Fluctuation-dissipation relations in critical coarsening: crossover from unmagnetized to magnetized initial states

TL;DR

This paper analyzes the non-equilibrium critical dynamics of the spherical ferromagnet with varying initial magnetization, deriving exact crossover functions for fluctuation-dissipation ratios and revealing complex, non-monotonic behaviors influenced by system dimensionality and fluctuations.

Contribution

It provides an exact solution for the crossover behavior of fluctuation-dissipation relations in critical coarsening, extending previous Gaussian results to include non-Gaussian effects for arbitrary initial magnetization.

Findings

Exact crossover functions for magnetization correlator and response.

Non-monotonic dependence of asymptotic FDR on system age.

Correction of previous errors aligning results with RG expansions.

Abstract

We study the non-equilibrium dynamics of the spherical ferromagnet quenched to its critical temperature, as a function of the magnetization of the initial state. The two limits of unmagnetized and fully magnetized initial conditions can be understood as corresponding to times that are respectively much shorter and much longer than a magnetization timescale, as in a recent field theoretical analysis of the $n$-vector model. We calculate exactly the crossover functions interpolating between these two limits, for the magnetization correlator and response and the resulting fluctuation-dissipation ratio (FDR). For $d>4$ our results match those obtained recently from a Gaussian field theory. For $d<4$, non-Gaussian fuctuations arising from the spherical constraint need to be accounted for. We extend our framework from the fully magnetized case to achieve this, providing an exact solution for the relevant integral kernel. The resulting crossover behaviour is very rich, with the asymptotic FDR $X^\infty$ depending non-monotonically on the scaled age of the system. This is traced back to non-monotonicities of the two-time correlator, themselves the consequence of large magnetization fluctuations on the crossover timescale. We correct a trivial error in our earlier calculation for fully magnetized initial states; the corrected FDR is {\em consistent} with renormalization group expansions to first order in $4-d$ for the longitudinal fluctuations of the O(n) model in the limit $n\to\infty$.