Spin, bond and global fluctuation-dissipation relations in the non-equilibrium spherical ferromagnet

Abstract

We study the out-of-equilibrium dynamics of the spherical ferromagnet after a quench to its critical temperature. We calculate correlation and response functions for spin observables which probe lengthscales much larger than the lattice spacing but smaller than the system size, and find that the asymptotic fluctuation-dissipation ratio (FDR) X^inf is the same as for local observables. This is consistent with our earlier results for the Ising model in dimension d=1 and d=2. We also check that bond observables, both local and long-range, give the same asymptotic FDR. In the second part of the paper the analysis is extended to global observables, which probe correlations among all N spins. Here non-Gaussian fluctuations arising from the spherical constraint need to be accounted for, and we develop a systematic expansion in 1/sqrt{N} to do this. Applying this to the global bond observable, i.e. the energy, we find that non-Gaussian corrections change its FDR to a nontrivial value which we calculate exactly for all dimensions d>2. Finally, we consider quenches from magnetized initial states. Here even the FDR for the global spin observable, i.e. the magnetization, is nontrivial. It differs from the one for unmagnetized states even in d>4, signalling the appearance of a distinct dynamical universality class of magnetized critical coarsening. For lower d the FDR is irrational even to first order in 4-d and d-2, the latter in contrast to recent results for the n-vector model.