Path-integral formulation of backward and umklapp scattering for 1d spinless fermions

TL;DR

This paper develops a path-integral approach to model 1D spinless fermions with various scattering processes, deriving an effective bosonic action that captures collective excitations and auxiliary fields.

Contribution

It introduces a novel fermionic quantum field theory with non-local couplings for spinless fermions, providing a bosonic representation of the vacuum functional.

Findings

Derived an effective bosonic action with three scalar fields.

Identified two fields as physical collective excitations.

Presented a method to handle auxiliary fields via approximation.

Abstract

We present a (1+1)-dimensional fermionic QFT with non-local couplings between currents. This model describes an ensemble of spinless fermions interacting through forward, backward and umklapp scattering processes. We express the vacuum to vacuum functional in terms of a non trivial fermionic determinant. Using path-integral methods we find a bosonic representation for this determinant. Thus we obtain an effective action depending on three scalar fields, two of which correspond to the physical collective excitations whereas the third one is an auxiliary field that is left to be integrated by means of an approximate technique.