Spin-glass dynamics with conserved magnetization

The dynamical response of a spin-glass is studied at the mean-field level for models in which the total magnetization is conserved. The first model considered is totally dissipative, like the conventional relaxational ones, but the Langevin equation is diffusive. In the $k\to 0$ limit, the diffusion constant is unaffected by the proximity of the spin-glass transition, but the region of $k$ space in which hydrodynamics is valid shrinks to the origin as $T\to T_g$. In the rest of $k$ space, the dynamics are effectively the same as in the relaxational model, with a relaxation rate $\propto T-T_g$. At $T_g$, spin correlations have the $t^{-\frac{1}{2}}$ behavior of the relaxational model. Mode coupling is added in the second model, and a self-consistent calculation gives a transport coefficient $\propto {\left[\ln \right(T-T_g\left)\right]}^{\frac{1}{2}}$ [and $\propto {\left(\ln \omega \right)}^{\frac{1}{2}} \mathrm{at} T_g$]. Sound propagation is also examined, and the sound-damping rate is found to have similar logarithmic behavior. The sound speed varies smoothly through $T_g$. Below $T_g$ we find new ${\omega }^{-\frac{1}{3}}$ singularities in the transport coefficient, spin-correlation function, and sound damping.