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

Abstract

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 $T_{n}$.

Figures (14)

• Figure 1: (color online) Sparseness of states between two special energies: (a) Data recorded during an energy-uniform random walk, shown as a 3D mesh plot of the logarithm of the density of microstates $\log p(e,S_n)$, over the plane of energy and order parameter $(e,S_n)$. There is an entropic saddle-point near $(e, S_n)= (-1.122,0.14)$. Over the interval from tunnel entry at $(-1.1, 0.05)$, the entropy drops by an order of magnitude. (b) The density of microstates $p(e, S_n)$ shown as a contour plot on the plane of ($e, S_{n}$). Superimposed on the plot are canonical-equilibrium averages from BMC (dashed line) and EAMC protocol (solid line) that match in the initial high energy region, Below some $e \simeq -1.11$, the BMC pathway parts company from the EAMC, that traverses a sparse microstate region to find richer states at a lower $e_{n} \simeq -1.16$.
• Figure 2: (color online) Two special temperatures: For $L = 128$, the 3D plot of free-energy per site $f(S_{n}, T)$ is projected as a contour map on the equilibrium $(S_{n}, T )$ plane, with a temperature resolution $\Delta T$ = 0.001, showing a narrow contour bottleneck. The superimposed $C_{v}(T)$ (red line with symbols) shows a peak at $T = T_{n} = 0.585$. The maximum-curvature point on the high temperature side (vertical dash line through the bottleneck) defines a precursor temperature $T= T_{p} = 0.590 (> T_{n})$.
• Figure 3: (color online) Density of microstates over entropy and temperature: 3D mesh plot of density of microstates $p (s,T_\text{eff})$ over the plane of the entropic variable pair ($s, T_\text{eff}$) in the entropy barrier region. There is an absence of latent heat (no jump in $s$) at the transition temperature $T_{n}= 0.585$ (dashed line). The bottleneck region shows an enhanced exploration.
• Figure 4: (color online) Density of microstates over defect density and temperature: 3D mesh plot of the density of microstates $p(\rho_{d},T_{\text{eff}})$ over the plane of defect density and temperature in the entropy barrier region. Dips (indicated by arrows) are at the same$T_{p}, T_{n}$ as previously, showing correlations between defect cores and short-range nematic order.
• Figure 5: (color online) Absence of hysteresis at transition: Equilibrium variation of nematic order and canonical entpy per site s, bracketing the transition at $T_{n}$. Data on both, collected during cooling and heating cycles, overlap respectively, ruling out hysteresis. The solid lines indicate equilibrium data from reweighting procedure of the random walk ensemble.