Complexity and line of critical points in a short-range spin-glass model

We investigate the critical behavior of a three-dimensional short-range spin glass model in the presence of an external field $\eps$ conjugated to the Edwards-Anderson order parameter. In the mean-field approximation this model is described by the Adam-Gibbs-DiMarzio approach for the glass transition. By Monte Carlo numerical simulations we find indications for the existence of a line of critical points in the plane $(\eps,T)$ which separates two paramagnetic phases and terminates in a critical endpoint. This line of critical points appears due to the large degeneracy of metastable states present in the system (configurational entropy) and is reminiscent of the first-order phase transition present in the mean-field limit. We propose a scenario for the spin-glass transition at $\eps=0$, driven by a spinodal point present above $T_c$, which induces strong metastability through Griffiths singularities effects and induces the absence of a two-step shape relaxation curve characteristic of glasses.