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:2006.07936 · Physical review. B./Physical review. B

Restoring the continuum limit in the time-dependent numerical renormalization group approach

Jan Böker, Frithjof B. Anders

Published Aug 31, 2020• 2 citations
arXivHTMLPDF

Abstract

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 renormalization group approach which combines an accurate numerical renormalization group treatment of the nonequilibrium dynamics on the finite-size Wilson chain with a Bloch-Redfield formalism to include the effect of these additional reservoirs. Our approach overcomes the intrinsic shortcoming of the time-dependent numerical renormalization group approach induced by the bath discretization with a Wilson parameter $\mathrm{\ensuremath{\Lambda}}>1$. We analytically prove that for a system with a single chemical potential, the thermal equilibrium reduced density operator is the steady-state solution of the Bloch-Redfield master equation. For the numerical solution of this master equation, a Lanczos method is employed which couples all energy shells of the numerical renormalization group. The presented hybrid approach is applied to the real-time dynamics in correlated fermionic QISs. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to nontrivial models, such as the interacting resonant-level model and the single-impurity Anderson model.

DOI
10.1103/physrevb.102.075149

Related papers

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

Open Wilson chain numerical renormalization group approach to steady-state nonequilibrium quantum transport

Anand Manaparambil, Frithjof B. Anders

The numerical renormalization group (NRG) approach was developed to identify and quantify different equilibrium regimes of quantum impurity systems (QISs) with unprecedented accuracy by a tailored finite size representation. Out of equilibrium, the steady-state density operator is not of the Boltzmannian form but one that is determined by the imposed boundary conditions. We extend the NRG to the nonequilibrium setting by augmenting each Wilson site with a reservoir, whose coupling functions are calculated via a continuous fraction expansion in order to recover the continuum limit exactly. The nonequilibrium parameters such as a finite bias as well as a finite temperature gradient enters through the Bloch-Redfield tensor (BRT), whose zero eigenvector gives the steady-state density operator. We used the resulting open chain full density matrix (OC-FDM) approach with an effective single lead description to investigate the charge and spin transport through a quantum dot (QD) under finite bias and temperature gradient. The influence of lead asymmetry and an external magnetic field on the transport properties are also studied. For completeness, we have also investigated local properties of the QD, such as charge fluctuations and find excellent agreement with real-time quantum Monte Carlo (RT-QMC) data. The OC-FDM approach was able to explore Kondo energy scales as low as $T_K/D \approx 10^{-8}$ in the non-equilibrium regime, as well as show convergence with the established equilibrium benchmarks, such as a quantitative agreement with full density matrix numerical renormalization group (FDM-NRG) and Fermi-liquid scaling at small bias and temperature. Owing to the effective single lead description, a single OC-FDM data point takes orders of magnitude less time on a standard laptop, compared to other state-of-the-art numerical methods.

arXiv:2606.14635•Physical review. B./Physical review. B•Oct 23, 2026•0 citations

Hybrid NRG-DMRG approach to real-time dynamics of quantum impurity systems

Fabian Güttge, Frithjof B. Anders, Ulrich Schollwöck +2 more

A hybrid approach to nonequilibrium dynamics of quantum impurity systems is presented. The numerical renormalization group serves as a means to generate a suitable low-energy Hamiltonian, allowing for an accurate evaluation of the real-time dynamics of the problem up to exponentially long times using primarily the time-adaptive density-matrix renormalization group. In particular, by constructing a suitable hybrid chain, discretization errors are essentially eliminated on all time scales of interest. We extract the decay time of the interaction-enhanced oscillations in the interacting resonant-level model and show their quadratic divergence with the interaction strength $U$. Our numerical analysis is in excellent agreement with analytic predictions based on an expansion in $1/U$.

arXiv:1206.2186•Physical Review B•Mar 11, 2013•43 citations

Auxiliary master equation approach to nonequilibrium correlated impurities

Antonius Dorda, Martin Nuss, Wolfgang von der Linden +1 more

We present a numerical method for the study of correlated quantum impurity problems out of equilibrium, which is particularly suited to address steady-state properties within dynamical mean field theory. The approach, recently introduced by Arrigoni et al. [Phys. Rev. Lett. 110, 086403 (2013)], is based upon a mapping of the original impurity problem onto an auxiliary open quantum system, consisting of the interacting impurity coupled to bath sites as well as to a Markovian environment. The dynamics of the auxiliary system is governed by a Lindblad master equation whose parameters are used to optimize the mapping. The accuracy of the results can be readily estimated and systematically improved by increasing the number of auxiliary bath sites, or by introducing a linear correction. Here, we focus on a detailed discussion of the proposed approach including technical remarks. To solve for the Green's functions of the auxiliary impurity problem, a non-Hermitian Lanczos diagonalization is applied. As a benchmark, results for the steady-state current-voltage characteristics of the single-impurity Anderson model are presented. Furthermore, the bias dependence of the single-particle spectral function and the splitting of the Kondo resonance are discussed. In its present form, the method is fast, efficient, and features a controlled accuracy.

