Anisotropic Friedel oscillations and isoperimetric transport in two-dimensional free lattice Fermi seas

Two physical consequences of the recent topological classification of 2d Fermi seas for free particles based on flat orbifolds ${\mathbb{R}}^2\mathrm{/}\mathrm{\Gamma }$ of discrete symmetry $\mathrm{\Gamma }$, are explored. The first is an study of the spatial density behavior inside the mesoscopic window enclosed by the Fermi and (infinite) thermal lengths. It is shown that the single-particle density matrix exhibits a sharp directional anisotropy, encoding a real space diffraction pattern, {\it i.e.}, a Fraunhofer-like diagram of the Fermi sea, that reveals the point-group symmetries of the background lattice $\mathrm{\Gamma }$ in the form of anisotropic Friedel oscillations. As examples, the special cases of the $p4m$ and $p6m$ crystallographic flat orbifolds are treated in some detail. The second physical consequence is the establishment of an explicit geometric correspondence between momentum-space packing and electrical transport. A mean-field expression for the longitudinal electrical conductivity ${\sigma }_{xx}$ as a function of the {\it isoperimetric parameter} $\mathcal{I}$ of the occupied orbifold domain by utilizing geometric measure theory is presented, satisfiying the scaling law ${\sigma }_{xx}\propto 1\mathrm{/}\sqrt{\mathcal{I}}$. The topological transport formula naturally incorporates the vanishing conductivity (${\sigma }_{xx}\to 0$) of the previously described ``geometric insulators,‘’ promoting the isoperimetric parameter as an efficient descriptor for topological and geometric phases in conductors.