Fig. 4.

Characterization of the apparent transition for $N=4$ site representations of 2ERL. (A) The dimensionless free energy, ${\beta }_{\mathcal{Q}}F$, at the transition temperature (black) and at temperatures above (red) and below (blue) the transition. The black X indicates the separatrix, ${\mathcal{Q}}_{*}$, for which $\mathcal{P}\left(\mathcal{Q}<{\mathcal{Q}}_{*}\right)=1/2$ at the transition temperature. (B and C) The averages and variances, respectively, for several metrics. The metric $d_0\left(\text{M}\right)$ quantifies the difference in the atomic groups defined by the map, $\text{M}$, and the ground state map, ${\text{M}}_0$, while ${\text{R}}_{\text{G}}\left(\text{M}\right)$ quantifies the compactness of the associated atomic groups. For convenience, we have shifted ${\text{R}}_{\text{G}}$ such that $\mathrm{\Delta }{\text{R}}_{\text{G}}\left(\text{M}\right)$ vanishes as $T_{\mathcal{Q}}\to 0$ and have normalized variances relative to their $T_{\mathcal{Q}}\to \infty$ limit. Error bars estimate statistical uncertainty. The dashed vertical line indicates the transition temperature, which is defined by the variance peak in $\mathcal{Q}$. T denotes the fictitious temperature, $T_{\mathcal{Q}}$, conjugate to $\mathcal{E}\left(\text{M}\right)$ = 1- $\mathcal{Q}\left(\text{M}\right)$.