Abstract
We present an exact mapping between the staggered six-vertex model and an integrable model constructed from the twisted affine {D}_2^2 Lie algebra. Using the known relations between the staggered six-vertex model and the antiferromagnetic Potts model, this mapping allows us to study the latter model using tools from integrability. We show that there is a simple interpretation of one of the known K -matrices of the {D}_2^2 model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works ([12, 13]) are discussed.
Highlights
The staggered six-vertex model and the D22 model2.1 Background The two-dimensional Q-state Potts model is defined by the classical Hamiltonian
The two-dimensional Q-state Potts model is defined by the classical HamiltonianH = −K δσiσj, ij (2.1)where σi = 1, 2, . . . , Q and ij denotes the set of nearest neighbours on the square lattice
By considering the antiferromagnetic Potts model in its formulation as a staggered sixvertex model we have shown that it is equivalent to the integrable vertex model constructed from the twisted affine D22 Lie algebra
Summary
2.1 Background The two-dimensional Q-state Potts model is defined by the classical Hamiltonian. It is well known that the Potts model can be reformulated as a height, loop and vertex model [16] where the partition functions are identical to that of the original Potts model described in terms of spins, but with different observables. It is another well-known result that when the correspondence between the Potts and the vertex model is carried out at the so-called “ferromagnetic critical point”, the resulting vertex model is the celebrated “six-vertex model”. We will be interested in “free” boundary conditions in the Potts model which corresponds in (2.1) to imposing no additional constraint on the Potts spins at the boundary so that the sum runs over all nearest neighbours as usual but boundary spins have fewer nearest neighbours
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