Geometric interpretation of the weak-field Hall conductivity in two-dimensional metals with arbitrary Fermi surface

The Hall conductivity ${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$ of a two-dimensional metal in the weak-field, semiclassical, limit has a simple geometric representation. ${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$ (normalized to ${\mathit{e}}^2$/h, where e is the electron charge and h is Planck’s constant), is equal to twice the number of flux quanta ${\mathrm{\phi }}_0$ threading the area ${\mathit{A}}_{\mathit{l}}$, where ${\mathit{A}}_{\mathit{l}}$ is the total ‘‘Stokes’’ area swept out by the scattering path length l(k) as k circumscribes the Fermi surface (FS). From this perspective, many properties of ${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$ become self-evident. The representation provides a powerful way to disentangle the distinct contributions of the three factors, FS area-to-circumference ratio, anisotropy in ${\mathit{l}}_{\mathbf{k}}$, and negative FS curvature. The analysis is applied to the Hall data on 2H-${\mathrm{NbSe}}_2$ and the cuprate perovskites. Previous model calculations of ${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$ are critically reexamined using the new representation.