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Journal of Physics A Mathematical and General

Bethe ansatz for spin-1 chain with long-range interaction

A G Choudhury, Asesh Roy Chowdhury

Published Jul 21, 1995• 1 citation
Publisher

Abstract

A fusion procedure for the R-matrices with disorder parameter mu i and the power-series expansion technique of de Vega (1984) have been used to construct a spin-1 chain with a long-range interaction. The energy eigenvalue and the equation for the momenta for the state containing m-excitations are determined with the help of the algebraic Bethe ansatz.

DOI
10.1088/0305-4470/28/14/020

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A fusion procedure for the R-matrices with disorder parameter mu i and the power-series expansion technique of de Vega (1984) have been used to construct a spin-1 chain with a long-range interaction. The energy eigenvalue and the equation for the momenta for the state containing m-excitations are determined with the help of the algebraic Bethe ansatz.
Abstract This chapter verifies the conjecture for the wave function, the Bethe ansatz wave function, of the anisotropic Heisenberg quantum spin chain by examining first the cases for one, two, and three spin deviations. The equations determining the quasi- momenta are the Bethe ansatz equations, now obtained from the coordinate Bethe ansatz. The Bethe ansatz equations derive from the eigenvalue equation in combination with boundary conditions, here periodic boundary conditions. These quasi-momenta also determine the energy eigenvalue. However, solving the Bethe ansatz equations to obtain a particular state requires more considerations. New variables, called rapidities, are useful. The consideration of the thermodynamic limit then allows to extract information about the ground state and low-lying excitations of the anisotropic quantum spin chain from the Bethe ansatz equations. Furthermore, complex solutions of the Bethe ansatz equations, called strings, are considered.
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It is shown that the description of the states of infinite S = 1/2 interacting spin systems with the Hamiltonian H s J Z a2 sinh-Za(J / ) (~ 1) 2 j# t , j , l~Z 2 can be performed by studying the hyperbolic Calogero-Sutherland eigenvalue problem. The construction of multimagnon wave functions in each N-magnon sector is based on solutions of the set of linear algebraic equations which determine also the structure of zonal spherical functions on symmetric spaces X ~ = SL(N, H)/Sp(N) of negative curvature. The usual Bethe Ansatz for the XXX Heisenberg model corresponds to asymptotic forms of these wave functions at small values of a 1 or large distances between spins turned over the ferromagnetic ground state.