Entropic bottlenecks to nematic ordering in an $RP^{2}$ apolar spin model

The Lebwohl-Lasher model of liquid crystals with (d = 2, n = 3) describes interacting apolar spins, with an $RP^{2}$ order-parameter topology. Simulations with a modified Wang-Landau Monte Carlo protocol, that includes a density of states (DoS) factor, had previously found a zero latent-heat transition at $T=T_{n}$ to a novel nematic order, coexisting with unbound defects whose binding is completed only on cooling. We find through this entropically augmented MC protocol, that there is a deep dip in the DoS at an energy preceding global ordering, reflecting 'sparse' intermediate configurations, or entropy barriers. The narrow entropic bottleneck induces a cusp in the initially rising nematic correlation length, at a micro-canonical 'precursor' temperature $T=T_{p}$. A finite-scale cooperativity of defects and nematic clusters penetrates the bottleneck at $T_{p}$ to enable a third-order phase transition at a lower $T_{n}$: a rare pathway, overlooked by energy-only acceptance protocols.