Discarded weight and entanglement spectra in the Numerical Renormalization Group

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.

12 pages, 10 figures

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