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Eurasian Physical Technical Journal

PARAMETRIZED EIGHT-VERTEX MODEL AND KNOT INVARIANT

T.K. Kassenova

Published Mar 28, 2022• 1 citation
Publisher

Abstract

The article discusses and expands the known elements of the eight-vertex model, paying special attention to the parameterization of the matrix. The matrix values are interconnected with the knot through the braids and this model is valid on finite square lattices in two-dimensional space. A new solution of the parametrized eight-vertex model of free fermions with a complex version of elliptic functions, which is valid on a finite lattice, will be constructed. The range of applicability of the eight-vertex model with elements of the Jacobi elliptic function and the construction of a knot invariant on its basis is discussed by comparing the results obtained analytically for the model. The construction of the knot invariant using the Clebsch-Gordan coefficients and the main tool of statistical mechanics of the Yang-Baxter equation will be studied in detail

DOI
10.31489/2022no1/119-126

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The article discusses and expands the known elements of the eight-vertex model, paying special attention to the parameterization of the matrix. The matrix values are interconnected with the knot through the braids and this model is valid on finite square lattices in two-dimensional space. A new solution of the parametrized eight-vertex model of free fermions with a complex version of elliptic functions, which is valid on a finite lattice, will be constructed. The range of applicability of the eight-vertex model with elements of the Jacobi elliptic function and the construction of a knot invariant on its basis is discussed by comparing the results obtained analytically for the model. The construction of the knot invariant using the Clebsch-Gordan coefficients and the main tool of statistical mechanics of the Yang-Baxter equation will be studied in detail
Baxter's method of solving the eight-vertex model in lattice statistical mechanics is examined from an elementary point of view. It is shown that the algebraic operations in the method can be carried out without recourse to elliptic functions. These include: construction of certain subspaces invariant uti.der the action of the transfer matrix; reduction of the transfer matrix eigenvalue problem to an equivalent ice-type problem and construction of certain matrices which commute with the transfer matrix and satisfy a functional matrix equation.
Abstract We survey and enlarge the known mappings of the 16-vertex model, with emphasis on mappings between the even and odd 8-vertex subcases of the general model, also giving new mappings between these models, valid on finite toroidal lattices. In particular, we find new mappings between the models by using their algebraic invariants with respect to the S L ( 2 ) × S L ( 2 ) symmetry of the 16-vertex model; we also find a larger set of weak-graph transformations. We show many examples of models with negative weights which map to models with only positive weights. Using the algebraic invariant relations of the even and odd 8-vertex models, we find the complete set of points in the complex field plane of the square lattice Ising model in a field which map to the even or odd 8-vertex models; these points also correspond to the set of free-fermionic points of the model. We do not find any new integrable points, but we find a new mapping between the odd 8-vertex model and the square lattice Ising model at magnetic field H = i π / ( 2 β ) , valid on finite toroidal lattices. We also show directly through various examples that mappings via algebraic invariants do not fully exhaust the possible mappings a model may have with another model. We construct a new solution to the odd 8-vertex free-fermion model which is valid on the finite lattice, since the previous known solution resulted from a mapping valid only in the thermodynamic limit. Finally, we detail for the first time the phase transitions of the column staggered free-fermion 8-vertex model, and show that it can be mapped to the bi-partite staggered free-fermion model.
A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid G\mathfrak{G}G together with a map π:G→End⁡(V⊗V)π:\mathfrak{G}\to\operatorname{End}(V\otimes V)π:G→End(V⊗V) for some vector space VVV such that the Yang-Baxter commutator [[π(u),π(w),π(v)]]=0[[ π(u),π(w),π(v)]]=0[[π(u),π(w),π(v)]]=0 if u,v∈Gu,v\in\mathfrak{G}u,v∈G are such that the groupoid composition w=u⋆vw=u\star vw=u⋆v is defined. An important role is played by an object map Δ:G→MΔ:\mathfrak{G}\to MΔ:G→M for some set MMM such that Δ(u)=Δ(v′)Δ(u)=Δ(v')Δ(u)=Δ(v′), Δ(w)=Δ(v)Δ(w)=Δ(v)Δ(w)=Δ(v) and Δ(w′)=Δ(u′)Δ(w')=Δ(u')Δ(w′)=Δ(u′), where v↦v′v\mapsto v'v↦v′ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group GL⁡(2)×GL⁡(1)\operatorname{GL}(2)\times\operatorname{GL}(1)GL(2)×GL(1). For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.
This chapter introduces the six-vertex model defined on a two-dimensional square lattice. It describes the model in detail, since it gives an important prototype of many solvable lattice models defined on two-dimensional lattices. The eight-vertex model, which generalizes the six-vertex model directly, may be considered as the most important exactly solvable model in statistical mechanics. The chapter explores some features of the six-vertex model defined on a square lattice. It reviews a method for diagonalizing the transfer matrix, which is called the coordinate Bethe ansatz, and provides the expressions of the free energy per site in the ferroelectric, the antiferroelectric and the disordered phases, respectively. The chapter discusses some mathematical theories associated with integrable models such as the braid group and the quantum groups. It outlines the Ising model, the Potts models and the chiral Potts model and then the eight-vertex model and the interaction-round-a-face models.
In this paper we study, in the manner of Baxter [R. J. Baxter Exactly Solved Models in Statistical Mechanics (Academic, New York, 1982)], the possible integrable manifolds of the asymmetric eight-vertex model. As expected, they occur when the Boltzmann weights are either symmetric or satisfy the free-fermion condition; but our analysis clarifies the reason why both manifolds need to share a universal invariant. We also show that the free-fermion condition implies three distinct classes of integrable models.