Universality classes of critical points in constrained glasses

We analyze critical points that can be induced in glassy systems by the presence of constraints. These critical points are predicted by the mean field thermodynamic approach and they are precursors of the standard glass transition in the absence of constraints. Through a deep analysis of the soft modes appearing in the replica field theory, we can establish the universality class of these points. In the case of the 'annealed potential' of a symmetric coupling between two copies of the system, the critical point is in the Ising universality class. More interesting is the case of the 'quenched potential', where a single copy is coupled with an equilibrium reference configuration, or the 'pinned particle' case, where a fraction of the particles is frozen in fixed positions. In these cases we find the random field Ising model (RFIM) universality class. The effective random field is a 'self-generated' disorder that reflects the random choice of the reference configuration. The RFIM representation of the critical theory predicts non-trivial relations governing the leading singular behavior of relevant correlation functions, that can be tested in numerical simulations.