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arXiv:2606.14635 · Physical review. B./Physical review. B

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

Anand Manaparambil, Frithjof B. Anders

Published Oct 23, 2026• 0 citations
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Abstract

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.

DOI
10.1103/hys8-c2rs

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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.
Scattering states fulfill the correct boundary conditions of a current carrying open quantum system. Discretizing the energy continuum of these states allows for employing Wilson's numerical renormalization group approach without violating the boundary conditions by using a finite size system. We evolve the analytically known steady-state density operator for a non-interacting quantum-system at finite bias to the full interacting problem by the time-dependent numerical renormalization group after switching on the local charging energy. Using a newly developed algorithm for steady-state nonequilibrium Green functions, we can calculate the current I as function of bias voltage V for arbitrary temperature and magnetic field. A comparison with second-order and GW Kadanoff-Baym-Keldysh results shows excellent agreement for weak interaction strength U .
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.
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.
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.
Numerical renormalization group (NRG) is formulated for nonequilibrium steady-state by converting finite-lattice many-body eigenstates into scattering states. Extension of the full-density-matrix NRG for a biased Anderson impurity model, simplified by formulating with the original orbital basis as the Hamiltonian, enables detailed studies of the sub-Kondo spectral evolution in the zero-temperature limit, confirming the double-resonance structure at bias of the Kondo energy scale TKT_KTK​. The distribution shows distinct multi-scale spectral features at energy ωωω below the Kondo scale (ω≲TKω\lesssim T_Kω≲TK​) and near the bias (ω≳Vω\gtrsim Vω≳V), leading to the nonequilibrium temperature TlocT_{\rm loc}Tloc​ local to the Kondo dot scaling as kBTloc≈Vk_BT_{\rm loc}\approx VkB​Tloc​≈V for V≫TKV\gg T_KV≫TK​. The current-voltage relation in the low-temperature limit (T≪TKT\ll T_KT≪TK​) deviates from the unitary limit as the bias exceeds the Kondo scale (V/2≳TKV/2\gtrsim T_KV/2≳TK​) and reaches the current saturation regime.
We present the real-time renormalization group (RTRG) method as a method to describe the stationary state current through generic multilevel quantum dots in nonequilibrium. The employed approach consists of a very rudimentary approximation for the renormalization group (RG) equations which neglects all vertex corrections while it provides a means to compute the effective dot Liouvillian self-consistently. Being based on a weak-coupling expansion in the tunneling between dot and reservoirs, the RTRG approach turns out to reliably describe charge fluctuations in and out of equilibrium for arbitrary coupling strength, even at zero temperature. We confirm this in the linear response regime with a benchmark against highly accurate numerical renormalization group data in the exemplary case of three-level quantum dots. For small to intermediate bias voltages and weak Coulomb interactions, we find an excellent agreement between RTRG and functional renormalization group data, which can be expected to be accurate in this regime. As a consequence, we advertise the presented RTRG approach as an efficient and versatile tool to describe charge fluctuations in quantum dot systems.