Kinetic Ising model in an oscillating field: Avrami theory for the hysteretic response and finite-size scaling for the dynamic phase transition

Hysteresis is studied for a two-dimensional, spin-$\frac{1}{2}$, nearest-neighbor, kinetic Ising ferromagnet in a sinusoidally oscillating field, using Monte Carlo simulations and analytical theory. Attention is focused on large systems and moderately strong field amplitudes at a temperature below $T_c.$ In this parameter regime, the magnetization switches through random nucleation and subsequent growth of many droplets of spins aligned with the applied field. Using a time-dependent extension of the Kolmogorov-Johnson-Mehl-Avrami theory of metastable decay, we analyze the statistical properties of the hysteresis-loop area and the correlation between the magnetization and the field. This analysis enables us to accurately predict the results of extensive Monte Carlo simulations. The average loop area exhibits an extremely slow approach to an asymptotic, logarithmic dependence on the product of the amplitude and the field frequency. This may explain the inconsistent exponent estimates reported in previous attempts to fit experimental and numerical data for the low-frequency behavior of this quantity to a power law. At higher frequencies we observe a dynamic phase transition. Applying standard finite-size scaling techniques from the theory of second-order equilibrium phase transitions to this nonequilibrium transition, we obtain estimates for the transition frequency and the critical exponents (β/ν≈0.11, γ/ν≈1.84, and ν≈1.1). In addition to their significance for the interpretation of recent experiments on switching in ferromagnetic and ferroelectric nanoparticles and thin films, our results provide evidence for the relevance of universality and finite-size scaling to dynamic phase transitions in spatially extended nonstationary systems.