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arXiv:cond-mat/0607494 · Physical Review B

Numerical renormalization group approach to Green’s functions for quantum impurity models

Robert Peters, Thomas Pruschke, Frithjof B. Anders

Published Dec 18, 2006• 254 citations
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Abstract

We present a technique for the calculation of dynamical correlation functions of quantum impurity systems in equilibrium with Wilson's numerical renormalization group. Our formulation is based on a complete basis set of the Wilson chain. In contrast to all previous methods, it does not suffer from overcounting of excitations. By construction, it always fulfills sum rules for spectral functions. Furthermore, it accurately reproduces local thermodynamic expectation values, such as occupancy and magnetization, obtained directly from the numerical renormalization group calculations.

DOI
10.1103/physrevb.74.245114

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We present a technique for the calculation of dynamical correlation functions of quantum impurity systems in equilibrium with Wilson's numerical renormalization group. Our formulation is based on a complete basis set of the Wilson chain. In contrast to all previous methods, it does not suffer from overcounting of excitations. By construction, it always fulfills sum rules for spectral functions. Furthermore, it accurately reproduces local thermodynamic expectation values, such as occupancy and magnetization, obtained directly from the numerical renormalization group calculations.
We present a method for the calculation of dynamical correlation functions of quantum impurity systems out of equilibrium using Wilson’s numerical renormalization group (NRG). 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 fulfils the spectral weight conserving sum-rule exactly by construction. A local Coulomb repulsion U >0 is switched on at t = 0, and the asymptotic steady-state spectral functions are obtained for various values of U as well as magnetic field strength H and temperature T . These benchmark tests show excellent agreement between the time-evolved and the directly calculated equilibrium NRG spectra for finite U . This method could be used for calculating steady-state non-equilibrium spectral functions at finite bias through interacting nanodevices.
By combining Wilson's numerical renormalization group (NRG) with a modified Bloch-Redfield approach (BRA), we are able to eliminate the artificial broadening of the Lehmann representation of quantum impurity spectral functions required by the standard NRG 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 additional reservoir. This open Wilson chain is constructed by a continuous fraction expansion, and the coupling function is treated in second order in the context of the BRA. The eigenvalues of the resulting Bloch-Redfield tensor generate a finite lifetime of the NRG excitations that leads to a natural broadening of the spectral functions. We combine this approach with z-averaging and an analytical exact expression for the correlation part of the self-energy to obtain an accurate representation of the spectral function of the original continuum model in the absence and presence of an external magnetic field.
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 intershell excitations. For the spin-boson model, we compare the spectral function obtained from the present method and the full density matrix (FDM) method, showing that while both guarantee rigorous sum rule, the full excitation approach avoids the causality problem of FDM method.
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 rules hold rigorously and spectral features at energies below the temperature can be described accurately.
We present a technique for calculating nonequilibrium 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 illustrated by calculating the nonequilibrium dynamics of the Ohmic two--state problem.
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 provides a unifying framework for calculating dynamics at any temperature and represents the correct extension of Wilson's original thermodynamic calculation.