Scaling of hysteresis dispersion in a model spin system

We present a calculation of the magnetic hysteresis and its area for a model continuum spin system based on three-dimensional (Φ²)² model with $O(N)$ symmetry in the limit N→∞, under a time-varying magnetic field. The frequency dependence of the hysteresis area $A(f),$ namely, hysteresis dispersion, is investigated in detail, predicting a single-peak profile which grows upwards and shifts rightwards gradually with increasing field amplitude H₀. We demonstrate that the hysteresis dispersion $A(f)$ over a wide range of H₀ can be scaled by scaling function W(η)∝τ₁A(f,H₀), where η=log₁₀(fτ₁) and τ₁ is the unique characteristic time for the spin reverse, as long as H₀ is not very small. The inverse characteristic time τ₁^-1 shows a linear dependence on amplitude H₀, supported by the well-established empirical relations for ferromagnetic ferrites and ferroelectric solids. This scaling behavior suggests that the hysteresis dispersion can be uniquely described by the characteristic time for the spin reversal once the scaling function is available.