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The electrodynamic response of ultra-pure materials at low temperaturesbecomes spatially non-local. This non-locality gives rise to phenomena such ashydrodynamic flow in transport and the anomalous skin effect in optics. Insystems characterized by an anisotropic electronic dispersion, the non-localdynamics becomes dependent on the relative orientation of the sample withrespect to the applied field, in ways that go beyond the usual, homogeneousresponse. Such orientational dependence should manifest itself not only intransport experiments, as recently observed, but also in optical spectroscopy.In this paper we develop a kinetic theory for the distribution function and thetransverse conductivity of two- and three-dimensional Fermi systems withanisotropic electronic dispersion. By expanding the collision integral into theeigenbasis of a collision operator, we include momentum-relaxing scattering aswell as momentum-conserving collisions. We examine the isotropic 2D case as areference, as well as anisotropic hexagonal and square Fermi-surface shapes. Weapply our theory to the quantitative calculation of the skin depth and thesurface impedance, in all regimes of skin effect. We find qualitativedifferences between the frequency dependence of the impedance in isotropic andanisotropic systems. Such differences are shown to persist even for morecomplex 2D Fermi surfaces, including the ''supercircle'' geometry and anexperimental parametrization for PdCoO$_2$, which deviate from an idealpolygonal shape. We study the orientational dependence of skin effect due toFermi-surface anisotropy, thus providing guidance for the experimental study ofnon-local optical effects.

Kinetic theory of the non-local electrodynamic response in anisotropic metals: skin effect in 2D systems