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arXiv:1708.06315 · Physical Review Letters

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, Andreas Weichselbaum

Published Sep 26, 2018• 81 citations
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Abstract

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.

DOI
10.1103/physrevlett.121.137702

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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.
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.
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.
Within the recently introduced auxiliary master equation approach it is possible to address steady state properties of strongly correlated impurity models, small molecules or clusters efficiently and with high accuracy. It is particularly suited for dynamical mean field theory in the nonequilibrium as well as in the equilibrium case. The method is based on the solution of an auxiliary open quantum system, which can be made quickly equivalent to the original impurity problem. In its first implementation a Krylov space method was employed. Here, we aim at extending the capabilities of the approach by adopting matrix product states for the solution of the corresponding auxiliary quantum master equation. This allows for a drastic increase in accuracy and permits us to access the Kondo regime for large values of the interaction. In particular, we investigate the nonequilibrium steady state of a single impurity Anderson model and focus on the spectral properties for temperatures T below the Kondo temperature T_K and for small bias voltages phi. For the two cases considered, with T=T_K/4 and T=T_K/10 we find a clear splitting of the Kondo resonance into a two-peak structure for phi close above T_K. In the equilibrium case (phi=0) and for T=T_K/4, the obtained spectral function essentially coincides with the one from numerical renormalization group.
The Anderson impurity model is a paradigmatic example in the study of strongly correlated quantum systems and describes an interacting quantum dot coupled to electronic leads. Here we investigate its dynamics following a quantum quench based on matrix product state simulations. We examine the behavior of its impurity magnetization. Its relaxation allows us to extract the predicted scaling of the Kondo temperature as a function of the impurity-lead hybridization and quantum dot repulsion. Additionally, our simulations provide estimates of the currents in the nonequilibrium quasisteady state appearing after the quench. Through their values, we examine the dependence of the conductance on the voltage bias Vb{V}_{b}Vb​ and on the impurity chemical potential Vg{V}_{g}Vg​, which displays a zero-bias Kondo peak. Our results are relevant for transport measurements in Coulomb blockaded devices, and, in particular, in quantum dots induced in nanowires.
We investigate nonequilibrium properties of the single-impurity Anderson model by means of the functional renormalization group (fRG) within Keldysh formalism. We present how the level broadening $\ensuremath{\Gamma}/2$ can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi-liquid behavior that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions $U\ensuremath{\lesssim}3\ensuremath{\Gamma}$. In equilibrium we find excellent agreement with numerical RG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with time-dependent density-matrix RG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second-order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.
We present an infinite Grassmann time-evolving matrix product operator method for quantum impurity problems, which directly works in the steady state. The method embraces the well-established infinite matrix product state algorithms with the recently developed GTEMPO method, and benefits from both sides: it obtains real-time Green's functions without sampling noises and bath discretization error, it is applicable for any temperature without the sign problem, and its computational cost is independent of the transient dynamics and does not scale with the number of baths. We benchmark the method on the finite-temperature equilibrium Green's function in the noninteracting limit against exact solutions and in the single-orbital Anderson impurity model against GTEMPO calculations. We also study the zero-temperature nonequilibrium steady state of an impurity coupled to two baths with a voltage bias, obtaining consistent particle currents with existing calculations. The method is ideal for studying steady-state quantum transport, and can be readily used as an efficient real-time impurity solver in the dynamical mean-field theory and its nonequilibrium extension.