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Physica A Statistical Mechanics and its Applications

The star-triangle transformation for self-dual spin models

H. Moraal

Published Sep 1, 1988• 3 citations
Publisher
DOI
10.1016/0378-4371(88)90067-2

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We study a duality analysis in conjunction with the star-triangle transformation for symmetric-group permutation models on the triangular and honeycomb lattices. The calculation is motivated by the permutation-model description of random tensor networks and by earlier duality analyses of replicated spin glasses. The essential point is that the finite-basis unit is not a bare bond but a star-triangle block. Our analysis estimates the critical bond dimension for the honeycomb lattice to be 2.634929344884, and the associated single-bond duality relation yields the triangular-lattice estimate 1.475661534848.

cond-mat.dis-nnCross-listedarXiv:2607.14917•Progress of Theoretical and Experimental Physics•Jul 17, 2026•0 citations
In this paper we present the solutions of the star-triangle equation for Boltzmann weights of the (Nα,Nβ) model. This model consists of two scalar Potts models (of Nα and Nβ states, respectively) coupled together by four-spin interaction terms and was introduced by Domany and Riedel to study the phase transitions in adsorbed thin films. If Nα,Nβ≠2, the star-triangle equations have nontrivial self-dual and non-self-dual solutions only for Nα=Nβ; whereas if Nα≠Nβ=2, we have other self-dual and non-self-dual solutions. After applying partial duality transformations several of our solutions can be identified with known solutions for Interaction-Round-a-Face (IRF) models. Our results also mean that our high-genus solutions found earlier are not the only solutions to the star-triangle equations in chiral Potts models with six or more states per site.
We study a duality analysis in conjunction with the star-triangle transformation for symmetric-group permutation models on the triangular and honeycomb lattices. The calculation is motivated by the permutation-model description of random tensor networks and by earlier duality analyses of replicated spin glasses. The essential point is that the finite-basis unit is not a bare bond but a star-triangle block. Our analysis estimates the critical bond dimension for the honeycomb lattice to be 2.634929344884, and the associated single-bond duality relation yields the triangular-lattice estimate 1.475661534848.