Reviews in Mathematical Physics

We construct, using fermionic functional integrals, thermodynamic Green's functions for a weakly coupled fermion gas whose Fermi energy lies in a gap. Estimates on the Green's functions are obtained that are characteristic of the size of the gap. This prepares the way for the analysis of single scale renormalization group maps for a system of fermions at temperature zero without a gap.

The first renormalization group map arising from the momentum space decomposition of a weakly coupled system of fermions at temperature zero differs from all subsequent maps. Namely, the component of momentum dual to temperature may be arbitrarily large — there is no ultraviolet cutoff. The methods of Part 1 are supplemented to control this special case.

The generic renormalization group map associated to a weakly coupled system of fermions at temperature zero is treated by supplementing the methods of Part 1. The interplay between position and momentum space is captured by "sectors". It is shown that the difference between the complete four-legged vertex and its "ladder" part is irrelevant for the sequence of renormalization group maps.

For a two-dimensional, weakly coupled system of fermions at temperature zero, one principal ingredient used to control the composition of the associated renormalization group maps is the careful counting of the number of quartets of sectors that are consistent with conservation of momentum. A similar counting argument is made to show that particle–particle ladders are irrelevant in the case of an asymmetric Fermi curve.