Mean field, two-site cluster, and computer simulation study of a nematogenic lattice-gas model

We have considered a classical lattice-gas model, consisting of a three-dimensional simple-cubic lattice, whose sites host three-component unit vectors; pairs of nearest-neighbouring sites interact via the nematogenic potential psi (jk) = -is an element of vjvkP(2)(T), T = Tjk = u(j) (.) u(k); here P-2(T) denotes the second Legendre polynomial, v(j) = 0, 1 are occupation numbers, uj are unit vectors (classical spins), and is an element of is a positive quantity setting energy and temperature scales (i.e. T* = k(B)T/is an element of); the total Hamiltonian is given by A = W/is an element of = - Sigma ({j < k}) (vjvk)P-2(uj (.) uk) - <mu>N, N = Sigma (vk)(k), where Sigma {j < k} denotes sum over all distinct nearest-neighbouring pairs of lattice sites. The saturated-lattice version of this model defines the extensively studied Lebwohl-Lasher model, possessing a transition to an orientationally ordered phase at low temperature; according to available rigorous results, there exists a <mu>o < 0, such that, for all <mu> > muo the system supports an ordering transition at a finite, mu -dependent, temperature. Continuing along the lines of our previous communication [S. Romano, Int. J. Mod. Phys. B14, 1195 (2000)], we present here a detailed study of the case mu = 0, using Monte Carlo simulation, Mean Field and Two Site Cluster treatments; the latter significantly improves the agreement with simulation results.