/N.
We study the spectrum of the integrable open XXX Heisenberg spin chain subject to non-diagonal boundary magnetic fields. The spectral problem for this model can be formulated in terms of functional equations obtained by separation of variables or, equivalently, from the fusion of transfer matrices. For generic boundary conditions the eigenvalues cannot be obtained from the solution of finitely many algebraic Bethe equations. Based on careful finite size studies of the analytic properties of the underlying hierarchy of transfer matrices we devise two approaches to analyze the functional equations. First we introduce a truncation method leading to Bethe-type equations determining the energy spectrum of the spin chain. In a second approach, the hierarchy of functional equations is mapped to an infinite system of nonlinear integral equations of TBA type. The two schemes have complementary ranges of applicability and facilitate an efficient numerical analysis for a wide range of boundary parameters. Some data are presented on the finite-size corrections to the energy of the state which evolves into the antiferromagnetic ground state in the limit of parallel boundary fields.
We analyze the ground state of the open spin-1/2 isotropic quantum spin chain with a non-diagonal boundary term using a recently proposed Bethe ansatz solution. As the coefficient of the non-diagonal boundary term tends to zero, the Bethe roots split evenly into two sets: those that remain finite, and those that become infinite. We argue that the former satisfy conventional Bethe equations, while the latter satisfy a generalization of the Richardson–Gaudin equations. We derive an expression for the leading correction to the boundary energy in terms of the boundary parameters.
A bstract The thermodynamic limits of the XYZ spin chain with periodic or twisted boundary conditions are studied. By using the technique of characterizing the eigenvalue of the transfer matrix by the T − Q relation and by the zeros of the associated polynomial, we obtain the constraints of the Bethe roots and the zeros for the eigenvalues. With the help of structure of Bethe roots, we obtain the distribution patterns of zeros. Based on them, the physical quantities such as the surface energy and excitation energy are calculated. We find that both of them depend on the parity of sites number due to the topological long-range Neel order on the Mobius manifold in the spin space. We also check our results with those obtaining by the density matrix renormalization group. The method provided in this paper can be applied to study the thermodynamic properties at the thermal equilibrium state with finite temperature.
We investigate the structure of Bethe-root configurations in the spin-1/2 XXZ chain with degenerate open boundaries. The physical solutions of the Bethe Ansatz equations are classified into three types in the fully degenerate case and two types in the partially degenerate case, for which we propose counting formulas for the number of physical solution sets of each type. Furthermore, we observe spectral degeneracies in the fully degenerate case and show that they can be naturally explained by the presence of specific phantom strings in the Bethe roots. The classification and resulting spectral degeneracies in the diagonal limit are also discussed.