Renormalization Group Theory of Hysteresis

We apply renormalization group theory directly to the first-order phase transition of the large-N model driven linearly by an external magnetic field $H=\dot{H}t$, where $\dot{H}$ is the sweeping rate. Novel dynamic scaling forms for the magnetization, the structure factor, and the area of hysteresis loop are $M(\dot{H},t,T)=f({\dot{H}}^{1/2}t,{\dot{H}}^{(d-2\mathrm{})/4\mathrm{}}T)$, $C(k,\dot{H},t,T)={\dot{H}}^{-d/4\mathrm{}}{f}^{\prime }({\dot{H}}^{-1/4\mathrm{}}k,{\dot{H}}^{1/2}t,{\dot{H}}^{\left(d-2\mathrm{}\right)/4\mathrm{}}T)$, and $A={\dot{H}}^{1/2}g\left({\dot{H}}^{\left(d-2\mathrm{}\right)/4\mathrm{}}T\right)$, respectively, where $T$ is the temperature of the system, $d$ the spatial dimensionality, $k$ the wave number, and $f$, $f^{\prime }$, and $g$ scaling functions. These results show that the rate of the external driving field can serve as a scaling parameter to study hysteresis.