Numerical Renormalization Group for the sub-ohmic spin-boson model: A conspiracy of errors

TL;DR

This paper critically examines the application of Numerical Renormalization Group (NRG) to the sub-ohmic spin-boson model, revealing how two specific errors can produce misleadingly consistent critical exponents that do not reflect the true physics.

Contribution

It identifies and explains how Hilbert-space truncation and mass-flow errors in NRG lead to incorrect conclusions about quantum critical behavior in the sub-ohmic spin-boson model.

Findings

NRG's critical exponents are affected by two key errors.

These errors cause artificial non-analytic terms in the free energy.

The errors lead to misleadingly consistent scaling laws.

Abstract

The application of Wilson's Numerical Renormalization Group (NRG) method to dissipative quantum impurity models, in particular the sub-ohmic spin-boson model, has led to conclusions regarding the quantum critical behavior which are in disagreement with those from other methods and which are by now recognized as erroneous. The errors of NRG remained initially undetected because NRG delivered an internally consistent set of critical exponents satisfying hyperscaling. Here we discuss how the conspiracy of two errors - the Hilbert-space truncation error and the mass-flow error - could lead to this consistent set of exponents. Remarkably, both errors, albeit of different origin, force the system to obey naive scaling laws even when the physical model violates naive scaling. In particular, we show that a combination of the Hilbert-space truncation and mass-flow errors induce an artificial non-analytic term in the Landau expansion of the free energy which dominates the critical behavior for bath exponents s<1/2.