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:1310.2724 · Physical Review B

Numerical evaluation of Green's functions based on the Chebyshev expansion

A. Braun, Peter Schmitteckert

Published Oct 9, 2014• 44 citations
arXivHTMLPDF

Abstract

We present a method to determine the impurity Green's function of the interacting resonant level model (IRLM) using numerical simulation techniques based on the expansion of a resolvent expression in terms of Chebyshev polynomials. The iterative determination of the contributions to the expansion is based on a density matrix renormalization group algorithm. The spectral function has Lorentzian shape, where we can show that the width grows monotonically with the interaction on the contact link. Moreover, for values of the interaction comparable to the bandwidth, there are additional peaks showing up for energies located outside the conduction band.

DOI
10.1103/physrevb.90.165112

Related papers

Chebyshev expansion for impurity models using matrix product states

Martin Ganahl, Patrik Thunström, Frank Verstraete +2 more

We improve a recently developed expansion technique for calculating real-frequency spectral functions of any one-dimensional model with short-range interactions, by postprocessing computed Chebyshev moments with linear prediction. This can be achieved at virtually no cost, and in sharp contrast to existing methods based on the dampening of the moments, improves the spectral resolution rather than lowering it. We validate the method for the exactly solvable resonating level model and the single impurity Anderson model. It is capable of resolving sharp Kondo resonances, as well as peaks within the Hubbard bands when employed as an impurity solver for dynamical mean-field theory. Our method works at zero temperature and allows for arbitrary discretization of the bath spectrum. It achieves similar precision as the dynamical density matrix renormalization group, at lower cost. We also propose an alternative expansion, of $\mathbb{1}\ensuremath{-}exp(\ensuremath{-}\ensuremath{\tau}H)$ instead of the usual $H$, which opens the possibility of using established methods for the time evolution of matrix product states to calculate the spectral functions directly.

arXiv:1403.1209•Physical Review B•Jul 30, 2014•80 citations

Nonequilibrium Green's functions and their relation to the negative differential conductance in the interacting resonant level model

Max Erich Sorantin, Wolfgang von der Linden, Roman Lucrezi +1 more

We evaluate the nonequilibrium single-particle Green's functions in the steady state of the interacting resonant level model (IRLM) under the effect of an applied bias voltage. Employing the so-called auxiliary master equation approach, we present accurate nonperturbative results for the nonequilibrium spectral and effective distribution functions, as well as for the current-voltage characteristics. We find a drastic change of these spectral properties between the regimes of low- and high-bias voltages and discuss the relation of these changes to the negative differential conductance (NDC), a prominent feature in the nonequilibrium IRLM. The anomalous evolution of the effective distribution function next to the impurity shown by our calculations suggests a mechanism whereby the impurity gets effectively decoupled from the leads at voltages where the NDC sets in, in agreement with previous renormalization group approaches. This scenario is qualitatively confirmed by a Hartree-Fock treatment of the model.

arXiv:1812.08618•Physical review. B./Physical review. B•Feb 20, 2019•2 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

Open Wilson chain numerical renormalization group approach to Green's functions

Jan Böker, Frithjof B. Anders

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.

arXiv:2204.03453•Physical review. B./Physical review. B•Jun 21, 2022•1 citation

Spectral functions of low-dimensional quantum systems

Piet E. Dargel

The focus of this thesis is set on the calculation of spectral functions for low-dimensional quantum systems. At first the quantum many-body problem is introduced and the related spectral functions of interest are defined. In a next step the applied numerical algorithms are presented: Exact diagonalization & Lanczos algorithm, the numerical renormalization group (NRG) and the density matrix renormalization group (DMRG) in the formulation of matrix product states (MPS). The main focus is set on the calculation of spectral functions with these algorithms. In particular for the DMRG a new algorithm is presented which combines the Lanczos algorithm with matrix product states. The presented algorithms are applied to different problems. The first system denotes a copper crystal with magnetic (cobalt/iron) impurities. These materials are known Kondo-systems, i.e. they show an untypical increase in the resistivity with decreasing temperature. At low temperatures local moments will develop, which will be screened from the surrounding conduction band electrons. The ground state of the system is a long-range strongly correlated many-body state between conduction band electrons and local moments. In the singleparticle impurity spectral function a so-called Kondo resonance at the Fermi energy is formed below the Kondo temperature. In scanning tunneling microscope measurements the spectral function at the surface - the so-called local density of states - of the copper crystals with buried impurities in varying depth was measured. The goal of this study is to verify the existence of long-range Kondo signatures via a comparison of the spectral functions for impurities with different depth below the surface. Therefore in this work numerical simulations are used to calculate the local density of states at the surface. The single impurity Anderson model (SIAM) is applied which shows the Kondo resonance in the impurity spectral function. In order to calculate the local density of states at the surface of the copper crystal the equations of motion of the SIAM and the Dyson equation are used and then solved by a combination of numerical renormalization group and band structure calculations of the copper crystal (linear combination of atomic orbitals). The simulation for impurities in different depth below the surface and the experimental data are in very good agreement and hence support the measurement of a long-range Kondo signature. Furthermore the simulations allow to determine the Kondo temperature of a single impurity. The focus of the second part is set on the one-dimensional Heisenberg model. The spectral function of interest is here the dynamic spin structure factor. The newly developed MPS-Lanczos algorithm within the DMRG is used to calculate the dynamic spin structure factor. The results are compared to analytic results from the Bethe ansatz. In the end the new algorithm is discussed in respect to existing methods within the DMRG to calculate spectral functions.

