Critical aging properties of the two-dimensional spherical model with the long-range interaction r−3

In this work we investigate critical aging properties of the two-dimensional spherical model with the long-range interaction, which decays at large distances r by a power-law as r−3. The model with an arbitrary initial order m0 is quenched from a very high temperature to the critical temperature Tc. In the short-time regime, the magnetization increases with a logarithmic power law. The logarithmic corrections, which are dependent of m0, enter into the scaling behavior of two-time response and correlation functions. The long-time limit of the fluctuation–dissipation ratio is calculated. Three distinct types of aging are found at both criticality and low temperatures. Some universal scaling relations are tested. The crossover from the disordered initial state to the ordered state is discussed. Our two-dimensional results can be extended to the general d-dimensional cases for the long-range interaction r−d−σ with d=2σ. It is shown that the inclusion of this kind of long range interaction tends to the mean field behavior.