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

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.