The mass-flow error in the Numerical Renormalization Group method and the critical behavior of the sub-ohmic spin-boson model

TL;DR

This paper identifies a systematic error in the Numerical Renormalization Group method affecting quantum criticality studies, demonstrates its impact on the spin-boson model, and proposes an extension to correct it, revealing expected mean-field critical behavior.

Contribution

The paper uncovers a mass-flow error in NRG calculations and introduces a correction method, enabling accurate analysis of quantum critical phenomena in impurity models.

Findings

Mass-flow error causes unphysical temperature dependence of the order parameter.

Corrected NRG reveals Gaussian critical fixed point for s<1/2 in the spin-boson model.

Results align with quantum-to-classical mapping predictions.

Abstract

We discuss a particular source of error in the Numerical Renormalization Group (NRG) method for quantum impurity problems, which is related to a renormalization of impurity parameters due to the bath propagator. At any step of the NRG calculation, this renormalization is only partially taken into account, leading to systematic variation of the impurity parameters along the flow. This effect can cause qualitatively incorrect results when studying quantum critical phenomena, as it leads to an implicit variation of the phase transition's control parameter as function of the temperature and thus to an unphysical temperature dependence of the order-parameter mass. We demonstrate the mass-flow effect for bosonic impurity models with a power law bath spectrum, J(w) ~ w^s, namely the dissipative harmonic oscillator and the spin-boson model. We propose an extension of the NRG to correct the mass-flow error. Using this, we find unambiguous signatures of a Gaussian critical fixed point in the spin-boson model for s<1/2, consistent with mean-field behavior as expected from quantum-to-classical mapping.