Magnetic hysteresis in two model spin systems

A systematic study of hysteresis in model continuum and lattice spin systems is undertaken by constructing a statistical-mechanical theory wherein spatial fluctuations of the order parameter are incorporated. The theory is used to study the shapes and areas of the hysteresis loops as functions of the amplitude (${\mathit{H}}_{0}$) and frequency (\ensuremath{\Omega}) of the magnetic field. The response of the spin systems to a pulsed magnetic field is also studied. The continuum model that we study is a three-dimensional (${\mathrm{\ensuremath{\Phi}}}^{2}$${)}^{2}$ model with O(N) symmetry in the large-N limit. The dynamics of this model are specified by a Langevin equation. We find that the area A of the hysteresis loop scales as A\ensuremath{\sim}${\mathit{H}}_{0}^{0.66}$${\mathrm{\ensuremath{\Omega}}}^{0.33}$ for low values of the amplitude and frequency of the magnetic field. The hysteretic response of a two-dimensional, nearest-neighbor, ferromagnetic Ising model is studied by a Monte Carlo simulation on 10\ifmmode\times\else\texttimes\fi{}10, 20\ifmmode\times\else\texttimes\fi{}20, and 50\ifmmode\times\else\texttimes\fi{}50 lattices. The framework that we develop is compared with other theories of hysteresis. The relevance of these results to hysteresis in real magnets is discussed.