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).

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

“…When moving away from criticality, on the other hand, ρ 4 and ρ 5 are expected 11 to develop nonvanishing values, thus producing deviations 12 from the off-critical O(N 2 − 1) behavior that vanish as T → 0. These conclusions parallel those we reached for the RP N −1 model in [29,30], whose basic findings we recall in the next section.…”

“…We briefly point out similarities and differences between the above results for the CP N −1 model and those obtained for the RP N −1 model in refs. [29,30], to which we refer the reader for the detailed derivation. The RP N −1 model, defined by the lattice Hamiltonian…”

“…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. Recently, however, we showed in [29,30] how the renormalization group fixed points of the RP N −1 model can be accessed in an exact way in the scale invariant scattering framework [31], which implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last years new results for pure and disordered systems [32,33,34,35,36,37,38] (see [39] for a review). We found that only O(N (N + 1)/2 − 1) fixed points exist for 1 N ≥ 3 and account for zero temperature criticality, while a line of fixed points yielding a BKT transition exists only for N = 2.…”

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 qstate Potts model, for which critical lines are obtained as the symmetry parameters N and q are varied (see [105,106] 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.…”

Two-dimensional lattice SU(Nc) gauge theories with multiflavor adjoint scalar fields

“…The zero-temperature critical behavior of 2D RP N −1 models is still debated; see, e.g., refs. [19][20][21][22][23][24][25]. Although 2D RP N −1 and O(N ) σ models have the same perturbative behavior [21], there is numerical evidence that their nonperturbative behavior differs.…”