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Physical Review

One-Dimensional Chain of Anisotropic Spin-Spin Interactions. I. Proof of Bethe's Hypothesis for Ground State in a Finite System

Chongming Yang, Chuan‐Zhang Yang

Published Oct 7, 1966• 1,096 citations
Publisher

Abstract

Bethe's hypothesis is proved for the ground state of a one-dimensional cyclic chain of anisotropic nearest-neighbor spin-spin interactions. The proof holds for any fixed number of down spins.Received 8 April 1966DOI:https://doi.org/10.1103/PhysRev.150.321©1966 American Physical Society

DOI
10.1103/physrev.150.321

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Bethe's hypothesis is proved for the ground state of a one-dimensional cyclic chain of anisotropic nearest-neighbor spin-spin interactions. The proof holds for any fixed number of down spins.Received 8 April 1966DOI:https://doi.org/10.1103/PhysRev.150.321©1966 American Physical Society
The two-reversed-spin problem for the anisotropic Heisenberg linear chain is solved in a straightforward and rigorous way. The method involves the solution of a second-order partial difference equation and does not necessitate the use of the Bethe hypothesis. It is shown that, although most of the eigenfunctions are of the Bethe form, there exist special states which, a priori, cannot be so written. In addition, a simple method is presented for enumerating the bound states. Detailed connections between the present solution and that given by Bethe, and later by Orbach, are discussed.
The effect of anisotropy in the nearest-neighbor spin interactions that couple N> 2N>~2N> 2 consecutive spin-12\frac{1}{2}21​ antiferromagnetic chains is studied theoretically by considering the limit where the coupling is purely of the Ising type. An analysis based on the equivalent Luttinger model reveals that the ground state is an Ising antiferromagnet in general.