Zero-frequency anomaly in quasiclassical ac transport: Memory effects in a two-dimensional metal with a long-range random potential or random magnetic field

We study the low-frequency behavior of the ac conductivity $\sigma \left(\omega \right)$ of a two-dimensional fermion gas subject to a smooth random potential (RP) or random magnetic field (RMF). We find a nonanalytic $\propto |\omega |$ correction to $\mathrm{Re}\sigma ,$ which corresponds to a ${1/t}^2$ long-time tail in the velocity correlation function. This contribution is induced by return processes neglected in Boltzmann transport theory. The prefactor of this $|\omega |$ term is positive and proportional to ${\mathrm{}(d/l)\mathrm{}}^2$ for the RP, while it is of opposite sign and proportional to $d/l$ in the weak RMF case, where l is the mean free path and d the disorder correlation length. This nonanalytic correction also exists in the strong RMF regime, when the transport is of a percolating nature. The analytical results are supported and complemented by numerical simulations.