Abstract
The nematic-isotropic phase transition in an external field is studied in the frame of Maier-Saupe model, where a global phase diagram including both cases of positive and negative susceptibility anisotropic is obtained. In the extreme case of infinitely large negative anisotropy, the system is reduced to the classical XY model because the rotational degree of freedom is restricted to lie in the plane perpendicular to the field, in which the second order phase transition occurs. The crossover of the directional degree of freedom between 3-dimension and 2-dimension is depicted. On the other hand, the critical point appears at a certain strength of the positive anisotropy, beyond which the transition disappears. The theory can be applied to the phase transition of very thin systems with homeotropic and planar anchorings.
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