Revisiting the replica theory of the liquid to ideal glass transition

The replica theory of the "Random First Order Transition" (RFOT) from a supercooled liquid to an "ideal" glass of a system of "soft spheres" is revisited. Following the seminal work of M'{e}zard and Parisi (J. Chem. Phys. 111, 1076 (1999)), the number $m$ of weakly interacting replicas of the system is varied continuously from $m=2$ to $m < 1$. Relevant order parameters and the free energy of the liquid and glass phases are calculated using the hyper-netted (HNC) approximation for the pair correlation functions. The scenario observed for all $m$ confirms the existence of two glass branches $G_{1}$ and $G_{2}$. The latter has the lowest free energy for all $m > 1$, while the former has a lower free energy for $m < 1$, but is shown to be unstable against spinodal decomposition for any non-zero value of the attractive inter-replica coupling. The critical temperature $T_{cr}$ of the RFOT turns out to depend on $m$, which may reflect the thermodynamic inconsistency of the HNC closure. The RFOT is predicted to be weakly first order, characterized by a small jump in density between the coexisting liquid and $G_{2}$ phases for all $m > 1$. Estimating $T_{cr}$ in the limit $mlongrightarrow 1$ requires a proper extrapolation of high resolution HNC calculations. The present protocol allows a direct access to the free energy of the ideal glass phase below $T_{cr}$.