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Physical Review

On the Bethe Hypothesis for the Anisotropic Heisenberg Chain

Michael Flicker, Harvey S. Leff

Published Apr 10, 1968• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.168.578

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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.
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.
We examine the question of whether Bethe's ansatz reproduces all states in the periodic Heisenberg XXZ and XXX spin chains. As was known to Bethe himself, there are states for which the Bethe momenta knk_nkn​ diverge: these are in fact the simplest examples of ``string'' solutions. The coefficients of the Bethe wavefunction, too, diverge. When there are only two down spins in the system (the case considered by Bethe), we can renormalize these coefficients to get a sensible (and correct) wavefunction. We show that this is not always possible when there are more than two down spins. The Bethe equations have several such divergent solutions, and some of these correspond to genuine eigenfunctions of the Hamiltonian, but several do not. Nor do they reproduce the correct energy eigenvalues. Moreover, we point out that the algebraic Bethe ansatz, an alternative way to construct the wavefunctions proposed by Faddeev, Takhtajan et al., leads to vanishing wavefunctions for all these solutions. Thus, the Bethe ansatz solution of the Heisenberg model must be regarded as either incomplete, or inaccurate.