Generalized Triangular Ising Lattice

Mills, R. G. J.; Hurst, C. A.

doi:10.1063/1.1665002

Cited by 1 publication

(3 citation statements)

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“…Beyond Ising models, dimers and free-fermion methods can also be used to solve vertex models. Hurst first remarked on this point from the perspective of his free-fermionic method, noting the solution of what he called the generalized square lattice [91,92], but without consideration of its physical interpretation; in [100], the triangular lattice is similarly considered. For the square lattice, the condition on the even weights such that it is solvable via this method is…”

Section: Free-fermion and Dimer-solvable Modelsmentioning

confidence: 99%

“…Any general 16-vertex model that has these same relations among its invariants will be fully equivalent to the odd 8-vertex model. We therefore define the class of odd 8-vertex models, invariant under the SL(2) × SL(2) symmetry, in terms of equations (99)(100)(101)(102) or (103)(104)(105)(106).…”

Section: Algebraic Invariants Of the 16-vertex Modelmentioning

confidence: 99%

“…This factorization property means that the free-fermion model can be expressed in terms of only four effective parameters. It would be interesting to see whether this factorization property, as well as a mapping to an Ising model on a different lattice, continue to hold for free-fermion vertex models defined on other lattices, for example the 32-vertex model on the triangular lattice [111,[115][116][117] or the 128-vertex model on the union-jack lattice [118].…”

Section: Free-fermion and Dimer-solvable Modelsmentioning

confidence: 99%

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The 16-vertex model and its even and odd 8-vertex subcases on the square lattice

Assis¹

2017

J. Phys. A: Math. Theor.

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We survey and enlarge the known mappings of the 16-vertex model, with emphasis on mappings between the even and odd 8-vertex subcases of the general model, also giving new mappings between these models, valid on finite toroidal lattices. In particular, we find new mappings between the models by using their algebraic invariants with respect to the SL(2) × SL(2) symmetry of the 16-vertex model; we also find a larger set of weak-graph transformations. We show many examples of models with negative weights which map to models with only positive weights. Using the algebraic invariant relations of the even and odd 8-vertex models, we find the complete set of points in the complex field plane of the square lattice Ising model in a field which map to the even or odd 8-vertex models; these points also correspond to the set of free-fermionic points of the model. We do not find any new integrable points, but we find a new mapping between the odd 8-vertex model and the square lattice Ising model at magnetic field H = iπ/(2β), valid on finite toroidal lattices. We also show directly through various examples that mappings via algebraic invariants do not fully exhaust the possible mappings a model may have with another model. We construct a new solution to the odd 8-vertex free-fermion model which is valid on the finite lattice, since the previous known solution resulted from a mapping valid only in the thermodynamic limit. Finally, we detail for the first time the phase transitions of the column staggered free-fermion 8-vertex model, and show that it can be mapped to the bi-partite staggered free-fermion model.

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Section: Free-fermion and Dimer-solvable Modelsmentioning

confidence: 99%