arXiv:1312.4586•Physical Review B•Apr 7, 2014•117 citations

Lindblad-driven discretized leads for nonequilibrium steady-state transport in quantum impurity models: Recovering the continuum limit

Frauke Schwarz, Moshe Goldstein, Antonius Dorda +3 more

The description of interacting quantum impurity models in steady-state nonequilibrium is an open challenge for computational many-particle methods: the numerical requirement of using a finite number of lead levels and the physical requirement of describing a truly open quantum system are seemingly incompatible. One possibility to bridge this gap is the use of Lindblad-driven discretized leads (LDDL): one couples auxiliary continuous reservoirs to the discretized lead levels and represents these additional reservoirs by Lindblad terms in the Liouville equation. For quadratic models governed by Lindbladian dynamics, we present an elementary approach for obtaining correlation functions analytically. In a second part, we use this approach to explicitly discuss the conditions under which the continuum limit of the LDDL approach recovers the correct representation of thermal reservoirs. As an analytically solvable example, the nonequilibrium resonant level model is studied in greater detail. Lastly, we present ideas towards a numerical evaluation of the suggested Lindblad equation for interacting impurities based on matrix product states. In particular, we present a reformulation of the Lindblad equation, which has the useful property that the leads can be mapped onto a chain where both the Hamiltonian dynamics and the Lindblad driving are local at the same time. Moreover, we discuss the possibility to combine the Lindblad approach with a logarithmic discretization needed for the exploration of exponentially small energy scales.

arXiv:1604.02050•Physical review. B./Physical review. B•Oct 27, 2016•72 citations

Nonequilibrium Steady-State Transport in Quantum Impurity Models: A Thermofield and Quantum Quench Approach Using Matrix Product States

Frauke Schwarz, Ireneusz Weymann, Jan von Delft +1 more

The numerical renormalization group (NRG) is tailored to describe interacting impurity models in equilibrium, but it faces limitations for steady-state nonequilibrium, arising, e.g., due to an applied bias voltage. We show that these limitations can be overcome by describing the thermal leads using a thermofield approach, integrating out high energy modes using NRG, and then treating the nonequilibrium dynamics at low energies using a quench protocol, implemented using the time-dependent density matrix renormalization group. This yields quantitatively reliable results for the current (with errors ≲3%) down to the exponentially small energy scales characteristic of impurity models. We present results of benchmark quality for the temperature and magnetic field dependence of the zero-bias conductance peak for the single-impurity Anderson model.

