Numerical renormalization group method for quantum impurity systems

TLDR

Wilson’s 1970s nonperturbative renormalization group concept was later generalized into the numerical renormalization group (NRG) method, which now applies to a wide range of quantum impurity problems. This review introduces the NRG method, offers calculation guidelines, and surveys its 30‑year development and applications. NRG is applied to variants of the Kondo problem, dissipative quantum systems such as the spin‑boson model, and lattice systems within dynamical mean‑field theory. NRG first revealed the full crossover from a high‑temperature free spin to a low‑temperature fully screened spin in the Kondo problem.

Abstract

In the early 1970s, Wilson developed the concept of a fully nonperturbative renormalization group transformation. When applied to the Kondo problem, this numerical renormalization group (NRG) method gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG method was later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method, including some guidelines for calculating physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi-liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean-field theory.