arXiv:1006.0651 · Advances in Theoretical and Mathematical Physics

A master solution of the quantum Yang–Baxter equation and classical discrete integrable equations

Vladimir V. Bazhanov, С. М. Сергеев

Published Jan 1, 2012• 77 citations

Abstract

We obtain a new solution of the star-triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known.This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters.If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on

DOI: 10.4310/atmp.2012.v16.n1.a3

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arXiv:1006.0651 · Advances in Theoretical and Mathematical Physics

A master solution of the quantum Yang–Baxter equation and classical discrete integrable equations

Vladimir V. Bazhanov, С. М. Сергеев

Published Jan 1, 2012• 77 citations

arXiv HTML PDF

Abstract

DOI: 10.4310/atmp.2012.v16.n1.a3

Related papers

A distant descendant of the six-vertex model

Vladimir V. Bazhanov, С. М. Сергеев

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.

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A new type of Yang-Baxter equation (YBE) for $R$ -operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the $Z_N$ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for $N>2$ , except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated $R$ -operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter $R$ -matrix---realized as an intertwiner of cyclic representations of $U_q(\mathfrak{sl}_2)$ at a root of unity---and Shastry's two-parameter $R$ -operator obtained from the decorated YBE.A new type of Yang-Baxter equation (YBE) for $R$-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the $Z_N$ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for $N>2$, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated $R$-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter $R$-matrix---realized as an intertwiner of cyclic representations of $U_q(\mathfrak{sl}_2)$ at a root of unity---and Shastry's two-parameter $R$-operator obtained from the decorated YBE.

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A new type of Yang-Baxter equation (YBE) for

R

-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the

Z_N

symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for

N>2

, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated

R

-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter

R

-matrix---realized as an intertwiner of cyclic representations of

U_q(\mathfrak{sl}_2)

at a root of unity---and Shastry's two-parameter

R

-operator obtained from the decorated YBE.

R

-matrix of the model (which satisfies a difference property Yang–Baxter equation) naturally splits into a product of a singular "classical" part and a finite dimensional quantum part. The classical part of the

R

-matrix itself satisfies the quantum Yang–Baxter equation, and therefore can be factored out producing, however, a certain "twist" of the quantum part. We show that the resulting equation exactly coincides with the star-triangle relation of the

N

-state chiral Potts model. The associated spin model on the whole lattice is, in fact, more general than the chiral Potts and reduces to the latter only for the simplest (constant) classical background. In a general case the model is inhomogeneous: its Boltzmann weights are determined by non-trivial background solutions of the equations of motion of the classical discrete sine-Gordon model.