Dynamic heterogeneity above and below the mode-coupling temperature

We study the temperature dependence of the spatial extend of the dynamic heterogeneity in a soft sphere system near the so-called mode-coupling temperature $T_c$. We utilize a recently introduced procedure\footnote{E. Flenner and G. Szamel, Phys. Rev. Lett. \textbf{105}, 217801 (2010)} to calculate the ensemble independent dynamic susceptibility $\chi_4(\tau_\alpha)$ and the dynamic correlation length $\xi(\tau_\alpha)$ at the alpha relaxation time $\tau_\alpha$. Above $T_c$, we find that $\chi_4(\tau_\alpha) \sim \xi(\tau_\alpha)^3$ and $\xi(\tau_\alpha) \sim \ln(\tau_\alpha)$, which is the same behavior found in a binary hard-sphere system. We track these relationships below $T_c$ to examine the recently reported non-monotonic temperature dependence of dynamic correlations found in the same system\footnote{W. Kob, S. Roland-Vargas and L. Berthier, Nat. Phys. DOI:10.1038/NPHYS2133}. Finally, we examine the relationship between dynamic susceptibilities that can be determined from experiments and the dynamic correlation length $\xi(\tau_\alpha)$.