Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0607497 · Physical Review Letters

Sum-Rule Conserving Spectral Functions from the Numerical Renormalization Group

Andreas Weichselbaum, Jan von Delft

Published Aug 16, 2007• 370 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.99.076402

Related papers

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

Robert Peters, Thomas Pruschke, Frithjof B. Anders

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.

arXiv:cond-mat/0607494•Physical Review B•Dec 18, 2006•254 citations

A numerical renormalization group approach to non-equilibrium Green functions for quantum impurity models

Frithjof B. Anders

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.

arXiv:0803.3004•Journal of Physics Condensed Matter•Apr 11, 2008•42 citations

Dynamical spectral function from numerical renormalization group: A full excitation approach

Ke Yang, Ning-Hua Tong

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.

arXiv:2006.16488•Physical review. B./Physical review. B•Aug 13, 2020•3 citations

Spectral functions for single- and multi-impurity models using density matrix renormalization group

Robert Peters

This article focuses on the calculation of spectral functions for single- and multi-impurity models using the density matrix renormalization group (DMRG). To calculate spectral functions in DMRG, the correction vector method is presently the most widely used approach. One, however, always obtains Lorentzian convoluted spectral functions, which in applications like the dynamical mean-field theory can lead to wrong results. In order to overcome this restriction, we use chain decompositions in which the resulting effective Hamiltonian can be diagonalized completely to calculate a discrete ``peak'' spectrum. We show that this peak spectrum is a very good approximation to a deconvolution of the correction vector spectral function. Calculating this deconvoluted spectrum directly from the DMRG basis set and operators is the most natural approach, because it uses only information from the system itself. Having calculated this excitation spectrum, one can use an arbitrary broadening to obtain a smooth spectral function or directly analyze the excitations. As a nontrivial test, we apply this method to obtain spectral functions for a model of three coupled Anderson impurities. Although we are focusing in this article on impurity models, the proposed method for calculating the peak spectrum can be easily adapted to usual lattice models.

arXiv:1103.5837•Physical Review B•Aug 11, 2011•29 citations

Padé approximant approach for obtaining finite-temperature spectral functions of quantum impurity models using the numerical renormalization group technique

Žiga Osolin, Rok Žitko

We introduce an alternative approach for obtaining smooth finite-temperature spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique. It is based on calculating first the Green's function on the imaginary-frequency axis, followed by an analytic continuation to the real-frequency axis using Pad\'e approximants. The arbitrariness in choosing a suitable kernel in the conventional broadening approach is thereby removed and, furthermore, we find that the Pad\'e method is able to resolve fine details in spectral functions with less artifacts on the scale of $\ensuremath{\omega}\ensuremath{\sim}T$. We discuss the convergence properties with respect to the NRG calculation parameters (discretization $\ensuremath{\Lambda}$, $z$-averaging, truncation cutoff) and the number of Matsubara points taken into account in the analytic continuation. We test the technique on the single-impurity Anderson model and the Hubbard model (within the dynamical mean-field theory). For the Anderson impurity model, we discuss the shape of the Kondo resonance and its temperature dependence. For the Hubbard model, we discuss the inner structure of the Hubbard bands in metallic and insulating solutions at half-filling, as well as in the doped Mott insulator. Based on these test cases, we conclude that the Pad\'e approximant approach provides improved results for spectral functions at low-frequency scales of $\ensuremath{\omega}\ensuremath{\lesssim}T$ and that it is capable of resolving sharp spectral features also at high frequencies.

arXiv:1302.3334•Physical Review B•Jun 28, 2013•35 citations

Full density-matrix numerical renormalization group calculation of impurity susceptibility and specific heat of the Anderson impurity model

