Dynamic heterogeneities in critical coarsening: exact results for correlation and response fluctuations in finite-sized spherical models

We study dynamic heterogeneities in the out-of-equilibrium coarsening dynamics of the spherical ferromagnet after a quench from infinite temperature to its critical point. A standard way of probing such heterogeneities is by monitoring the fluctuations of correlation and susceptibility, coarse-grained over mesoscopic regions. We discuss how to de. ne such fluctuating coarse-grained correlations and susceptibilities in models where no quenched disorder is present. Our focus for the spherical model is on coarse-graining over the whole volume of N spins, which requires accounting for O(N(-1/2)) non-Gaussian fluctuations of the spin variables. The latter are treated as a perturbation about the leading order Gaussian statistics. We obtain exact results for these quantities, which enable us to characterize the joint distribution of correlation and susceptibility fluctuations. We find that this distribution is qualitatively different, even for equilibrium above criticality, from the spin-glass scenario where correlation and susceptibility fluctuations are linked in a manner akin to the fluctuation-dissipation relation between the average correlation and susceptibility. Our results show that coarsening at criticality is clearly heterogeneous above the upper critical dimension and suggest that, as in other glassy systems, there is a well-defined timescale on which fluctuations across thermal histories are largest. Surprisingly, however, neither this timescale nor the amplitude of the heterogeneities increases with the age of the system, as would be expected from the growing correlation length. Below the upper critical dimension, the strength of correlation and susceptibility fluctuations varies on a timescale proportional to the age of the system; the corresponding amplitude also grows with age, but does not scale with the correlation volume as might have been expected naively.