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Physical review. B, Solid state

Determination of an Operator Algebra for the Two-Dimensional Ising Model

Leo P. Kadanoff, Horacio Ceva

Published Jun 1, 1971• 555 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.3.3918

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Journal of the Physical Society of Japan•May 15, 1995•1 citation
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" \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ such that the average is nonzero only for \ensuremathΓ=0\ensuremath{\Gamma}=0\ensuremathΓ=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.
We study products of spin and disorder variables placed along a diagonal of the Ising model, using the path formulation of correlation functions. A number $\ensuremath{\Gamma}$ used in a previous publication, associated with their ordering on the line, characterizes the correlation, in the sense that only those arrangements having $\ensuremath{\Gamma}=0$, \ifmmode\pm\else\textpm\fi{} 1/2 can have nonvanishing average for all temperatures $T\ensuremath{\ne}{T}_{c}$; at Tc{T}_{c}Tc​, only those with $\ensuremath{\Gamma}=0$ are different from zero. A formal, exact expression for the 2n2n2n-spins correlation function is presented.
The partition function ZZZ of a two-dimensional Ising lattice is evaluated by expressing it in the form $Z=\ensuremath{\Sigma}{i}^{}\ensuremath{\lambda}_{i}^{}{}_{}{}^{\mathrm{nm}}$ where ${\ensuremath{\lambda}}_{i}$ are the eigenvalues of a 2n{2}^{n}2n-dimensional matrix M, nnn is the number of rows and mmm the number of columns of the lattice. M is a generalization of the V-shaped matrix studied by Kramers and Wannier in connection with the same problem. It is shown that M can be expressed in terms of 2n{2}^{n}2n-dimensional representations of 2n2n2n-dimensional orthogonal matrices. The eigenvalues of M are determined for nnn large by making use of the known relations between the eigenvalues of a 2n2n2n-dimensional orthogonal matrix and the eigenvalues of its 2n{2}^{n}2n-dimensional representative.The lattice considered differs only slightly from that treated by Onsager and Kaufman. In the lattice considered here, the boundary effects are eliminated by introducing an interaction between the last lattice point of each row and the first point of the next row. The matrix V considered by Onsager and Kaufman is, except for the boundary effects, equivalent to the matrix Mn{\mathbf{M}}^{n}Mn. ZZZ is shown to differ from the partition function calculated by Onsager and Kaufman only in order ${n}^{\ensuremath{-}2}$.
The 2n2n2n-spin correlation function for the two-dimensional Ising model at T=TcT={T}_{c}T=Tc​ is evaluated for the special case in which all the spins lie along a straight line, separated by many lattice constants. The resulting 2n2n2n-spin function is simply a quotient of products of two-spin correlations. A hypothesis of reducibility of fluctuations in the critical state is introduced. This hypothesis asserts that the product of any two local fluctuating quantities in the same neighborhood of space may be effectively replaced by a finite sum of local fluctuating quantities in this neighborhood. As a result, the previously found form for the 2n2n2n-spin function may be used to evaluate the correlation function of nnn energy densities when all nnn points lie on a line. The nnn-energy correlation function is simply a sum of products of two-energy correlations. The quotient form for the spin correlation plus scaling is shown to immediately imply the logarithmic specific heat.