arXiv:1708.06315•Physical Review Letters•Sep 26, 2018•81 citations
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 renormalization group approach which combines an accurate numerical renormalization group treatment of the nonequilibrium dynamics on the finite-size Wilson chain with a Bloch-Redfield formalism to include the effect of these additional reservoirs. Our approach overcomes the intrinsic shortcoming of the time-dependent numerical renormalization group approach induced by the bath discretization with a Wilson parameter \ensuremathΛ>1\mathrm{\ensuremath{\Lambda}}>1\ensuremathΛ>1. We analytically prove that for a system with a single chemical potential, the thermal equilibrium reduced density operator is the steady-state solution of the Bloch-Redfield master equation. For the numerical solution of this master equation, a Lanczos method is employed which couples all energy shells of the numerical renormalization group. The presented hybrid approach is applied to the real-time dynamics in correlated fermionic QISs. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to nontrivial models, such as the interacting resonant-level model and the single-impurity Anderson model.
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 numerical renormalization group (NRG) approach was developed to identify and quantify different equilibrium regimes of quantum impurity systems (QISs) with unprecedented accuracy by a tailored finite size representation. Out of equilibrium, the steady-state density operator is not of the Boltzmannian form but one that is determined by the imposed boundary conditions. We extend the NRG to the nonequilibrium setting by augmenting each Wilson site with a reservoir, whose coupling functions are calculated via a continuous fraction expansion in order to recover the continuum limit exactly. The nonequilibrium parameters such as a finite bias as well as a finite temperature gradient enters through the Bloch-Redfield tensor (BRT), whose zero eigenvector gives the steady-state density operator. We used the resulting open chain full density matrix (OC-FDM) approach with an effective single lead description to investigate the charge and spin transport through a quantum dot (QD) under finite bias and temperature gradient. The influence of lead asymmetry and an external magnetic field on the transport properties are also studied. For completeness, we have also investigated local properties of the QD, such as charge fluctuations and find excellent agreement with real-time quantum Monte Carlo (RT-QMC) data. The OC-FDM approach was able to explore Kondo energy scales as low as TK/D≈10−8T_K/D \approx 10^{-8}TK​/D≈10−8 in the non-equilibrium regime, as well as show convergence with the established equilibrium benchmarks, such as a quantitative agreement with full density matrix numerical renormalization group (FDM-NRG) and Fermi-liquid scaling at small bias and temperature. Owing to the effective single lead description, a single OC-FDM data point takes orders of magnitude less time on a standard laptop, compared to other state-of-the-art numerical methods.
A hybrid approach to nonequilibrium dynamics of quantum impurity systems is presented. The numerical renormalization group serves as a means to generate a suitable low-energy Hamiltonian, allowing for an accurate evaluation of the real-time dynamics of the problem up to exponentially long times using primarily the time-adaptive density-matrix renormalization group. In particular, by constructing a suitable hybrid chain, discretization errors are essentially eliminated on all time scales of interest. We extract the decay time of the interaction-enhanced oscillations in the interacting resonant-level model and show their quadratic divergence with the interaction strength UUU. Our numerical analysis is in excellent agreement with analytic predictions based on an expansion in 1/U1/U1/U.
We present a numerical method for the study of correlated quantum impurity problems out of equilibrium, which is particularly suited to address steady-state properties within dynamical mean field theory. The approach, recently introduced by Arrigoni et al. [Phys. Rev. Lett. 110, 086403 (2013)], is based upon a mapping of the original impurity problem onto an auxiliary open quantum system, consisting of the interacting impurity coupled to bath sites as well as to a Markovian environment. The dynamics of the auxiliary system is governed by a Lindblad master equation whose parameters are used to optimize the mapping. The accuracy of the results can be readily estimated and systematically improved by increasing the number of auxiliary bath sites, or by introducing a linear correction. Here, we focus on a detailed discussion of the proposed approach including technical remarks. To solve for the Green's functions of the auxiliary impurity problem, a non-Hermitian Lanczos diagonalization is applied. As a benchmark, results for the steady-state current-voltage characteristics of the single-impurity Anderson model are presented. Furthermore, the bias dependence of the single-particle spectral function and the splitting of the Kondo resonance are discussed. In its present form, the method is fast, efficient, and features a controlled accuracy.
The description of interacting quantum impurity models in steady-state nonequilibrium is an open challenge for computational many-particle methods: the numerical requirement of using a finite number of lead levels and the physical requirement of describing a truly open quantum system are seemingly incompatible. One possibility to bridge this gap is the use of Lindblad-driven discretized leads (LDDL): one couples auxiliary continuous reservoirs to the discretized lead levels and represents these additional reservoirs by Lindblad terms in the Liouville equation. For quadratic models governed by Lindbladian dynamics, we present an elementary approach for obtaining correlation functions analytically. In a second part, we use this approach to explicitly discuss the conditions under which the continuum limit of the LDDL approach recovers the correct representation of thermal reservoirs. As an analytically solvable example, the nonequilibrium resonant level model is studied in greater detail. Lastly, we present ideas towards a numerical evaluation of the suggested Lindblad equation for interacting impurities based on matrix product states. In particular, we present a reformulation of the Lindblad equation, which has the useful property that the leads can be mapped onto a chain where both the Hamiltonian dynamics and the Lindblad driving are local at the same time. Moreover, we discuss the possibility to combine the Lindblad approach with a logarithmic discretization needed for the exploration of exponentially small energy scales.
The numerical renormalization group (NRG) is tailored to describe interacting impurity models in equilibrium, but it faces limitations for steady-state nonequilibrium, arising, e.g., due to an applied bias voltage. We show that these limitations can be overcome by describing the thermal leads using a thermofield approach, integrating out high energy modes using NRG, and then treating the nonequilibrium dynamics at low energies using a quench protocol, implemented using the time-dependent density matrix renormalization group. This yields quantitatively reliable results for the current (with errors ≲3%) down to the exponentially small energy scales characteristic of impurity models. We present results of benchmark quality for the temperature and magnetic field dependence of the zero-bias conductance peak for the single-impurity Anderson model.