A Numerical Renormalization Group approach to Green's Functions for Quantum Impurity Models

We present a method for the calculation of dynamical correlation functions of quantum impurity systems out of equilibrium using Wilson's numerical renormalization group. Our formulation is based on a complete basis set of the Wilson chain and embeds the recently derived algorithm for equilibrium spectral functions. Our method fulfills the spectral weight conserving sum-rule exactly by construction. A local Coulomb repulsion $U>0$ is switched on at $t=0$, and the asymptotic st...

By combining Wilson's numerical renormalization group with a modified Bloch-Redfield approach we are able to eliminate the artificial broadening of the Lehmann representation of quantum impurity spectral functions required by the standard numerical renormalization group algorithm. Our approach is based on the exact reproduction of the continuous coupling function in the original quantum impurity model. It augments each chain site of the Wilson chain by a coupling to an additi...

In this paper we introduce a new approach for calculating dynamical properties within the numerical renormalization group. It is demonstrated that the method previously used fails for the Anderson impurity in a magnetic field due to the absence of energy scale separation. The problem is solved by evaluating the Green function with respect to the reduced density matrix of the full system, leading to accurate spectra in agreement with the static magnetization. The new procedure...

For a given quantum impurity model, Wilson's numerical renormalization group (NRG) naturally defines a NRG Hamiltonian whose exact eigenstates and eigenenergies are obtainable. We give exact expressions for the free energy, static, as well as dynamical quantities of the NRG Hamiltonian. The dynamical spectral function from this approach contains full excitations including intra- and inter-shell excitations. For the spin-boson model, we compare the spectral function obtained f...

We show how spectral functions for quantum impurity models can be calculated very accurately using a complete set of ``discarded'' numerical renormalization group eigenstates, recently introduced by Anders and Schiller. The only approximation is to judiciously exploit energy scale separation. Our derivation avoids both the overcounting ambiguities and the single-shell approximation for the equilibrium density matrix prevalent in current methods, ensuring that relevant sum rul...

The continuous coupling function in quantum impurity problems is exactly partitioned into a part represented by a finite size Wilson chain and a part represented by a set of additional reservoirs, each coupled to one Wilson chain site. These additional reservoirs represent high-energy modes of the environment neglected by the numerical renormalization group and are required to restore the continuum limit of the original problem. We present a hybrid time-dependent numerical re...

We present a technique for calculating non-equilibrium Green functions for impurity systems with local interactions. We use an analogy to the calculation of response functions in the x-ray problem.The initial state and the final state problems, which correspond to the situations before and after the disturbance (an electric or magnetic field, for example) is suddenly switched on, are solved with the aid of Wilson's momentum shell renormalization group. The method is illustrat...

In the beginning of the 1970's, Wilson developed the concept of a fully non-perturbative renormalization group transformation. Applied to the Kondo problem, this numerical renormalization group method (NRG) 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 has been later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief...

We present a unified framework for renormalization group methods, including Wilson's numerical renormalization group (NRG) and White's density-matrix renormalization group (DMRG), within the language of matrix product states. This allows improvements over Wilson's NRG for quantum impurity models, as we illustrate for the one-channel Kondo model. Moreover, we use a variational method for evaluating Green's functions. The proposed method is more flexible in its description of s...

The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required ...