Absence of nematic quasi-long-range order in two-dimensional liquid crystals with three director components

Abstract

The Lebwohl–Lasher model describes the isotropic–nematic transition in liquid crystals. In two dimensions, where its continuous symmetry cannot break spontaneously, it is investigated numerically since decades to verify, in particular, the conjecture of a topological transition leading to a nematic phase with quasi-long-range order. We use scale invariant scattering theory to exactly determine the renormalization group fixed points in the general case of N director components (RP N−1 model), which yields the Lebwohl–Lasher model for N = 3. For N > 2 we show the absence of quasi-long-range order and the presence of a zero temperature critical point in the universality class of the O(N(N + 1)/2 − 1) model. For N = 2 the fixed point equations yield the Berezinskii–Kosterlitz–Thouless transition required by the correspondence RP 1 ∼ O(2).