Evidence for a topological transition in nematic - to - isotropic phase transition in two dimensions

The nematic-to-isotropic orientational phase transition, or equivalently the RP 2 model, is considered in two dimensions and the question of the nature of the phase transition is addressed. Using powerful conformal techniques adapted to the investigation of critical properties of two-dimensional scale-invariant systems, we report strong evidences for a transition governed by topological defects analogous to the Berezinskii-Kosterlitz-Thouless transition in two-dimensional XY model. Liquid crystals may be seen as constituted of molecules essentially represented by long rigid rods. From maximization of entropy at high temperatures , all the molecule orientations are equally probable , independently of the neighbouring molecule directions and the system exists in an isotropic phase. At low temperatures a preferential orientation is more favourable in order to minimize interaction terms, and an ordered structure emerges. When order occurs along one space dimension only, the system is said to be nematic. Still at lower temperatures, other ordered phases can appear, e.g. smectic phases. In a lattice model, each molecule may be represented by a unit vector σ w at site w of an hyper-cubic lattice Λ of linear extent L. The σ's live in a three-dimensional space attached to each lattice site. In the nematic phase, the preferential direction defines a unit vector, n, called the director, and one can measure the deviation of molecule σ w with respect to the director by the scalar product σ w · n = cos θ w. Due to the local Z 2 symmetry (the rods are not oriented), one cannot distinguish between opposite directions θ w and θ w + π, and cos θ w vanishes on average while cos 2 θ w does not. In the disordered phase on the other hand, the angles are measured with respect to any arbitrary direction , and the thermal average of course leads to cos θ w = 0 and cos 2 θ w = 1 3 , so that cos 2 θ w − 1 3 represents a convenient order parameter. In the literature on liquid crystals, one usually defines the local order parameter by the second Legendre polynomial , m(w) = P 2 (σ w · n) = P 2 (cos θ w). (1) This definition suggests to consider the following Hamiltonian to describe the nematic transition, − H k B T = J k B T w µ P 2 (σ w · σ w+µ), (2) where µ stands …