Time reparametrization symmetry in spin glass models

We study the long-time aging dynamics of spin-glass models with two-spin interactions by performing a Renormalization Group transformation on the time variable in the non-equilibrium dynamical generating functional. We obtain the RG equations and find that the flow converges to an exact fixed point. We show that this fixed point is invariant under reparametrizations of the time variable. This continuous symmetry is broken, as evidenced by the fact that the observed correlations and responses are not invariant under it. We argue that this gives rise to the presence of Goldstone modes, and that those Goldstone modes shape the behavior of fluctuations in the nonequilibrium dynamics.