LONG-RANGE ANISOTROPIC ANTIFERROMAGNETIC INTERACTIONS IN ONE-DIMENSIONAL CLASSICAL LATTICE-SPIN MODELS

The present paper considers a classical system, consisting of n-component unit vectors (n = 2,3), associated with a one-dimensional lattice {u(k)\k is-an-element-of Z}, and interacting via a translationally invariant pair potential of the long-range, antiferromagnetic and anisotropic form W = W(jk) = + epsilon \j-k\-2(au(j,n)u(k,n)+b SIGMA(lambda<n u(j,lambda)u(k,lambda). Here epsilon is a positive quantity setting energy and temperature scales (i.e., T* = k(B)T/epsilon), a and b are positive numbers, and u(k,lambda) denotes the Cartesian components of the unit vectors. Available rigorous results exclude the existence of order at finite temperature in the isotropic case a = b, whereas spin-wave arguments imply its existence in the anisotropic one a > b greater-than-or-equal-to 0, for which no such theorems axe known; we report here a simulation study of the extremely anisotropic case a = 1, b = 0. Results obtained over a range of sample sizes suggested the existence of antiferromagnetic order in the thermodynamic limit at finite temperature; we have estimated the transition temperatures to be T(c)* = 0.19 +/- 0.01 (n = 2) and T(c)* = 0.15 +/- 0.01 (n = 3).