Response of non-equilibrium systems at criticality: ferromagnetic models in dimension two and above

TLDR

The study investigates the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. The authors analyze the spherical model in any dimension D > 2 and perform numerical simulations of the two‑dimensional Ising model to probe these dynamics. They find that the systems age, with two‑time autocorrelation, response, and fluctuation‑dissipation ratio exhibiting non‑trivial scaling, and that the universal amplitude X∞ equals 1 − 2/D for 2 < D < 4, 1/2 for D > 4, and ≈0.26 for the 2D Ising model.

Abstract

We study the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. We show that these systems are aging in the long-time regime, i.e., their two-time autocorrelation and response functions and associated fluctuation-dissipation ratio are non-trivial scaling functions of both time variables. This is exemplified by the exact analysis of the spherical model in any dimension D>2, and by numerical simulations on the two-dimensional Ising model. We show in particular that, for $1\ll s$ (waiting time) $\ll t$ (observation time), the fluctuation-dissipation ratio possesses a non-trivial limit value $X_\infty$, which appears as a dimensionless amplitude ratio, and is therefore a novel universal characteristic of non-equilibrium critical dynamics. For the spherical model, we obtain $X_\infty=1-2/D$ for 2<D<4, and $X_\infty=1/2$ for D>4 (mean-field regime). For the two-dimensional Ising model we measure $X_\infty\approx0.26\pm0.01$.