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Journal of Physics A Mathematical Nuclear and General

Baxter's quantum number in the XY model

Robert B. Jones

Published Jan 21, 1974• 3 citations
Publisher

Abstract

In the particular case of the asymmetric XY model an explicit form for the operator corresponding to Baxter's quantum number n is obtained. It is shown that this operator is closely associated with a transformation of spin operators which converts the asymmetric XY hamiltonian into a symmetric XY hamiltonian with alternating nearest-neighbour interactions.

DOI
10.1088/0305-4470/7/2/013

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In the particular case of the asymmetric XY model an explicit form for the operator corresponding to Baxter's quantum number n is obtained. It is shown that this operator is closely associated with a transformation of spin operators which converts the asymmetric XY hamiltonian into a symmetric XY hamiltonian with alternating nearest-neighbour interactions.
The Baxter diagonalization of the XYZ hamiltonian is considered in the special case of the XY model. For this particular model the normalization properties of the basic Baxter states are obtained showing that the Baxter families form an overcomplete set of states. After choosing a simple subset of states which is complete, Bethe-type eigenstates of the XY hamiltonian are formed explicitly. By fixing a free parameter in Baxter's states an operator expression for these states is obtained. The XY hamiltonian is then diagonalized in terms of fermion operators closely related to the quasiparticle operators of the standard solution due to Lieb, Schultz, and Mattis. Finally relations are given expressing the Baxter states in terms of the standard quasiparticle states of this model.
The authors obtain the associated Hamiltonian for the symmetric eight-vertex model by taking the time-continuous limit in an equivalent Ashkin-Teller (1943) model. The result is a Heisenberg Hamiltonian with coefficients J x , J y and J z identical to those found by Sutherland (1970) for choices of the parameters a, b, c and d that bring the model close to the transition. The change in the operators is accomplished explicitly, the relation between the crossover operator for the Ashkin-Teller model and the energy operator for the eight-vertex model being obtained in a transparent form.
It is shown that Baxter's method when applied to the XXZ hamiltonian gives a set of eigenvectors and wavenumbers different from those of the familiar Bethe solution. In this limiting case of Baxter's results for the eight-vertex model the author finds that one of Baxter's parameters corresponds to the freedom of making arbitrary rotations about the z axis. It is suggested that for the general XXZ hamiltonian there may be a generalization of the ordinary z component of spin which would be conserved in addition to Baxter's quantum number.
It is shown that a solvable Hamiltonian can be obtained from a series of operators satisfying specific commutation relations. A transformation that diagonalize the Hamiltonian is obtained simultaneously. The two-dimensional Ising model with periodic interactions, the one-dimensional XY model with period 2, the transverse Ising chain, the one-dimensional Kitaev model and the cluster model, and other composite quantum spin chains are diagonalized following this procedure. The Jordan–Wigner transformation, the transformation from the Pauli spin operators to the Majorana fermion used by Shankar and Murthy, and the transformation introduced by Nambu, are special cases of this treatment.