Sh. Khachatryan

Characteristics of two-dimensional lattice models from a fermionic realization: Ising and X Y Z models

We develop a field theoretical approach to the classical two-dimensional (2D) models, particularly to 2D Ising model (2DIM) and

On the Solutions of the Yang-Baxter Equations with General Inhomogeneous Eight-Vertex R -Matrix: Relations with Zamolodchikov’s Tetrahedral Algebra

XYZ

Network models: Action formulation

model, which is simple to apply for calculation of various correlation functions. We calculate the partition function of 2DIM and

Solutions to the Yang–Baxter equations with ospq(1|2) symmetry: Lax operators

XY

New solutions to the slq(2)-invariant Yang–Baxter equations at roots of unity: Cyclic representations

model within the developed framework. Determinant representation of spin-spin correlation functions is derived using fermionic realization for the Boltzmann weights. The approach also allows formulation of the partition function of 2DIM in the presence of an external magnetic field.

Fusion rules of the lowest weight representations of {\rm {osp}}_q(1|2) at roots of unity: polynomial realization

We present most general one-parametric solutions of the Yang-Baxter equations (YBE) for one spectral parameter dependent Rij(u)-matrices of the six- and eight-vertex models, where the only constraint is the particle number conservation by mod(2). A complete classification of the solutions is performed. We have obtained also two spectral parameter dependent particular solutions Rij(u,v) of YBE. The application of the non-homogeneous solutions to construction of Zamolodchikov’s tetrahedral algebra is discussed.

Simplified tetrahedron equations: Fermionic realization

Abstract We develop a technique to formulate quantum field theory on an arbitrary network, based on different randomly disposed sets of scattering points. We define the R -matrix of the whole network as a product of R -matrices attached to each scattering node. Then an action is formulated for a network in terms of fermionic fields, which allows to calculate the transition amplitudes as Green functions. On so-called bubble and triangle diagrams it is shown that the method produces the same results as the one which uses the generalized star product. The approach allows to extend network models by including multiparticle interactions at the scattering nodes.

New solutions to the slq(2)-invariant Yang–Baxter equations at roots of unity

We find a new 4×4 solution to the ospq(1|2)-invariant Yang–Baxter equation with simple dependence on the spectral parameter and propose 2×2 matrix expressions for the corresponding Lax operator. The general inhomogeneous universal spectral-parameter dependent R-matrix is derived. It is proven, that there are two independent solutions to the homogeneous ospq(1|2)-invariant YBE, defined on the fundamental three-dimensional representations. One of them is the particular case of the universal matrix, while the second one does not admit generalization to the higher-dimensional cases. Also the 3×3 matrix expression of the Lax operator is found, which have a well-defined limit at q→1.

Polynomial Realization of sℓq(2) and Fusion Rules at Exceptional Values of q

Abstract We find all the solutions to the sl q ( 2 ) -invariant multi-parametric Yang–Baxter equations (YBE) at q = i defined on the cyclic (semi-cyclic, nilpotent) representations of the algebra. We derive the solutions in form of the linear combinations over the sl q ( 2 ) -invariant objects — projectors. The direct construction of the projector operators at roots of unity gives us an opportunity to consider all the possible cases, including also degenerated one, when the number of the projectors becomes larger, and various types of solutions are arising, and as well as the inhomogeneous case. We give full classification of the YBE solutions for the considered representations. A specific character of the solutions is the existence of the arbitrary functions.

An integrable model with a non-reducible three particle R-matrix

The degeneracy of the lowest weight representations of the quantum superalgebra and their tensor products at exceptional values of q is studied. The main features of the structures of the finite-dimensional lowest weight representations and their fusion rules are illustrated using realization of group generators as finite-difference operators acting in the space of the polynomials. The complete fusion rules for the decompositions of the tensor products at roots of unity are presented. The appearance of indecomposable representations in the fusions is described using Clebsh?Gordan coefficients derived for general values of q and at roots of unity.