Magnetic hysteresis loops as Lissajous plots of relaxationally delayed response to periodic field variation

We have studied the nature of the response function (magnetisation) arising in response to a periodic perturbation (magnetic field) on Ising systems, using the Monte Carlo simulation (for a three dimensional Ising system) and also solving the mean field equation of Ising dynamics. The hysteresis loop is identified here as the Lissajous plot of the response function versus the perturbing field. It is observed that the response function is always periodic with the same frequency as that of the perturbing field and gets delayed in general due to relaxation. This equality in the frequency gives rise to the quadratic equation of the Lissajous figures (double valuedness and stability of magnetisation in the hysteresis loop) and the delay gives rise to the nonvanishing width of the Lissajous figures (nonvanishing hysteresis loss). We have also studied the scaling behaviour for the hysteresis loop area A, obtained from the mean field solution of the Ising dynamics and from the Monte Carlo study of a three dimensional Ising system. The scaling function is observed to be Lorentzian in form in the mean field case, while the scaling function is an exponentially decaying one multiplied by a power law function, in the case of (Monte Carlo results for) the three dimensional Ising systems. Additionally, we have studied the dynamic phase transition for the three dimensional Ising system and the tricritical point in the phase diagram has been located.