The numerical renormalization group (NRG) approach was developed to identify and quantify different equilibrium regimes of quantum impurity systems 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 continued fraction expansion to recover the continuum limit exactly. The nonequilibrium parameters such as a finite bias as well as a finite temperature gradient enter through the Bloch-Redfield tensor, whose zero eigenvector gives the steady-state density operator. We used the resulting open-chain full-density matrix (OC-FDM) approach in combination with an effective 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 data. The OC-FDM approach was able to explore Kondo energy scales as low as $T_K/D\sim {10}^{-8}$ in the nonequilibrium regime, as well as show convergence with the established equilibrium benchmarks, such as a quantitative agreement with full-density matrix numerical renormalization group 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.

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• Kondo effect
• Numerical Renormalization Group
• Quantum transport