ONE-DIMENSIONAL CLASSICAL SPIN MODELS WITH LONG-RANGE ANISOTROPIC INTERACTION - AN EXTREME NEMATOGENIC LATTICE MODEL

We consider a classical system, consisting of n-component unit vectors (classical spins, 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, ferromagnetic, and anisotropic form [GRAPHICS] where epsilon > 0 is a positive quantity setting energy and temperature scales (i.e. T* = kT/epsilon), a greater-than-or-equal-to 0. b greater-than-or-equal-to 0, p > 1 and u(k, lambda) denotes the cartesian components of the unit vector u(k). Available rigorous results entail the existence of an ordering transition at a finite temperature for 1 < p < 2, and for the borderline cases p = 2, b = 0 (thus a = 1), studied here by Monte Carlo simulation. Moreover, the case n = 2 can be interpreted both as a ferromagnet and as an extreme case of a nematogenic lattice model, and was investigated accordingly, also by calculating the singlet orientational distribution function at one temperature in the ordered region. Simulation results showed a broad, qualitative similarity between the two models. The estimated transition temperatures are T(C)* = 1.04 + 0.02 (n = 2) and T(C)* = 0.735 +/- 0.015 (n = 3); we conjecture them to be of second order, although a Thouless effect (as in the Ising counterpart) cannot be completely ruled out. The molecular field approximation overestimates the transition temperature by about 50 per cent.