Bosonization and the eikonal expansion: similarities and differences

TL;DR

This paper compares bosonization and eikonal expansion techniques for calculating Green's functions in interacting Fermi systems, highlighting their similarities in Fermi liquids and differences in non-Fermi liquids like the Tomonaga-Luttinger model.

Contribution

It demonstrates that both methods agree for Fermi liquids but reveals limitations of the eikonal expansion in capturing non-Fermi liquid behavior accurately.

Findings

Both methods produce identical quasi-particle residue for Fermi liquids.

The eikonal method fails to reproduce correct spectral function scaling in non-Fermi liquids.

Eikonal expansion does not predict correct singularities in the Tomonaga-Luttinger model.

Abstract

We compare two non-perturbative techniques for calculating the single-particle Green's function of interacting Fermi systems with dominant forward scattering: our recently developed functional integral approach to bosonization in arbitrary dimensions, and the eikonal expansion. In both methods the Green's function is first calculated for a fixed configuration of a background field, and then averaged with respect to a suitably defined effective action. We show that, after linearization of the energy dispersion at the Fermi surface, both methods yield for Fermi liquids exactly the same non-perturbative expression for the quasi-particle residue. However, in the case of non-Fermi liquid behavior the low-energy behavior of the Green's function predicted by the eikonal method can be erroneous. In particular, for the Tomonaga-Luttinger model the eikonal method neither reproduces the correct scaling behavior of the spectral function, nor predicts the correct location of its singularities.