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 R 2/Γ of discrete symmetry Γ, 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, i.e., a Fraunhofer-like diagram of the Fermi sea, that reveals the point-group symmetries of the background lattice Γ 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 σ xx as a function of the isoperimetric parameter of the occupied orbifold domain by utilizing geometric measure theory is presented, satisfiying the scaling law σ xx proportional to 1/sqrt. The topological transport formula naturally incorporates the vanishing conductivity (σ xx to 0) of the previously described ``geometric insulators,'' promoting the isoperimetric parameter as an efficient descriptor for topological and geometric phases in conductors.