Tunneling through two resonant levels: fixed points and conductances

TL;DR

This paper analyzes quantum tunneling through two resonant levels in a Tomonaga-Luttinger liquid, revealing how asymmetry affects conductance and identifying a unique fixed point with distinct conductance properties.

Contribution

It introduces a mapping to a generalized Coulomb model to study the effects of asymmetry and identifies a non-trivial fixed point in the symmetric case.

Findings

Asymmetry in tunneling amplitudes leads to vanishing conductance at low temperatures.

A non-trivial fixed point exists with conductance different from a single resonant level.

Symmetric tunneling case exhibits unique fixed point behavior.

Abstract

We study point contact tunneling between two leads of a Tomonaga-Luttinger liquid through two degenerate resonant levels in parallel. This is one of the simplest cases of a quantum junction problem where the Fermi statistics of the electrons plays a non-trivial role through the Klein factors appearing in bosonization. Using a mapping to a `generalized Coulomb model' studied in the context of the dissipative Hofstadter model, we find that any asymmetry in the tunneling amplitudes from the two leads grows at low temperatures, so that ultimately there is no conductance across the system. For the symmetric case, we identify a non-trivial fixed point of this model; the conductance at that point is generally different from the conductance through a single resonant level.