Dynamics of spins interacting with quenched random impurities

The critical dynamics of the time-dependent Ginzburg-Landau model for a system with quenched random impurities and nonconserved order parameter is studied in the framework of the $\epsilon$ expansion. In contrast to the situation in pure systems, the dynamic critical exponent $z$ deviates from its conventional value at first order in $\epsilon \equiv 4-d$. The impurities cause an enhancement of the shape function $f_x\left(\nu \right)$ at small frequencies $\nu$; $f_x(\nu =0)$ diverges as $T\to T_c$. Below $T_c$ the equation of state, static susceptibility $\chi$, and dynamic response function $G$, are studied. A new, purely static correlated function, $C^{\mathrm{}\left(s\right)\mathrm{}}$, whose existence is unique to the random system is introduced. The coexistence curve singularities of $C^{\mathrm{}\left(s\right)\mathrm{}}$, $\chi$, and $G$ in systems with continuously broken symmetry are explored. The connection of the quenched-impurity model with "model $C$" of Halperin, Hohenberg, and Ma is discussed.