Dynamic scaling behavior in the presence of a periodic magnetic driving across Ising continuous transitions

Abstract

We study the critical dynamics arising from a time-dependent periodic homogenous source coupled to the order-parameter field, which drives a classical ferromagnetic system across a continuous transition. For this purpose, we consider the paradigmatic two-dimensional (2D) Ising model in the presence of a periodic magnetic field $h(t)=-A\, \cos (2\pi t/P)$, evolving under a purely relaxational dynamics at the critical temperature. We show that the periodic driving gives rise to a peculiar dynamic scaling behavior in the thermodynamic limit, arising from a nontrivial interplay among the time $t$, the amplitude $A$ and period $P$ of $h(t)$. The relevant scaling variables are $\tau=t/P$ and $\sigma=A P^\kappa$, with $\kappa = y_h/z$, where $y_h=(d+2-\eta)/2$ is the critical dimension of the magnetic field, and $z$ is dynamic exponent for the critical relaxational dynamics ($\kappa\approx 0.865$ for the 2D Ising model). The dynamic scaling behaviors of the magnetization and bond-energy density show an oscillatory behavior around a smooth curve which approaches a large-$\tau$ stationary behavior. We also briefly discuss the dynamic behavior of an Ising system driven across the critical point by a periodic time-varying temperature at zero magnetic field.