The vertex formulation of the Bazhanov-Baxter model

2006

J. Phys. A: Math. Gen.

109

348

The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finitedimensional highest weight representations for all quantum affine algebra U q ( sl(n)), where the rank n coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the sl(n)-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2 + 1 dimensions.

Section: Introductionmentioning

confidence: 63%

Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

Bazhanov

2006

J. Phys. A: Math. Gen.

109

348

“…The form (33) which we shall use has first been given in [11] several years ago. Introduce a two component vector p = (x, y) which is restricted to the Fermat curve x N + y N = 1.…”

Section: Proposition 1 the Finite Dimensional Matrices R Satisfy The mentioning

confidence: 99%

“…Later, Bazhanov and Baxter [5] showed how to generalize the Zamolodchikov model to N > 2, using the interaction-round-a-cube language (IRC-ZBB-model). Then, Sergeev, Mangazeev and Stroganov found the vertex-formulation of the ZBB-model [11]. These models share the feature the ZBB-Boltzmann weights are not positive definite, but at least in the thermodynamic limit the partition function becomes real.…”

Section: Introductionmentioning

confidence: 99%

The Modified Tetrahedron Equation and Its Solutions

Gehlen

Pakuliak

2004

Int. J. Mod. Phys. A

A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator R 1,2,3 in the space of a triple Weyl algebra. R 1,2,3 is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for R 1,2,3 follows without further calculation. If the Weyl parameter is taken to be a root of unity, R 1,2,3 decomposes into a matrix conjugation operator R 1,2,3 and a c-number functional mapping R (f ) 1,2,3 . The operator R 1,2,3 satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equations. R 1,2,3 can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.

“…We may prove the first two formulas considering the poles and zeros of the left-and right-hand sides. Details may be found in the appendix of [25]. Let the normalization factor for R m,φ be ρ R , …”

Section: )mentioning

confidence: 99%

“…It means the consideration of the trivial classical dynamics. Algebraically, the conditions of the trivialization are the origin of, for example, Baxter's curve for the chiral Potts model [1,4], or of the spherical triangle parameterization for the Zamolodchikov-Bazhanov-Baxter model [25].…”

Section: Introductionmentioning

confidence: 99%

Quantum relativistic Toda chain at root ofunity: isospectrality, modified Q‐operator, and functional Bethe ansatz

Pakuliak

International Journal of Mathematics and Mathematical Sciences

2002

We investigate an N-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra with q being Nth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart correspond to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter's Q-operators. The classical counterpart of the modified Q-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modified Q-operators.