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 H0ex0ex=0ex0ex ̇ \.Ht, where ̇ \.H is the sweeping rate. Novel dynamic scaling forms for the magnetization, the structure factor, and the area of hysteresis loop are $M({ ̇ \.{}{}}{H},t,T){{0ex}{0ex}}={{0ex}{0ex}}f({{ ̇ \.{}{}}{H}}1/2t,{{ ̇ \.{}{}}{H}}^{(d{-}2)/4}T)$, $C(k,{ ̇ \.{}{}}{H},t,T){{0ex}{0ex}}={{0ex}{0ex}}{{ ̇ \.{}{}}{H}}^{{-}d/4}{f}^{{'}}({{ ̇ \.{}{}}{H}}^{{-}1/4}k,{{ ̇ \.{}{}}{H}}1/2t,{{ ̇ \.{}{}}{H}}^{(d{-}2)/4}T)$, and $A{{0ex}{0ex}}={{0ex}{0ex}}{{ ̇ \.{}{}}{H}}1/2g({{ ̇ \.{}{}}{H}}^{(d{-}2)/4}T)$, respectively, where $T$ is the temperature of the system, $d$ the spatial dimensionality, $k$ the wave number, and $f$, ${f}^{{'}}$, and $g$ scaling functions. These results show that the rate of the external driving field can serve as a scaling parameter to study hysteresis.