Experimental Determination of Spin Glass Lower Critical Dimension

Zero field cooled (ZFC) measurements on thin film Ge:Mn spin glass can explore the lower critical dimension $d_l$. The correlation length $\xi(t, T)$ is nucleated upon a rapid quench into the spin glass phase, and grows to the thickness of the film, $L$, resulting in a transition for dynamics from $d=3$ to $d=2$ at a crossover time $t_{co}$. Our experiments demonstrate that conventional ZFC dynamics vanish at $t=t_{co}$, but there remain spins within a length scale $\leq L$ for which $d=3$ dynamics remain. Because of the ultrametric distribution of states, the rise of the remaining ZFC magnetization exhibits an exponential time dependence determined by the highest barrier surmounted at $t_{co}$, $\Delta_{\rm max}(t_{co}, T)$. By carefully choosing a temperature region where the dynamics fall within experimental time scales, both regimes are observed. Further, there is a direct relationship between the magnitude of $\xi(t_{co}, T)$ and $\Delta_{\rm max}({t_{co}}, T)$. This relationship is satisfied, determining the parameters controlling the growth of $\xi(t, T)$ without arbitrary parameters. The existence of the crossover establishes that $2 < d_l < 3$ for spin glass dynamics, in agreement with theory for Ising (Franz {\it et al.}) and Heisenberg (Lee and Young) spin glasses.