Scaling of energy barriers for flux lines and other random systems

Using a combination of analytic arguments and numerical simulations, we determine lower and upper bounds for the energy barriers to the motion of a defect line in a random potential at low temperatures. We study the cases of magnetic flux lines in high-${\mathit{T}}_{\mathit{c}}$ superconductors in two and three dimensions, and of domain walls in two-dimensional random-field Ising models. The results show that, under fairly general conditions, energy barriers have the same scaling as the fluctuations in free energy, except for possible logarithmic factors. This holds not only for barriers between optimal configurations of the line, but also for barriers separating any metastable configuration from a configuration of minimal energy. Similar arguments may be applicable to other elastic media with impurities, such as bunches of flux lines.