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Journal of Physics A Mathematical and General

On representations of an operator algebra and the transfer matrix spectrum of the q-state Potts model

Paul Martin

Published Dec 21, 1986• 6 citations
Publisher

Abstract

The author shows that the Potts and Temperley-Lieb representations of the projection generators of a von Neumann algebra are, in general, reducible. The author writes down a new representation with non-vanishing product of odd generators R and gives evidence that it is a common element in these reductions. The operators in this representation may be interpreted as giving the bond transfer matrices of the square lattice Whitney polynomial. The author shows that the reduction of the Temperley-Lieb representation also contains irreducible representations with R=0 which are responsible for eigenvalues in the ice-model spectrum independent of those of the Potts model.

DOI
10.1088/0305-4470/19/18/001

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The author shows that the Potts and Temperley-Lieb representations of the projection generators of a von Neumann algebra are, in general, reducible. The author writes down a new representation with non-vanishing product of odd generators R and gives evidence that it is a common element in these reductions. The operators in this representation may be interpreted as giving the bond transfer matrices of the square lattice Whitney polynomial. The author shows that the reduction of the Temperley-Lieb representation also contains irreducible representations with R=0 which are responsible for eigenvalues in the ice-model spectrum independent of those of the Potts model.
We discuss generalizations of the Temperley–Lieb algebra in the Potts and XXZ models. These can be used to describe the addition of integrable boundary terms of different types. We use the Temperley–Lieb algebra and its one-boundary, two-boundary, and periodic extensions to classify different integrable boundary terms in the two-, three-, and four-state Potts models. The representations always lie at critical points where the algebras becomes non-semisimple and possess indecomposable representations. In the one-boundary case we show how to use representation theory to extract the Potts spectrum from an XXZ model with particular boundary terms and hence obtain the finite size scaling of the Potts models. In the two-boundary case we find that the Potts spectrum can be obtained by combining several XXZ models with different boundary terms. As in the Temperley–Lieb case, there is a direct correspondence between representations of the lattice algebra and those in the continuum conformal field theory.
The author writes down the regular representation of the Temperley-Lieb algebra T k (q) in the basis of reduced words on the k Temperley-Lieb generators. When k=2n-1, representations of these generators may be used to construct the transfer matrices for statistical mechanical models on an n-site wide lattice. He shows that the generically irreducible representation of T 2n-1 (q) responsible for the unique free energy in such models may be restricted to the regular representation of T n-1 (q). He gives equivalent forms for the regular representation, derived from lattice models, which manifest its indecomposable structure when k and q are such that T k (q) is semisimple. He hence generalises to obtain the structure of T k (q) when not semisimple. A transfer matrix built with generators in the regular representation gives the long distance properties of all possible such models. He shows how to find the semisimple quotient algebra which gives these long distance properties when T k (q) itself is not semisimple. He hence classifies the long distance properties of these statistical mechanical models.
We deal with quantum spin chains whose Hamiltonian arises from a representation of the Temperley-Lieb algebra, and we consider the mean values of those local operators which are generated by the Temperley-Lieb algebra. We present two key conjectures which relate these mean values to existing literature about factorized correlation functions in the XXZ spin chain. The first conjecture states that the finite volume mean values of the current and generalized current operators are given by the same simple formulas as in the case of the XXZ chain. The second conjecture states that the mean values of products of Temperley-Lieb generators can be factorized: they can expressed as sums of products of current mean values, such that the coefficients in the factorization depend neither on the eigenstate in question, nor on the selected representation of the algebra. The coefficients can be extracted from existing work on factorized correlation functions in the XXZ model. The conjectures should hold for all eigenstates that are non-degenerate with respect to the local charges of the models. We consider concrete representations, where we check the conjectures: the so-called golden chain, the Q Q -state Potts model, and the trace representation. We also explain how to derive the generalized current operators from concrete expressions for the local charges.