Hall effect of noninteracting electrons as a Fermi-surface property: A rigorously derived gauge-independent formula in the on-shell form

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Abstract
A rigorously derived formula for the Hall conductivity of a one-electron Hamiltonian is obtained. The formula is written in terms of the correlation functions with four vertices. For degenerate electrons, the only relevant energy parameter is found to be the Fermi energy, which shows clearly that the Hall coefficient for independent electrons in a weak magnetic field is in general, determined only by states on the Fermi surface. When this formula is evaluated in the ladder approximation, we recover the result previously derived by Fukuyama et al. and by Itoh.

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