•Jan 1, 2013•3 citations

Chebyshev matrix product state impurity solver for dynamical mean-field theory

F. Alexander Wolf, Ian P. McCulloch, Olivier Parcollet +1 more

We compute the spectral functions for the two-site dynamical cluster theory and for the two-orbital dynamical mean-field theory in the density-matrix renormalization group (DMRG) framework using Chebyshev expansions represented with matrix product states (MPS). We obtain quantitatively precise results at modest computational effort through technical improvements regarding the truncation scheme and the Chebyshev rescaling procedure. We furthermore establish the relation of the Chebyshev iteration to real-time evolution and discuss technical aspects as computation time and implementation in detail.

arXiv:1407.1622•Physical Review B•Sep 11, 2014•100 citations
We present a method to determine the impurity Green's function of the interacting resonant level model (IRLM) using numerical simulation techniques based on the expansion of a resolvent expression in terms of Chebyshev polynomials. The iterative determination of the contributions to the expansion is based on a density matrix renormalization group algorithm. The spectral function has Lorentzian shape, where we can show that the width grows monotonically with the interaction on the contact link. Moreover, for values of the interaction comparable to the bandwidth, there are additional peaks showing up for energies located outside the conduction band.
We improve a recently developed expansion technique for calculating real-frequency spectral functions of any one-dimensional model with short-range interactions, by postprocessing computed Chebyshev moments with linear prediction. This can be achieved at virtually no cost, and in sharp contrast to existing methods based on the dampening of the moments, improves the spectral resolution rather than lowering it. We validate the method for the exactly solvable resonating level model and the single impurity Anderson model. It is capable of resolving sharp Kondo resonances, as well as peaks within the Hubbard bands when employed as an impurity solver for dynamical mean-field theory. Our method works at zero temperature and allows for arbitrary discretization of the bath spectrum. It achieves similar precision as the dynamical density matrix renormalization group, at lower cost. We also propose an alternative expansion, of $\mathbb{1}\ensuremath{-}exp(\ensuremath{-}\ensuremath{\tau}H)$ instead of the usual HHH, which opens the possibility of using established methods for the time evolution of matrix product states to calculate the spectral functions directly.
We evaluate the nonequilibrium single-particle Green's functions in the steady state of the interacting resonant level model (IRLM) under the effect of an applied bias voltage. Employing the so-called auxiliary master equation approach, we present accurate nonperturbative results for the nonequilibrium spectral and effective distribution functions, as well as for the current-voltage characteristics. We find a drastic change of these spectral properties between the regimes of low- and high-bias voltages and discuss the relation of these changes to the negative differential conductance (NDC), a prominent feature in the nonequilibrium IRLM. The anomalous evolution of the effective distribution function next to the impurity shown by our calculations suggests a mechanism whereby the impurity gets effectively decoupled from the leads at voltages where the NDC sets in, in agreement with previous renormalization group approaches. This scenario is qualitatively confirmed by a Hartree-Fock treatment of the model.
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.
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.
The focus of this thesis is set on the calculation of spectral functions for low-dimensional quantum systems. At first the quantum many-body problem is introduced and the related spectral functions of interest are defined. In a next step the applied numerical algorithms are presented: Exact diagonalization & Lanczos algorithm, the numerical renormalization group (NRG) and the density matrix renormalization group (DMRG) in the formulation of matrix product states (MPS). The main focus is set on the calculation of spectral functions with these algorithms. In particular for the DMRG a new algorithm is presented which combines the Lanczos algorithm with matrix product states. The presented algorithms are applied to different problems. The first system denotes a copper crystal with magnetic (cobalt/iron) impurities. These materials are known Kondo-systems, i.e. they show an untypical increase in the resistivity with decreasing temperature. At low temperatures local moments will develop, which will be screened from the surrounding conduction band electrons. The ground state of the system is a long-range strongly correlated many-body state between conduction band electrons and local moments. In the singleparticle impurity spectral function a so-called Kondo resonance at the Fermi energy is formed below the Kondo temperature. In scanning tunneling microscope measurements the spectral function at the surface - the so-called local density of states - of the copper crystals with buried impurities in varying depth was measured. The goal of this study is to verify the existence of long-range Kondo signatures via a comparison of the spectral functions for impurities with different depth below the surface. Therefore in this work numerical simulations are used to calculate the local density of states at the surface. The single impurity Anderson model (SIAM) is applied which shows the Kondo resonance in the impurity spectral function. In order to calculate the local density of states at the surface of the copper crystal the equations of motion of the SIAM and the Dyson equation are used and then solved by a combination of numerical renormalization group and band structure calculations of the copper crystal (linear combination of atomic orbitals). The simulation for impurities in different depth below the surface and the experimental data are in very good agreement and hence support the measurement of a long-range Kondo signature. Furthermore the simulations allow to determine the Kondo temperature of a single impurity. The focus of the second part is set on the one-dimensional Heisenberg model. The spectral function of interest is here the dynamic spin structure factor. The newly developed MPS-Lanczos algorithm within the DMRG is used to calculate the dynamic spin structure factor. The results are compared to analytic results from the Bethe ansatz. In the end the new algorithm is discussed in respect to existing methods within the DMRG to calculate spectral functions.
We compute the spectral functions for the two-site dynamical cluster theory and for the two-orbital dynamical mean-field theory in the density-matrix renormalization group (DMRG) framework using Chebyshev expansions represented with matrix product states (MPS). We obtain quantitatively precise results at modest computational effort through technical improvements regarding the truncation scheme and the Chebyshev rescaling procedure. We furthermore establish the relation of the Chebyshev iteration to real-time evolution and discuss technical aspects as computation time and implementation in detail.