  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.
