Bosonization rules in $1/2 +1$ dimensions

TL;DR

This paper derives bosonization rules for free fermions on a half-line with boundary conditions, analyzing boundary effects on fermionic currents, Green's functions, and boundary degrees of freedom using path-integral methods.

Contribution

It provides explicit bosonization rules for fermions with boundaries, including boundary effects and degrees of freedom, using path-integral and determinant calculations.

Findings

Boundary degrees of freedom do not modify bulk commutation relations.

Explicit bosonization rules incorporating boundary effects are derived.

Boundary influences on the Green's function's analytic structure are characterized.

Abstract

We derive the bosonization rules for free fermions on a half-line with physically sensible boundary conditions for Luttinger fermions. We use path-integral methods to calculate the bosonized fermionic currents on the half-line and derive their commutation relations for a system with a boundary. We compute the fermion determinant of the fermionic fluctuations for a system with a boundary using Forman's approach. We find that the degrees of freedom induced at the boundary do not to modify the commutation relations of the bulk. We give an explicit derivation of the bosonization rules for the fermion operators for a system with boundaries. We derive a set of bosonization rules for the Fermi operators which include the explicit effect of the boundaries and of boundary degrees of freedom. As a byproduct, we calculate the one-particle Green's function and determine the effects of the boundaries on its analytic structure.