Fermi-surface harmonics: A general method for nonspherical problems. Application to Boltzmann and Eliashberg equations

A set of functions called Fermi-surface harmonics (FSH's) are defined to be polynomials of the Cartesian components of the electronic velocity orthonormalized on the Fermi surface. These functions have many desirable properties, such as cell periodicity and a simple correspondence to spherical harmonics. However, there are many more linearly independent polynomials on a general surface than occur on a sphere. It is shown that this set is complete for simple Fermi surfaces; on general surfaces the mathematical question is not resolved but seems unlikely to cause physical difficulties. In the FSH representation, many problems take a particularly simple form. In particular the Boltzmann and Eliashberg equations are studied. By truncating at first-order polynomials, a slightly improved version of the usual variational solution for dc electrical conductivity is found. A convenient definition of Landau Fermi-liquid coefficients for antisotropic metals is suggested.