One dimensional classical spin models with long range anisotropic interaction. An extreme nematogenic lattice model

Abstract

We consider a classical system, consisting of n-component unit vectors (classical spins, n = 2, 3), asssociated with a one dimensional lattice {u _k /kεZ}, and interacting via a translationally invariant pair potential of the long range, ferromagnetic, and anisotropic form

where ε>0 is a positive quantity setting energy and temperature scales (i.e. T* = kT/ε), a ⩾0. b ⩾0, p>1 and u_k,λ A 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.