Reduction formula for fermion loops and density correlations of the 1D Fermi gas

TL;DR

This paper derives a reduction formula for fermion loops with multiple density vertices and demonstrates that symmetrized N-loops in a 1D Fermi gas do not diverge at low energies, providing explicit scaling behaviors.

Contribution

It introduces a reduction formula for fermion loops with many density vertices and applies it to derive non-divergent symmetrized N-loops in 1D Fermi gases.

Findings

Symmetrized N-loops are finite at low energies and momenta.

Explicit scaling laws for N-loops in infrared limits.

Reduction formula simplifies analysis of density correlations.

Abstract

Fermion N-loops with an arbitrary number of density vertices N > d+1 in d spatial dimensions can be expressed as a linear combination of (d+1)-loops with coefficients that are rational functions of external momentum and energy variables. A theorem on symmetrized products then implies that divergencies of single loops for low energy and small momenta cancel each other when loops with permuted external variables are summed. We apply these results to the one-dimensional Fermi gas, where an explicit formula for arbitrary N-loops can be derived. The symmetrized N-loop, which describes the dynamical N-point density correlations of the 1D Fermi gas, does not diverge for low energies and small momenta. We derive the precise scaling behavior of the symmetrized N-loop in various important infrared limits.