A dynamical method is introduced to study the effect of dimensionality on phase transitions. Direct experimental measurements for the lower critical dimension for spin glasses is provided as an example. The method makes use of the spin glass correlation length $\xi (t,T)$. Once nucleated, it can become comparable to sample dimensions in convenient time and temperature ranges. Thin films of amorphous Ge:Mn alloys were prepared with thickness $\mathcal{L}\approx 15.5\mathrm{nm}$. Conventional behavior is observed as long as $\xi (t,T)<\mathcal{L}$. At the measurement time $t_{\text{co}}$, when $\xi (t_{\text{co}},T)\approx \mathcal{L}$, the time dependence is observed to cross over to exponential. These results are interpreted using spin glass dynamics, and are consistent with a lower critical dimension for spin glasses, $d_{\ell }$, between $2<d_{\ell }<3$.

• Received 20 November 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.126401