Response of non-equilibrium systems at criticality: Ferromagnetic models in dimension two and above

We study the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. We show that these systems are aging in the long-time regime, i.e., their two-time autocorrelation and response functions and associated fluctuation-dissipation ratio are non-trivial scaling functions of both time variables. This is exemplified by the exact analysis of the spherical model in any dimension D>2, and by numerical simulations on the two-dimensional Ising model. We show in particular that, for $1\ll s$ (waiting time) $\ll t$ (observation time), the fluctuation-dissipation ratio possesses a non-trivial limit value $X_{\mathrm{\infty }}$, which appears as a dimensionless amplitude ratio, and is therefore a novel universal characteristic of non-equilibrium critical dynamics. For the spherical model, we obtain $X_{\mathrm{\infty }}=1-2\mathrm{/}D$ for 2<D<4, and $X_{\mathrm{\infty }}=1\mathrm{/}2$ for D>4 (mean-field regime). For the two-dimensional Ising model we measure $X_{\mathrm{\infty }}\approx 0.26\pm 0.01$.