Numerical Renormalization Group at marginal spectral density: application to quantum tunneling in Luttinger liquids

TL;DR

This paper develops a Numerical Renormalization Group method tailored for systems with a marginal spectral density, enabling accurate low-energy analysis of quantum tunneling in Luttinger liquids with defects.

Contribution

It introduces a NRG approach for singular spectral densities, allowing non-perturbative analysis of quantum tunneling in Luttinger liquids with a defect.

Findings

Successfully applied NRG to compute dynamical conductance over multiple energy decades.

Demonstrated reliable low-energy physics capture despite high-energy scale violations.

Provided insights into impurity effects in Luttinger liquids.

Abstract

Many quantum mechanical problems (such as dissipative phase fluctuations in metallic and superconducting nanocircuits, or impurity scattering in Luttinger liquids) involve a continuum of bosonic modes with a marginal spectral density diverging as the inverse of energy. We construct a Numerical Renormalization Group in this singular case, with a manageable violation of scale separation at high energy, capturing reliably the low energy physics. The method is demonstrated by a non-perturbative solution over several energy decades for the dynamical conductance of a Luttinger liquid with a single static defect.