Discarded weight and entanglement spectra in the Numerical Renormalization Group

TL;DR

This paper introduces a quantitative criterion based on discarded weight and entanglement spectra to assess convergence and accuracy in the Numerical Renormalization Group (NRG), linking entanglement analysis with traditional energy flow diagrams.

Contribution

It presents a novel method to analyze NRG convergence using reduced density matrix spectra, connecting entanglement spectra with physical regimes.

Findings

Discarded weight effectively indicates numerical accuracy.

Entanglement spectra resemble energy flow diagrams.

Method provides site-specific accuracy assessment.

Abstract

A quantitative criterion to prove and analyze convergence within the numerical renormalization group (NRG) is introduced. By tracing out a few further NRG shells, the resulting reduced density matrices carry relevant information on numerical accuracy as well as entanglement. Their spectra can thus be analyzed twofold. The smallest eigenvalues provide a sensitive estimate of how much weight is discarded in the low energy description of later iterations. As such, the discarded weight is a clear indicator of the accuracy of a specific NRG calculation. In particular, it indicates in a site-specific manner whether sufficiently many states have been kept within a single NRG run. The largest eigenvalues of the reduced density matrices, on the other hand, lend themselves to a straightforward analysis in terms of entanglement spectra, which can be combined into entanglement flow diagrams. The latter show strong similarities with the well-known standard energy flow diagram of the NRG, supporting the prevalent usage of entanglement spectra to characterize different physical regimes.