Full dynamical solution for a spherical spin-glass model

We present a detailed analysis for the Langevin dynamics of a spherical spin-glass model (the spherical Sherrington-Kirkpatrick model). All the spins in the system are coupled by pairs via a random interaction matrix taken from the Gaussian ensemble. One finds that for a general initial configuration the system never reaches an equilibrium state and the theorems associated to `equilibrium dynamics' are violated. Only very particular initial conditions drive the system to equilibrium. The weak ergodicity breaking scenario is demonstrated for general `non-equilibrium' initial conditions. Two-time quantities such as the auto-correlation function explicitly depend on both times. When the time difference is short compared to the smaller time one finds stationary dynamics with time translation invariance and the fluctuation-dissipation theorem satisfied. Instead, when the time difference is of the order of the smaller time one finds non-stationary dynamics with aging phenomena, the system {\em remembers} the time spent after the initial time (the quench below the critical temperature). Interestingly enough the short time-difference dynamics (FDT regime) for non-equilibrium initial conditions is identical to the relaxation within the equilibrium states obtained for the particular `equilibrium' initial conditions. This points to a self-similaririty of the energy landascape. In addition we analyse the effect of temperature variations on the behaviour of the correlation function and energy density. In this way we are able to make some comparisons between experimentally observed results and exact calculations for the model. Somewhat surprisingly, this simple model captures some of the effects seen in laboratory spin-glasses.