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The Lebwohl-Lasher model describes the isotropic-nematic transition in liquidcrystals. In two dimensions, where its continuous symmetry cannot breakspontaneously, it is investigated numerically since decades to verify, inparticular, the conjecture of a topological transition leading to a nematicphase with quasi-long-range order. We use scale invariant scattering theory toexactly determine the renormalization group fixed points in the general case of$N$ director components ($RP^{N-1}$ model), which yields the Lebwohl-Lashermodel for $N=3$. For $N>2$ we show the absence of quasi-long-range order andthe presence of a zero temperature critical point in the universality class ofthe $O(N(N+1)/2-1)$ model. For $N=2$ the fixed point equations yield theBerezinskii-Kosterlitz-Thouless transition required by the correspondence$RP^1\sim O(2)$.

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