Existence of a line of critical points in a two-dimensional Lebwohl Lasher model

Critical points in the $RP^{N-1}$ model

“…which takes into account the present way of indexing the particles. The different possible choices of external indices (see table 2) then yield the equations 25)- (35).…”

“…In the latter respect, the fact that the RP N −1 fixed point solutions do not allow for free parameters at fixed N > 2 excludes the presence in this range of BKT-like transitions yielding quasi-long-range order. The possibility of such a transition driven by disclination defects had been debated in numerical studies [28,29,30,31,32,33,34,35,36].…”

Critical points in the $CP^{N-1}$ model

“…For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions [5] generically suggests that criticality is limited to zero temperature, and numerical studies for T → 0 show a fast growth of the correlation length which makes particularly hard to reach the asymptotic limit and draw conclusions about universality classes [6,7,8,9,10,11,12]. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one [13] -which should definitely occur for RP 1 ∼ O(2) -and mediated by "disclination" defects [14,15] has also been debated in numerical studies [16,17,18,19,20,21,22,23,24]. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability [25,26] and conformal field theory [27,28], models with local symmetries traditionally remained outside the range of application of these methods.…”

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

“…In absence of analytical approaches, the matter has been investigated experimentally [5,6,7] and, more extensively, through numerical simulations within the Lebwohl-Lasher (LL) lattice model [8], which encodes head-tail symmetry and successfully accounts for the weak first order transition in 3D [9]. The possibility in the 2D model of a topological transition driven by "disclination" defects [10] and leading to a nematic phase with quasi-long-range order (QLRO) received support by some numerical studies [11,12,13,14], with others concluding for the absence of a true transition [15,16,17,18,19]. It was also argued [20,21,22] that in 2D the head-tail symmetry is not relevant for the critical behavior of the LL model, which should then coincide with that of the O(3) model, with a zero-temperature critical point and exponentially diverging correlation length [4,23].…”

“…which can be used to express the amplitudes S i≥7 in terms of S i≤6 . In this way the unitarity equations ( 4), where now µ = ab and Kronecker deltas are replaced by (14), take the form…”