L. Merker, Andreas Weichselbaum, Theo A. Costi

Recent developments in the numerical renormalization group (NRG) allow the construction of the full density matrix (FDM) of quantum impurity models [see A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99, 076402 (2007)] by using the completeness of the eliminated states introduced by F. B. Anders and A. Schiller [F. B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801 (2005)]. While these developments prove particularly useful in the calculation of transient response and finite-temperature Green's functions of quantum impurity models, they may also be used to calculate thermodynamic properties. In this paper, we assess the FDM approach to thermodynamic properties by applying it to the Anderson impurity model. We compare the results for the susceptibility and specific heat to both the conventional approach within NRG and to exact Bethe ansatz results. We also point out a subtlety in the calculation of the susceptibility (in a uniform field) within the FDM approach. Finally, we show numerically that for the Anderson model, the susceptibilities in response to a local and a uniform magnetic field coincide in the wide-band limit, in accordance with the Clogston-Anderson compensation theorem.

arXiv:1207.2631•Physical Review B•Aug 30, 2012•30 citations
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 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.
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.
This article focuses on the calculation of spectral functions for single- and multi-impurity models using the density matrix renormalization group (DMRG). To calculate spectral functions in DMRG, the correction vector method is presently the most widely used approach. One, however, always obtains Lorentzian convoluted spectral functions, which in applications like the dynamical mean-field theory can lead to wrong results. In order to overcome this restriction, we use chain decompositions in which the resulting effective Hamiltonian can be diagonalized completely to calculate a discrete ``peak'' spectrum. We show that this peak spectrum is a very good approximation to a deconvolution of the correction vector spectral function. Calculating this deconvoluted spectrum directly from the DMRG basis set and operators is the most natural approach, because it uses only information from the system itself. Having calculated this excitation spectrum, one can use an arbitrary broadening to obtain a smooth spectral function or directly analyze the excitations. As a nontrivial test, we apply this method to obtain spectral functions for a model of three coupled Anderson impurities. Although we are focusing in this article on impurity models, the proposed method for calculating the peak spectrum can be easily adapted to usual lattice models.
We introduce an alternative approach for obtaining smooth finite-temperature spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique. It is based on calculating first the Green's function on the imaginary-frequency axis, followed by an analytic continuation to the real-frequency axis using Pad\'e approximants. The arbitrariness in choosing a suitable kernel in the conventional broadening approach is thereby removed and, furthermore, we find that the Pad\'e method is able to resolve fine details in spectral functions with less artifacts on the scale of $\ensuremath{\omega}\ensuremath{\sim}T$. We discuss the convergence properties with respect to the NRG calculation parameters (discretization $\ensuremath{\Lambda}$, zzz-averaging, truncation cutoff) and the number of Matsubara points taken into account in the analytic continuation. We test the technique on the single-impurity Anderson model and the Hubbard model (within the dynamical mean-field theory). For the Anderson impurity model, we discuss the shape of the Kondo resonance and its temperature dependence. For the Hubbard model, we discuss the inner structure of the Hubbard bands in metallic and insulating solutions at half-filling, as well as in the doped Mott insulator. Based on these test cases, we conclude that the Pad\'e approximant approach provides improved results for spectral functions at low-frequency scales of $\ensuremath{\omega}\ensuremath{\lesssim}T$ and that it is capable of resolving sharp spectral features also at high frequencies.
Recent developments in the numerical renormalization group (NRG) allow the construction of the full density matrix (FDM) of quantum impurity models [see A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99, 076402 (2007)] by using the completeness of the eliminated states introduced by F. B. Anders and A. Schiller [F. B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801 (2005)]. While these developments prove particularly useful in the calculation of transient response and finite-temperature Green's functions of quantum impurity models, they may also be used to calculate thermodynamic properties. In this paper, we assess the FDM approach to thermodynamic properties by applying it to the Anderson impurity model. We compare the results for the susceptibility and specific heat to both the conventional approach within NRG and to exact Bethe ansatz results. We also point out a subtlety in the calculation of the susceptibility (in a uniform field) within the FDM approach. Finally, we show numerically that for the Anderson model, the susceptibilities in response to a local and a uniform magnetic field coincide in the wide-band limit, in accordance with the Clogston-Anderson compensation theorem.