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Journal of Physics A Mathematical and General

Temperley-Lieb algebra, group theory and the Potts model

Paul Martin

Published Feb 7, 1988• 15 citations
Publisher

Abstract

The author shows that a scheme based on the critical Andrews-Baxter-Forrester model may be used to construct all known representations of operators obeying the Temperley-Lieb relations. Using associated groups he shows, for a small number of operators, that this scheme gives the complete set of irreducible representations. He obtains the degeneracies of the transfer matrix spectrum for the q-state Potts model.

DOI
10.1088/0305-4470/21/3/014

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The author shows that a scheme based on the critical Andrews-Baxter-Forrester model may be used to construct all known representations of operators obeying the Temperley-Lieb relations. Using associated groups he shows, for a small number of operators, that this scheme gives the complete set of irreducible representations. He obtains the degeneracies of the transfer matrix spectrum for the q-state Potts model.
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.
It is shown that ( r -1)-state restricted eight-vertex SOS model (8VSOS) by Andrews, Baxter and Forrester is equivalent to q -state Potts model with q =4·cos 2 (π/ r ) ( r =4, 5, 6···) at criticality. The equivalence follows from the Temperley-Lieb algebra shared by the models which is nothing but C * -algebra A q , n defined by Jones in von Neumann algebra theory. This implies the correspondence of indices for subfactors of type II 1 factors and discrete series of Virasoro algebra which labels the universality classes of the 8VSOS models.