We extend our study of hysteresis to the O(N) symmetric (${\mathrm{\Phi }}^2$${)}^3$ model in three dimensions with the dynamics of the nonconserved order parameter given by a Langevin equation. We analyze both thermal and magnetic hysteresis within this model. We study the systematics of the shapes and areas of hysteresis loops as functions of the amplitude and frequency of the applied field and other parameters in the theory. In the case of magnetic hysteresis we obtain pinched loops for a range of values in parameter space and demonstrate a scaling behavior of the area of the hysteresis loop with the amplitude ${\mathit{H}}_0$ of the magnetic field for low amplitudes: A≊${\mathit{H}}_0^{\mathrm{\alpha }}$, where the exponent α=2/3. This puts the magnetic hysteresis behavior of the (${\mathrm{\Phi }}^2$${)}^3$ model in the same universality class as that of the (${\mathrm{\Phi }}^2$${)}^2$ model. Thermal hysteresis, obtained by cycling the temperature in the presence of a small magnetic field, is characterized by asymmetric loops. We find that the area of the thermal hysteresis loops scales as a function of the amplitude of the periodic temperature (for low amplitudes): A≊${\mathit{r}}_0^{\mathrm{\alpha }}$, where α=1. we show that our study is relevant to the physics of ferroelectric materials and charge-density-wave systems. Our observations are consistent with existing experimental data on ferroelectrics and charge-density waves.

• Received 1 August 1990

DOI:https://doi.org/10.1103/PhysRevB.43.3373