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Sh. Khachatryan

Publications and source records attributed to Sh. Khachatryan.

16 recordsLinked to original sources
arXiv2026

Critical couplings of two dimensional Ising model on various lattices

We develop a unified fermionic-field formulation of the two-dimensional Ising model on several planar lattices using the Kac--Ward representation. Grassmann fields are associated with directed lattice links, while the turning of a fermionic trajectory at a lattice vertex is encoded by the corresponding Kac--Ward phase factor. Within this approach the partition function is expressed through the determinant of a finite-dimensional momentum-space matrix, whose zeros determine the excitation spectrum and the critical coupling. We apply the method to the regular square, honeycomb, triangular, kagom\'e, and dual kagom\'e (dice or $T_3$) lattices. In all cases the known exact critical couplings are reproduced. Particular attention is given to the anisotropic kagom\'e lattice, for which the fermionic determinant yields the complete critical surface and the low-energy spectral equation. We also construct the fermionic action for the dual kagom\'e lattice and derive its anisotropic critical condition. In the isotropic dice model the spectrum reduces at low energy and momentum to a relativistic massive form, with the mass vanishing at $\cosh(2J_c)=(1+\sqrt3)/2$. The results demonstrate that the same fermionic construction provides a compact description of criticality and low-energy excitations for Ising models on lattices with different local geometries and coordination numbers.

cond-mat.stat-mech↗
arXiv2021

Explicit R-matrices for inhomogeneous 3D chiral Potts models: Integrability and the action formulation for

We construct the exact spectral parameter dependent vertex R-matrix for the classical 3D $\mathcal{N}$-state chiral Potts models, convenient for considering the model in context of the Bethe ansatz. The R-matrix is defined on the $\mathcal{N}^4$ dimensional space $V_\mathcal{N}\otimes V_\mathcal{N}\otimes V_\mathcal{N}\otimes V_\mathcal{N}$, appropriate for consideration by means of the cube-equations defined in [14]. We present the 2D quantum spin Hamiltonians for general case and, at $\mathcal{N}=2$, a fermionic lattice action representation corresponding to 3D Ising's statistical model.

math-ph↗
arXiv2015

Integrability of three dimensional models: cubic equations

We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang-Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.

math-ph↗
arXiv2014

A matrix model for strings beyond the c=1 barrier: the spin-s Heisenberg model on random surfaces

We consider a spin-s Heisenberg model coupled to two-dimensional quantum gravity. We quantize the model using the Feynman path integral, summing over all possible two-dimensional geometries and spin configurations. We regularize this path integral by starting with the R-matrices defining the spin-s Heisenberg model on a regular 2d Manhattan lattice. 2d quantum gravity is included by defining the R-matrices on random Manhattan lattices and summing over these, in the same way as one sums over 2d geometries using random triangulations in non-critical string theory. We formulate a random matrix model where the partition function reproduces the annealed average of the spin-s Heisenberg model over all random Manhattan lattices. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.

hep-th↗
arXiv2014

On the solutions to the multi-parametric Yang-Baxter equations

A unified approach is applied in the consideration of the multi-parametric (colored) Yang-Baxter equations (YBE) and the usual YBE with two-parametric R-matrices, relying on the existence of the arbitrary functions in the general solutions. The colored YBE are considered with the R-matrices defined on two and three dimensional states. We present an exhaustive study and the overall solutions for the YBE with $4 \times 4$ colored R-matrices. The established classification includes new multi-parametric free fermionic solutions. In the context of the given approach there are obtained the colored solutions to the YBE with $9 \times 9$ R-matrices having 15 non-zero elements.

math-ph↗
arXiv2012

New solutions to the $s\ell_q(2)$-invariant Yang-Baxter equations at roots of unity: cyclic representations

We find the all 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 are deriving 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 type of solutions are arising, and as well as the inhomogeneous case. We are giving a full classification of the YBE solutions for the considered representations. A specific character of the solutions is the existence of the arbitrary functions.

math-ph↗
arXiv2012

On the solutions of the Yang-Baxter equations with general inhomogeneous eight-vertex $R$-matrix: Relations with Zamolodchikov's tetrahedral algebra

