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Journal of Physics C Solid State Physics

Bethe ansatz for two-deviation states in quantum spin chains of arbitrary S with anisotropic Heisenberg exchange

R P Hodgson, John B. Parkinson

Published Dec 10, 1985• 30 citations
Publisher

Abstract

The Bethe ansatz is used to calculate the eigenstates of Heisenberg spin chains of arbitrary S with two deviations from the completely aligned state. The states are analysed into bound (class A/B) and 'free' (class C). States with a high probability of having two deviations on the same site are found to be class C. The authors also obtain asymptotic expressions for the magnetisation and the autocorrelation function in the spin-flop phase as the field approaches the spin-flop critical field.

DOI
10.1088/0022-3719/18/34/009

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The Bethe ansatz is used to calculate the eigenstates of Heisenberg spin chains of arbitrary S with two deviations from the completely aligned state. The states are analysed into bound (class A/B) and 'free' (class C). States with a high probability of having two deviations on the same site are found to be class C. The authors also obtain asymptotic expressions for the magnetisation and the autocorrelation function in the spin-flop phase as the field approaches the spin-flop critical field.
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.
States of the pure biquadratic quantum spin chain with up to four deviations are calculated exactly for arbitrary chain length. The four-deviation states have a Bethe ansatz form and consist of two interacting two-strings. They have the same form as two-deviation states of an integrable S=1/2 XXZ model. It is found that states with more than four deviations also map onto corresponding S=1/2 states. The ground states of the two systems are not the same for finite N, but appear to become the same in the limit N to infinity .
States of the pure biquadratic S=1 quantum spin chain with more than one deviation are considered. An expression for the wave function for an interacting pair of 2-strings has been calculated in the Bethe ansatz form. It is shown that there is a partial mapping of states of multiple 2-strings onto the states of the anisotropic S= 1 / 2 quantum spin chain (XXZ model) which is known to be integrable. The mapping does not include the ground state, however. Other types of state with multiple deviations are also discussed.
The integrable S=1 isotropic quantum spin chain of N atoms, introduced by Takhtajan (1982) and Bubujian (1983), is studied as a function of magnetisation and hence of magnetic field. The methods used are (i) direct numerical diagonalisation for N<or=12, (ii) numerical solution of the Bethe ansatz equations for N=6, 12, 16, 20, 40, (iii) analytic solution of the Bethe ansatz equations for N to infinity . The authors obtain results for the multiple string structure using the first two of these methods, showing the dominance of two-strings in the ground state and elementary excitations. Based on this they calculate the magnetisation as a function of field in the thermodynamic limit, over the whole spin-flop regime. They observe a tracking soft mode in the finite N results and also an energy gap which tends to zero as N to infinity . Their conclusion is that the system behaves much more like the integrable S= 1 / 2 Heisenberg model than the nonintegrable S=1 Heisenberg model.