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Journal of Physics A Mathematical Nuclear and General

XY model and algebraic methods

D K Jha, Jean G. Valatin

Published Nov 1, 1973• 14 citations
Publisher

Abstract

A systematic study of properties of a set of operators leads to a new approach to solve the XY model. The algebraic solution refers only to a subalgebra of spin operators. Through the connection with the algebra of related fermion operators is apparent, these need not be referred to. As a by-product, Onsager's method for the Ising model results in a very simple form, and its relationship to the approach of Schultz, Mattis and Lieb (1961) becomes apparent.

DOI
10.1088/0305-4470/6/11/007

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A systematic study of properties of a set of operators leads to a new approach to solve the XY model. The algebraic solution refers only to a subalgebra of spin operators. Through the connection with the algebra of related fermion operators is apparent, these need not be referred to. As a by-product, Onsager's method for the Ising model results in a very simple form, and its relationship to the approach of Schultz, Mattis and Lieb (1961) becomes apparent.
An algebraic method of spin operators was developed by Jha and Valatin to solve (a) the Hamiltonian of the isotropic and anisotropic xy model in a one-dimensional lattice of N spin 1/2 particles and (b) the partition function of the Ising model in the absence of magnetic field in two dimensions. The pair of fermion operators used to explain BCS theory in superconductivity were shown to be related to a set of spin operators of Jha and Valatin in a very simple way. Onsager’s Lie algebra for diagonalizing the partition function of the Ising model was found to be included within the said commutator algebra of the spin operators. The structure constants of the algebra are so simple as to allow the entire algebra to be casted in one general commutator equation. In the present paper, the author presents the proof of a general equation from which all sets of commutator relationships existing among the elements of the algebra directly follow.
A previous publication showed how the critical indices for the two-dimensional Ising model could be derived from an assumed form of an operator algebra which describes how the product of two fluctuating variables may be reduced to a linear combination of the basic fluctuating variables. In this paper, the previously used algebra is derived from the Onsager solution of the two-dimensional Ising model. The calculation makes use of a "disorder" variable which is mathematically the result of applying the Kramers-Wannier transformation to the Ising-model spin variable. The average of products of spin and disorder variables are evaluated at the critical point for the special case in which all the variables lie on a single straight line. The ordering of these variables on the line determines a "quantum number" $\ensuremath{\Gamma}$ such that the average is nonzero only for $\ensuremath{\Gamma}=0$. Composition rules for this quantum number are derived and used to develop an algebra for the multiplication of complex variables at the critical point. Arguments are given to suggest the identifications of elements of the algebra as the spin, the energy density, the Kaufman spinors, and a stress density. The result of this calculation is the operator algebra which formed the starting point of the previous paper.
So far, we have primarily focused on the spin or angular momentum of an individual quantum particle. In this chapter, we examine the simplest models of spins interacting with each other. The Ising model was created by Wilhelm Lenz—he gave the problem to his Ph. D. student Ernst Ising, who solved the one-dimensional model. Hence, a statistical-mechanical model for a simplified model of quantum spins was solved even before the quantum theory for spin was fully developed! In this chapter, we investigate a number of different spin models. They are all exactly solvable, but some are more complicated than others. One of the key techniques we use is fermionization, which relates the spins to fermions. This allows us to introduce the ideas of second quantization and of fermions. Specifically, we look at the Ising model, the XY model, the transverse-field Ising model, the Lipkin-Meshkov-Glick model, and the Dicke model.
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.