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 Mézard 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 hypernetted chain (HNC) approximation for the pair correlation functions. The scenario observed for all m confirms the existence of two glass branches G₁ and G₂. 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 nonzero value of the attractive inter-replica coupling. The critical temperature T_cr of the RFOT turns out to depend on m, which may be a by-product of the approximation inherent in the HNC closure. The RFOT is predicted to be weakly first order, characterized by a small jump in density between the coexisting liquid and G₂ phases for all m > 1. Estimating T_cr in the limit m → 1 requires a proper extrapolation of high resolution HNC calculations. The present protocol explores the behavior of the free energy of the ideal glass phase below T_cr as a function of m.