Tensor networks and the numerical renormalization group

TL;DR

This paper discusses the integration of tensor network methods with the full-density-matrix numerical renormalization group (NRG) to improve calculations of thermodynamic quantities and spectral functions across temperatures.

Contribution

It introduces a tensor network framework for NRG, enhancing the understanding and implementation of the full-density-matrix approach in thermodynamic and spectral calculations.

Findings

Tensor network formalism simplifies NRG algorithms.

Enhanced accuracy in thermodynamic quantity calculations.

Unified framework for spectral and time-dependent NRG methods.

Abstract

The full-density-matrix numerical renormalization group (NRG) has evolved as a systematic and transparent setting for the cal- culation of thermodynamical quantities at arbitrary temperatures within the NRG framework. It directly evaluates the relevant Lehmann representations based on the complete basis sets intro- duced by Anders and Schiller (2005). In addition, specific attention is given to the possible feedback from low energy physics to high energies by the explicit and careful construction of the full thermal density matrix, naturally generated over a distribution of energy shells. Specific examples are given in terms of spectral functions (fdmNRG), time-dependent NRG (tdmNRG), Fermi-Golden rule calculations (fgrNRG), as well as the calculation of plain thermodynamic expectation values. Furthermore, based on the very fact that, by its iterative nature, the NRG eigenstates are naturally described in terms of matrix product states, the language of tensor networks has proven enormously convenient in the description of the underlying algorithmic procedures. This paper therefore also provides a detailed introduction and discussion of the prototypical NRG calculations in terms of their corresponding tensor networks.