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On the algebraic approach to cubic lattice Potts models

Srinandan Dasmahapatra, Paul Martin

Published Dec 13, 1995• 6 citations
OpenAlex

Abstract

We consider Diagram algebras, DG(Q) (generalized Temperley-Lieb algebras) defined for a large class of graphs G, including those of relevance for cubic lattice Potts models, and study their structure for generic Q. We find that these algebras are too large to play the precisely analogous role in three dimensions to that played by the Temperley-Lieb algebras for generic Q in the planar case. We outline measures to extract the quotient algebra that would illuminate the physics of three dimensional Potts models. PACS: 75.10.H 1

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We consider Diagram algebras, DG(Q) (generalized Temperley-Lieb algebras) defined for a large class of graphs G, including those of relevance for cubic lattice Potts models, and study their structure for generic Q. We find that these algebras are too large to play the precisely analogous role in three dimensions to that played by the Temperley-Lieb algebras for generic Q in the planar case. We outline measures to extract the quotient algebra that would illuminate the physics of three dimensional Potts models. PACS: 75.10.H 1
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.
Abstract The Hamiltonian of the N-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by N − 1 types of Temperley–Lieb generators. This generalises a previous result for N = 3 obtained by Fjelstad and Månsson (2012 J. Phys. A: Math. Theor. 45 155208). A pictorial representation of a related coupled algebra is given for the N = 3 case which involves a generalisation of the pictorial presentation of the Temperley–Lieb algebra to include a pole around which loops can become entangled. For the two known representations of this algebra, the N = 3 SICP chain and the staggered spin-1/2 XX chain, closed (contractible) loops have weight 3 and weight 2, respectively. For both representations closed (non-contractible) loops around the pole have weight zero. The pictorial representation provides a graphical interpretation of the algebraic relations. A key ingredient in the resolution of diagrams is a crossing relation for loops encircling a pole which involves the parameter ρ = e 2πi/3 for the SICP chain and ρ = 1 for the staggered XX chain. These ρ values are derived assuming the Kauffman bracket skein relation.
The Onsager algebra is one of the cornerstones of exactly solvable models in statistical mechanics. Starting from the generalised Clifford algebra, we demonstrate its relations to the graph Temperley--Lieb algebra, and a generalisation of the Onsager algebra. We present a series of quantum lattice models as representations of the generalised Clifford algebra, possessing the structure of a special type of the generalised Onsager algebra [1]. The integrability of those models is presented, analogous to the free fermionic eight-vertex model. We also mention further extensions of the models and physical properties related to the generalised Onsager algebras, hinting at a general framework that includes families of quantum lattice models possessing the structure of the generalised Onsager algebras.