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 {\it ac} conductivity $\sigma (\omega )$ of a two-dimensional fermion gas subject to a smooth random potential (RP) or random magnetic field (RMF). We find a non-analytic $\propto \mathrm{\mid }\omega \mathrm{\mid }$ correction to $\mathrm{R}\mathrm{e}\sigma$, which corresponds to a $1\mathrm{/}{t}^2$ long-time tail in the velocity correlation function. This contribution is induced by return processes neglected in Boltzmann transport theory. The prefactor of this $\mathrm{\mid }\omega \mathrm{\mid }$-term is positive and proportional to $(d\mathrm{/}l{)}^2$ for RP, while it is of opposite sign and proportional to $d\mathrm{/}l$ in the weak RMF case, where $l$ is the mean free path and $d$ the disorder correlation length. This non-analytic 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.