Lattice System of Interacting Spins in the Thermodynamical Limit


In this paper we investigate some particular spin lattice (a higher dimensional generalization of a spin chain) related to Zamolodchikov model, in the limit when both sizes of the lattice tend to infinity. An infinite set of bilinear equations, describing a distribution of eigenvalues of infinite set of mutually commuting operators, is derived. The distribution for the maximal eigenvalues is obtained explicitly. The way to obtain the excitations is discussed. Introduction Integrability of Zamolodchikov-Baxter three dimensional spin model [2, 1] is based on the existence of commutative set of transfer-matrices T (θ1, θ2, θ3), (1) [ T (θ1, θ2, θ3), T (θ1, θ ′ 2, θ ′ 3) ] = 0 , where θj are Zamolodchikov’ dihedral angles, and we understand T as an operator in the vertex formulation [3] of Zamolodchikov-Bazhanov-Baxter model [4] with two spin states. In contrast to two-dimensional integrable models, T is layer-to-layer transfer matrix [5], that means • it is associated with a rectangular lattice with the size N ×M , therefore matrix T has the dimension 2 × 2 , • two parameters θ2 and θ3 are varied in eq. (1). Matrix elements of R-matrix of the Zamolodchikov–Baxter model are not positively defined: it is the obstacle for the decent interpretation of it as the model of statistical mechanics. The quantum mechanical interpretation is preferable. Relation (1) implies the existence of a set of commutative operators {tm,n(θ1)} such that (2) [tm,n(θ1), T (θ1, θ2, θ3)] = 0 ∀ θ2, θ3,m, n , i.e. the problem of diagonalization of T for any θ2, θ3 and the problem of simultaneous diagonalization of {tm,n} are equivalent. In what follows, we mean a determined definition of {tm,n} with 0 ≤ m ≤ M and 0 ≤ n ≤ N related to an auxiliary problem of the model. It is well known, the Zamolodchikov model and its generalization – Bazhanov-Baxter model [4] – are related to the generalized chiral Potts model [6]. The set of {tm,n(θ1)} may be produced by the expansion of 1991 Mathematics Subject Classification. 37K15.

Cite this paper

@inproceedings{Sergeev2008LatticeSO, title={Lattice System of Interacting Spins in the Thermodynamical Limit}, author={Sergeı̆ Sergeev}, year={2008} }