Entropic bottlenecks to nematic ordering in an $RP^{2}$ apolar spin model
The Lebwohl-Lasher model of uniaxial liquid crystals with (\textit{n} = 3, \textit{d} = 2) was reported earlier to undergo a crossover transition to a novel nematic phase at a temperature $T=T_{n}$. This phase has unbound topological defects in a nematic background, that pair at a lower $T_{\text{BKT}} < T_{n}$. The transition has zero latent heat, and a specific heat and correlation length that remain finite. We discover here a significant sparseness of states or an entropy barrier `bottleneck', between the isotropic and novel nematic phases. Passage through these sparse configurations is enabled by short-range nematic clusters dressing the defect cores. The free energy temperature derivatives, along with energy derivatives of the micro-canonical entropy, determine that this is a {\it third-order} transition in the Ehrenfest classification. The local transformation to dressed defects induces a sharp downward cusp in the correlation length, at a precursor temperature $T_{p} > T_{n}$. The entropic bottleneck manifests as a rippling of the free energy landscape, over mutually modifying nematic order and defect density. Cooling through $T_{p}$ yields an itinerant para-nematic fluid of dressed defects with macroscopically occupied local polar angle tilts, that catalyse a common global tilt or nematic phase at $T_{n}$.