We present most general one-parametric solutions of the Yang-Baxter equations (YBE) for one spectral parameter dependent $R_{ij}(u)$-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 $R_{ij}(u,v)$ of YBE. The application of the non-homogeneous solutions to construction of Zamolodchikov's tetrahedral algebra is discussed.

math-ph↗
arXiv2011

New solutions to the $s\ell_q(2)$-invariant Yang-Baxter equations at roots of unity

We find new solutions to the Yang-Baxter equations with the $R$-matrices possessing $sl_q(2)$ symmetry at roots of unity, using indecomposable representations. The corresponding quantum one-dimensional chain models, which can be treated as extensions of the XXZ model at roots of unity, are investigated. We consider the case $q^4=1$. The Hamiltonian operators of these models as a rule appear to be non-Hermitian. Taking into account the correspondence between the representations of the quantum algebra $sl_q(2)$ and the quantum super-algebra $osp_t(1|2)$, the presented analysis can be extended to the latter case for the appropriate values of the deformation parameter.

math-ph↗
arXiv2009

Fusion Rules of the Lowest Weight Representations of osp_q(1|2) at Roots of Unity: Polynomial Realization and Degeneration at Roots of Unity

The degeneracy of the lowest weight representations of the quantum superalgebra $osp_q(1|2)$ and their tensor products at exceptional values of %when deformation parameter $q$ takes exceptional values 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.

math-ph↗
arXiv2009

Network Models: Action formulation

We develop a technique to formulate quantum field theory on arbitrary network, based on different, randomly disposed sets of scattering's. We define R-matrix of the whole network as a product of R-matrices attached to each of scattering nods. Then an action for a network in terms of fermionic fields is formulated, which allows to calculate the transition amplitudes as their 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 nods.

cond-mat.mes-hall↗
arXiv2009

Grassmann-Gaussian integrals and generalized star products

In quantum scattering on networks there is a non-linear composition rule for on-shell scattering matrices which serves as a replacement for the multiplicative rule of transfer matrices valid in other physical contexts. In this article, we show how this composition rule is obtained using Berezin integration theory with Grassmann variables.

math-ph↗
arXiv2008

Solutions to the Yang-Baxter equations with $osp_q(1|2)$ symmetry: Lax operators

We find a new $4\times4$ solution to the $osp_q(1|2)$-invariant Yang-Baxter equation with simple dependence on the spectral parameter and propose $2\times 2$ 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 $osp_q(1|2)$-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 $3 \times 3$ matrix expression of the Lax operator is found, which have a well defined limit at $q \to 1$.

math-ph↗
arXiv2005

Simplified tetrahedron equations: Fermionic realization

The natural generalization of the (two-dimensional) Yang-Baxter equations to three dimensions is known as the Zamolodchikov's tetrahedron equations. We consider a simplified version of these equations which still ensures the commutativity of the transfer matrices with different spectral parameters and we present a family of free fermionic solutions.

cond-mat.stat-mech↗
arXiv2004

An Integrable Model with non-reducible three particle R-Matrix

We define an integrable lattice model which, in the notation of Yang, in addition to the conventional 2-particle $R$-matrices also contains non-reducible 3-particle $R$-matrices. The corresponding modified Yang-Baxter equations are solved and an expression for the transfer matrix is found as a normal ordered exponential of a (non-local) Hamiltonian.

cond-mat.stat-mech↗
arXiv2001

3D Ising Model on Dual BCC Lattice: the Sign-Factor

The 3d Ising model on a regular cubic lattice can be expressed in terms of an SU(2) 2d fermionic model with a $Z_2$-fluxes. We modify the model such that it is defined on the dual to a body centered cubic lattice. The advantage of this lattice is that 2d embedded surfaces have no selfintersections, thus partially avoiding the Sign-factor problem associated with the 2d fermionoc models related to the 3d Ising model. Rather than solving the full SU(2) fermionic theory on this lattice we study the simpler model of scalar fermions and find the spectrum of excitations. The model has no mass gap. We reformulate the model using the R-formalism and a new interesting structure appears due to the necessity of introducing a three-paticle matrix $R^{(3)}_{ijk}$. It encodes the essential character of the Sign-factor. We analyse the integrability properties of this class of models.

cond-mat↗