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Canadian Journal of Physics

Elementary Bethe Ansatz solution of two lattice spin models

Paul Bracken

Published Jan 1, 1997• 0 citations
Publisher

Abstract

A simple detailed derivation is given of the Bethe Ansatz solution and the raising operator condition on the amplitudes of the wave function that demands that this state be a highest weight state for the one-dimensional Heisenberg model. This is extended to the case of a single spin impurity on the lattice. The equivalent form of the Ansatz is derived, and several properties of the associated eigenvalue problem are developed. R´ esum´

DOI
10.1139/cjp-75-4-247

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A simple detailed derivation is given of the Bethe Ansatz solution and the raising operator condition on the amplitudes of the wave function that demands that this state be a highest weight state for the one-dimensional Heisenberg model. This is extended to the case of a single spin impurity on the lattice. The equivalent form of the Ansatz is derived, and several properties of the associated eigenvalue problem are developed. R´ esum´
We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω=±1 with Ω the eigenvalue of T. We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω, ϒ)=(±1, ∓1) where ϒ is the eigenvalue of V. Therefore, the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω, ϒ)=(±1, ∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.
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.
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