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).

Asymptotic low-temperature behavior of two-dimensional RP$^{N-1}$ models

“…Ref. [29] suggested that the critical behavior should be related to that of the O(5) vector model. We have verified that our curve differs from that computed in the O(5) model.…”

“…Their behavior has been for long controversial, and at present, it is not yet fully understood, see, e.g., Refs. [24][25][26][27][28][29]. For N ≥ 3, these models are not expected to undergo finite-temperature continuous transitions related to the breaking of the O(N ) symmetry, because of the Mermin-Wagner theorem [30].…”

Particles, conformal invariance and criticality in pure and disordered systems

“…The infinite-dimensional character of conformal symmetry in d = 2 induces essential simplifications in the scattering formalism, which then yields exact equations whose solutions provide a classification of RG fixed points with a given internal symmetry. We illustrated the method for two main models of the theory of critical phenomena, namely the O(N ) vector model and the q-state Potts model, for which critical lines are obtained as the symmetry parameters N and q are varied (see [98,99] for the study of other symmetries). In the case of pure systems, for which many exact results are already known, the formalism allows to obtain the different critical lines with the given symmetry from a single set of equations, and to gain a global view of their location in the space of parameters.…”