Duality relation for 32-vertex model on the triangular lattice

Kelland has solved a restricted ice-type model on the triangular lattice. Here it is shown that this is equivalent to a restricted six-vertex model on the Kagomé lattice, and to the g-state triangular (or hexagonal) Potts model at its transition temperature T c . This enables us to obtain the free energy, internal energy and latent heat of the Potts model at T c . The relation of this work to the operator method of Temperley and Lieb is explained, and this method is used to consider a generalized triangular Potts model which includes a three-site interaction on alternate triangles. It is shown that this model is self-dual. The results for the bond percolation problem on the triangular lattice give an excellent verification of series expansion predictions.

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Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings

The Electronic Journal of Combinatorics ◽

10.37236/8809 ◽

2020 ◽

Vol 27 (2) ◽

Author(s):

Philippe Di Francesco ◽

Emmanuel Guitter

Keyword(s):

Domain Wall ◽

Triangular Lattice ◽

Roots Of Unity ◽

Vertex Model ◽

Domino Tilings ◽

Enumeration Problems ◽

Shaped Hole ◽

Type Boundary ◽

Type 3 ◽

Ice Model

We consider the triangular lattice ice model (20-Vertex model) with four types of domain-wall type boundary conditions. In types 1 and 2, the configurations are shown to be equinumerous to the quarter-turn symmetric domino tilings of an Aztec-like holey square, with a central cross-shaped hole. The proof of this statement makes extensive use of integrability and of a connection to the 6-Vertex model. The type 3 configurations are conjectured to be in same number as domino tilings of a particular triangle. The four enumeration problems are reformulated in terms of four types of Alternating Phase Matrices with entries $0$ and sixth roots of unity, subject to suitable alternation conditions. Our result is a generalization of the ASM-DPP correspondence. Several refined versions of the above correspondences are also discussed.

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Twenty-vertex Model on a Triangular Lattice

Australian Journal of Physics ◽

10.1071/ph740813 ◽

1974 ◽

Vol 27 (6) ◽

pp. 813 ◽

Cited By ~ 21

Author(s):

SB Kelland

Keyword(s):

Phase Transition ◽

Order Phase Transition ◽

Triangular Lattice ◽

Infinite Order ◽

Vertex Model ◽

Ferroelectric State ◽

First Order ◽

First Order Phase Transition ◽

First Order Phase

An ice-type vertex model is formulated on a triangular lattice and solved exactly for restricted values of the vertex activities. It is found to undergo an infinite-order phase transition into an ordered antiferroelectric state and a first-order phase transition into a 'frozen' ferroelectric state.

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