Exact solution of the Maier-Saupe model for a nematic liquid crystal on a one-dimensional lattice

Abstract

The planar and spatial lattice versions of the Maier-Saupe model (1959, 1960) for a nematic liquid crystal are exactly solved for a one-dimensional lattice, without periodic boundary conditions. The two-molecule correlation functions are studied, and it is shown that these two models do not exhibit an order-disorder phase transition except at T=0, as is the case for the classical Heisenberg model in one dimension.

Computer simulation of two continuous spin models using Wang–Landau-Transition-Matrix Monte Carlo algorithm

“…As is to be expected in a low-dimensional model with nn interaction, the 1-d LL model does not exhibit any finite temperature phase transition. The results obtained in [24] are quoted below. The partition function Z N ( K ) for the N particle system is given by (14) where K = 3 2T is a dimensionless quantity.…”

“…The 3-d Lebwohl-Lasher (LL) model is a lattice version of a 3-d nematic, described in the mean field approximation by the Maier-Saupe theory [34], and exhibits an orientational order-disorder transition. The 1-d LL model has been simulated by [35] and has also been solved exactly by Vuillermot and Romerio [24] using a group theoretic method. As is to be expected in a low-dimensional model with nn interaction, the 1-d LL model does not exhibit any finite temperature phase transition.…”

“…Fig. 2 is a plot of the error in ln Z ( Z being the partition function) and a comparison has been made with the exact results of [24]. Here we have presented the results of simulation where the construction of the C-matrix was started at different stages of iteration.…”

“…3 is shown the error in ln Z for the three lattice sizes, achieved with the WLTM algorithm and C-matrix updating corresponding to λ 2 . Since the theoretical estimate of ln Z [24] is for the thermodynamic limit, it is natural to expect that the error should decrease with the increase in system size. In Fig.…”

“…We have chosen (i) the one-dimensional Lebwohl-Lasher (1-d LL) model [23] and (ii) the two-dimensional XY (2-d XY) model for this purpose. The reason for the choice of the former is that the model is exactly solvable [24] and hence one can compare the performance of the WLTM algorithm with the exact results. This is analogous to the practice where the results of a new MC algorithm is being compared with the exact results on a 2d Ising model.…”