Functional Bosonization of Non-Relativistic Fermions in $(2+1)$ Dimensions

TL;DR

This paper demonstrates that bosonization rules for non-relativistic fermions in (2+1)d are universal for linear and certain non-linear dispersions, mapping fermionic theories to gauge theories and analyzing collective excitations.

Contribution

It establishes the universality of bosonization rules in (2+1)d fermionic systems and applies the formalism to analyze collective modes in a non-local Thirring-like model.

Findings

Bosonization rules map fermionic density to magnetic flux and current to electric field.

In the large mass limit, the spectrum of collective excitations is exactly calculable.

Massless case exhibits no collective excitations, but current interactions can induce a gapless mode.

Abstract

We analyze the universality of the bosonization rules in non-relativistic fermionic systems in $(2+1)d$. We show that, in the case of linear fermionic dispersion relations, a general fermionic theory can be mapped into a gauge theory in such a way that the fermionic density maps into a magnetic flux and the fermionic current maps into a transverse electric field. These are universal rules in the sense that they remain valid whatever the interaction considered. We also show that these rules are universal in the case of non-linear dispersion relations provided we consider only density-density interactions. We apply the functional bosonization formalism to a non-relativistic and non-local massive Thirring-like model and evaluate the spectrum of collective excitations in several limits. In the large mass limit, we are able to exactly calculate this spectrum for arbitrary density-density and current-current interactions. We also analyze the massless case and show that it has no collective excitations for any density-density potential in the Gaussian approximation. Moreover, the presence of current interactions may induce a gapless mode with a linear dispersion relation.