Explicit thermostatics of certain classical one-dimensional lattice models by harmonic analysis

A certain class of one-dimensional classical lattice models is considered. Using the method of abstract harmonic analysis explicit thermostatic properties of such models are derived. In particular, we discuss the low-temperature behavior of some of these models. 1 A class of one-dimensional models In this section we will characterize a certain class of lattice models in one dimension. This class of models has already been considered by Romerio and Vuillermot [1] in connection with the transfer-matrix method. See also the book by Moraal [2] where discrete spin models of this class are discussed. First, we begin with the definition of the “spin space” denoted by M . We will assume that M is a homogeneous space and thus can be identified with a group quotient G/H, i.e. M = G/H. Here G is the transformation group acting transitively on M , i.e. for each pair (S, S0) ∈ M ×M there exists a group element g ∈ G such that S = gS0 (see for example ref. [3] for details). The subgroup H ⊂ G is the stability group of some spin-direction in M . We will keep this direction fixed throughout this paper and denote it by S0. Hence, hS0 = S0 for all h ∈ H. For simplicity we will assume that M has a finite volume and hence G is a compact group. However, all results presented below can be generalized to the case of non-compact unimodular groups. With the help of the unique normalized invariant Haar measure dg on G we can define a G-invariant probability measure dS on the spin space M [4, 5]: