Residual Energies after Slow Cooling of Disordered Systems

The residual energy, $\epsilon \left(\tau \right)$, left after cooling to zero temperature in a finite time $\tau$ is analyzed for various disordered systems, including spin-glasses and random-field magnets. We argue that the generic behavior for such frustrated systems is $\epsilon \left(\tau \right)\sim {\left(\ln \tau \right)}^{-\zeta }$ for large $\tau$, with the exponent $\zeta$ depending on the system. This result is dominated in some cases by a distribution of classical two-level systems with low excitation energies, and in other cases by large-scale nonequilibrium effects.