Exact Green's functions and Bosonization of a Luttinger liquid coupled to impedances

TL;DR

This paper derives exact Green's functions for a finite-size Luttinger Liquid connected to impedances, proving bosonization under impedance boundary conditions and exploring limits to open boundaries and infinite impedance cases.

Contribution

It provides a rigorous proof of bosonization with impedance boundary conditions and explicitly connects finite and infinite Luttinger liquids through impedance limits.

Findings

Exact Green's functions for finite-size LL with impedances

Bosonization holds under impedance boundary conditions

Finite LL with impedance equal to characteristic impedance behaves like an infinite LL

Abstract

The exact Green's functions of a finite-size Luttinger Liquid (LL) connected to impedances are computed at zero and finite temperature. Bosonization for a LL with Impedance boundary conditions (IBC) is proven to hold. The LL with open boundary conditions (for both Neumann and Dirichlet cases) is explicitly recovered as a special limit when one has infinite impedances. Additionally when the impedances are equal to the characteristic impedance of the Luttinger liquid then the finite Luttinger liquid is shown to be effectively equivalent to an infinite Luttinger liquid.