Baxter-Bazhanov Model Research Articles

Theta-function parametrization and fusion for 3D integrable Boltzmann weights

We report progress in constructing Boltzmann weights for integrable threedimensional lattice spin models. We show that a large class of vertex solutions to the modified tetrahedron equation (MTE) can be conveniently parametrized in terms of Nth roots of theta functions on the Jacobian of a compact algebraic curve. Fay’s identity guarantees the Fermat relations and the classical equations of motion for the parameters determining the Boltzmann weights. Our parametrization allows us to write a simple formula for fused Boltzmann weights R which describe the partition function of an arbitrary open box and which also obey the modified tetrahedron equation. Imposing periodic boundary conditions we observe that the R satisfy the normal tetrahedron equation. The scheme described contains the Zamolodchikov– Baxter–Bazhanov model and the chessboard model as special cases.

Three-Dimensional Vertex Model Related BCC Model in Statistical Mechanics

In this paper, a three-dimensional vertex model is obtained. It is a duality of the three-dimensional integrable lattice model with N states proposed by Boos, Mangazeev, Sergeev and Stroganov. The Boltzmann weight of the model is dependent on four spin variables, which are the linear combinations of the spins on the corner sites of the cube, and obeys the modified vertex-type tetrahedron equation. This vertex model can be regarded as a deformation of the one related to the three-dimensional Baxter–Bazhanov model. The constrained conditions of the spectrums are discussed in detail and the symmetry properties of weight functions of the vertex model are presented.

Baxter-Bazhanov Model, Frenkel-Moore Equation and the Braid Group11The project partially supported by the China Center of Advanced Science and Technology

In this paper the three-dimensional vertex model is given, which is the duality of the three-dimensional Baxter–Bazhanov (BB) model. The braid group corresponding to Frenkel–Moore equation is constructed and the transformations are found. These maps act on the group and denote the rotations of the braids through the angles about some special axes. The weight function of another three-dimensional vertex model related the lattice integrable model proposed by Boos, Mangazeev, Sergeev and Stroganov is presented also, which can be interpreted as the deformation of the vertex model corresponding to the BB model.

Three-dimensional vertex model in statistical mechanics from Baxter-Bazhanov model

We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube, and the Wu-Kadanoff-Wegner duality between the cube- and vertex-type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which corresponds to the three-dimensional Baxter-Bazhanov model. The vertex-type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. We write down the symmetry relations of the weight functions under the actions of the symmetry groupG of the cube. The six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected, with the six spaces in which the vertextype tetrahedron equation is defined.

Knot invariants associated with a particularN→∞ continuous limit of the Baxter-Bazhanov model

First we briefly recall the definition of the three-dimensional Baxter-Bazhanov lattice model. The spins of this model are elements ofZN and theR-matrix is associated to the algebraUqsl(n) ifq is a primitiveNth root of unity. Then we construct a particularN→∞ limit of the model, in which it is meaningful to interpret the spins as elements ofR and which gives the free Gaussian boson model. Finally, we study special limits of the rapidity variables in which we obtain braid group representations and we show that forn odd the associated knot invariants are given by the inverse of products of Alexander polynomials, evaluated at certain roots of unity.

On the Star-Triangle Relation and the Star-Square Relation in the Baxter-Bazhanov Model

The connection is found between the star-triangle relation and the star-square relation in the Baxter-Bazhanov model which is an Interaction-Round-a-Cube model. This gives a proof of the guess suggested by Bazhanov and Baxter.

Remarks on the star-triangle relation in the Baxter-Bazhanov model

We show that the restricted star-triangle relation introduced by Bazhanov and Baxter can be obtained either from the star-triangle relation of the chiral Potts model or from the star-square relation proposed by Kashaev, Mangazeev, and Stroganov, and give a response to a guess of Bazhanov and Baxter.

Cyclic quantum dilogarithm, shift operator and star-square relation of the BB model

From the cyclic quantum dilogarithm the shift operator is constructed with a root of unity, q, and the representation for the current algebra introduced by Faddeev et al. is given. It is shown that the theta function is factorizable in this case by using the star-square equation of the Baxter-Bazhanov (BB) model.

THE STAR SQUARE IN THE BAXTER–BAZHANOV MODEL AND THE STAR–TRIANGLE RELATION IN THE CHIRAL POTTS MODEL

In this letter, the connection is found between the "star–square" relation in the Baxter–Bazhanov model and the "star–triangle" relation in the chiral Potts model, which means that the tetrahedron equation of the Baxter–Bazhanov model is a consequence of the latter. The four additional constraints in the tetrahedron equation given by Kashaev et al. hold naturally in respect to the spherical trigonometry parametrizations.

Three-dimensional integrable models and associated tangle invariants

In this paper we show that the Boltzmann weights of the three-dimensional Baxter-Bazhanov model give representations of the braid group, if some suitable spectral limits are taken. In the trigonometric case we classify all possible spectral limits which produce braid group representations. Furthermore we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones.

STAR-SQUARE AND TETRAHEDRON EQUATIONS IN THE BAXTER-BAZHANOV MODEL

Spatially symmetrical Baxter-Bazhanov model’s Boltzmann weights are proved to satisfy the tetrahedron equation. The latter is a consequence of the “inversion” and “star-square” relations for the Boltzmann weight elementary building blocks.

SPATIAL SYMMETRY, LOCAL INTEGRABILITY AND TETRAHEDRON EQUATIONS IN THE BAXTER-BAZHANOV MODEL

We show that the Baxter-Bazhanov model is invariant under the action of the cube’s symmetry group. The three-dimensional star-star relations, proposed by Baxter and Bazhanov as local integrability conditions, correspond to a particular transformation from this group. Invariant Boltzmann weights, parametrized in terms of the Zamolodchikov angle variables, apparently satisfy the tetrahedron equations.