A General Approach to Bosonization

TL;DR

This paper introduces a comprehensive formalism for bosonization in one dimension, utilizing a new 'singular complex number' concept, enabling the calculation of Green functions for various interaction regimes in electron gases.

Contribution

The authors develop a general bosonization formula using singular complex numbers, extending the method to finite temperatures and overcoming limitations of traditional approaches.

Findings

Derived a new formula for the field operator in 1D bosonization.

Computed Green functions for short- and long-range interactions.

Showed the formalism handles essential singularities better than traditional methods.

Abstract

We summarize recent developments in the field of higher dimensional bosonization made by the authors and collaborators and propose a general formula for the field operator in terms of currents and densities in one dimension using a new ingredient known as a `singular complex number'. Using this formalism, we compute the Green function of the homogeneous electron gas in one spatial dimension with short-range interaction leading to the Luttinger liquid and also with long-range interactions that leads to a Wigner crystal whose momentum distribution computed recently exhibits essential singularities. We generalize the formalism to finite temperature by combining with the author's hydrodynamic approach. The one-particle Green function of this system with essential singularities cannot be easily computed using the traditional approach to bosonization which involves the introduction of momentum cutoffs, hence the more general approach of the present formalism is proposed as a suitable alternative.