Renormalization Group Methods for Quantum Impurity Systems

T M Nghiem Hoa., A Costi Theo

Rong-Qiang He, Zhong-Yi Lu

One of the most important and accurate approaches for studying interacting many-electron correlated systems is the quantum renormalization group (RG) procedure, upon which we propose a concept in the framework of natural orbitals so that we can generalize the RG into general orbital space. To demonstrate its effectiveness, we apply this new approach on a quantum cluster-impurity model with four impurities in comparison with the exact diagonalization method. We also find a shortcut to dramatically improving this RG procedure. A profound impact on the cluster dynamical mean-field theory and multi-impurity/orbital Kondo problems is further discussed.

L. Nunes de Oliveira, Valter L. Líbero, H.O. Frota +1 more

M. Kolb

R. Bulla, Theo A. Costi, Thomas Pruschke

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.

Tao Yang, Z. Y. Xie, Rui Wang +1 more

Quantum impurities can host exotic many-body states that serve as sensitive probes of bath correlations. However, quantitative and non-perturbative methods for determining impurity thermodynamics in such settings remain scarce. Here, we introduce an impurity renormalization group approach that merges the tensor-network representation with the numerical renormalization group cutoff scheme. This method overcomes conventional limitations by treating bath correlations and impurity interactions on an equal footing. Applying our approach to the finite-temperature quantum critical regime of quantum spin systems, we uncover striking impurity-induced phenomena. In a coupled Heisenberg ladder, the impurity triggers a fractionalization of the local magnetic moment. Moreover, the derivative of the impurity susceptibility develops cusps that mark the crossover into the quantum critical regime. We also observe an exotic evolution of the spin correlation function driven by the interplay between bath correlations and the impurity. Our results demonstrate that this method can efficiently solve correlated systems with defects, opening new pathways to discovering novel impurity physics beyond those in non-interacting thermal baths.