Duality of Weak and Strong Scatterer in Luttinger liquid Coupled to Massless Bosons

TL;DR

This paper investigates how coupling a Luttinger liquid with massless bosons affects impurity scattering, revealing a preserved duality relation and complex phase behavior including metal-insulator transitions.

Contribution

It demonstrates that the duality between weak scatterer and weak link persists with boson coupling, and uncovers a rich phase diagram with potential metal-insulator transitions.

Findings

Duality relation $\Delta_{ws}\Delta_{wl}=1$ holds with boson coupling.

System remains either an ideal insulator or metal at low temperatures.

Correlated fermion and boson scattering lead to a metal-insulator transition.

Abstract

We study electronic transport in a Luttinger liquid (LL) with an embedded impurity, which is either a weak scatterer (WS) or a weak link (WL), when interacting electrons are coupled to one-dimensional massless bosons (e.g., acoustic phonons). We find that the duality relation, $\Delta_{ws}\Delta_{wl}=1 $, between scaling dimensions of the electron backscattering in the WS and WL limits, established for the standard LL, holds in the presence of the additional coupling for an arbitrary fixed strength of boson scattering from the impurity. This means that at low temperatures such a system remains either an ideal insulator or an ideal metal, regardless of the scattering strength. On the other hand, when fermion and boson scattering from the impurity are correlated, the system has a rich phase diagram that includes a metal-insulator transition at some intermediate values of the scattering.