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Journal of Physics Condensed Matter

The two-element spin chain related to the Heisenberg spin model

Lai Yun-zhong, Zhan-Ning Hu, J.-Q. Liang, Fu‐Cho Pu

Published Jan 1, 1999• 0 citations
Publisher

Abstract

We construct a two-element spin-chain model based on the periodic Heisenberg spin model. This model describes a system containing two kinds of particle. The Bethe ansatz equations are obtained by using the quantum inverse scattering method. The high- and low-temperature limits are discussed, and we find that the mean energy per site increases with increase of the next-nearest-neighbour coupling in the high-temperature limit.

DOI
10.1088/0953-8984/11/12/002

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We construct a two-element spin-chain model based on the periodic Heisenberg spin model. This model describes a system containing two kinds of particle. The Bethe ansatz equations are obtained by using the quantum inverse scattering method. The high- and low-temperature limits are discussed, and we find that the mean energy per site increases with increase of the next-nearest-neighbour coupling in the high-temperature limit.
Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.
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Two physical models are discussed. We start out with the Heisenberg spin chain, which consists of a number of atoms in a circular con?guration, each with spin up or down. We set up a Hamiltonian of this system considering only nearest-neighbour interactions, and determine the energy of the spin chain. The mathematical description of the state in which these atoms will be, i.e. the wave functions of these atoms, satis?es a certain template, which is called the Bethe Ansatz. Using this Ansatz, we can solve the eigenvalue equation of the Hamiltonian and thus calculate the energy of the system. What follows is a discussion on a model for water ice, of which we want to determine the entropy at absolute zero. This leads to a set of equations that is very similar to that encountered in the Heisenberg model, and it turns out that the Bethe Ansatz is useful in this model as well: it enables us to explicitly calculate the entropy of an infinitely big system.
The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The Heisenberg XXZ model is a chain model with nearest-neighbor interactions. Its thermodynamics is exactly obtained via Bethe ansatz. Recently, we developed a method to derive the high-temperature expansion of the grand potential per site of translationally invariant chain models, with periodic boundary conditions. Here we apply this approach to the XXZ model with periodic boundary conditions for the Ising limit case (t = 0) and the free fermion case (delta = 0 and h = 0), obtaining results in agreement with the literature. In this new way of obtaining the coefficients of the high-temperature expansion of the grand potential, the coefficients are derived from an auxiliary function written only in terms of open connected sub-chains.