Fermion loops, loop cancellation and density correlations in two dimensional Fermi systems

TL;DR

This paper derives explicit formulas for fermion loops with multiple density vertices in two-dimensional Fermi systems, showing how divergences cancel and the symmetrized loops behave at low energies and momenta.

Contribution

It provides explicit expressions for fermion loops with arbitrary vertices and demonstrates divergence cancellation in symmetrized density correlation functions.

Findings

3-loop is an elementary function of external variables

Divergences cancel when summing permuted loops

Symmetrized N-loop vanishes as the (2N-2)-th power in the dynamical limit

Abstract

We derive explicit results for fermion loops with an arbitrary number of density vertices in two dimensions at zero temperature. The 3-loop is an elementary function of the three external momenta and frequencies, and the N-loop can be expressed as a linear combination of 3-loops with coefficients that are rational functions of momenta and frequencies. We show that the divergencies of single loops for low energy and small momenta cancel each other when loops with permuted external variables are summed. The symmetrized N-loop, i.e. the connected N-point density correlation function of the Fermi gas, does not diverge for low energies and small momenta. In the dynamical limit, where momenta scale to zero at fixed finite energy variables, the symmetrized N-loop vanishes as the (2N-2)-th power of the scale parameter.