Real-time dynamics in quantum-impurity systems: a time-dependent numerical renormalization-group approach

Quantum Monte Carlo Method in the Steady State

We present a numerically exact steady-state inchworm Monte Carlo method for nonequilibrium quantum impurity models. Rather than propagating an initial state to long times, the method is directly formulated in the steady state. This eliminates any need to traverse the transient dynamics and grants access to a much larger range of parameter regimes at vastly reduced computational costs. We benchmark the method on equilibrium Green's functions of quantum dots in the noninteracting limit and in the unitary limit of the Kondo regime. We then consider correlated materials described with dynamical mean field theory and driven away from equilibrium by a bias voltage. We show that the response of a correlated material to a bias voltage differs qualitatively from the splitting of the Kondo resonance observed in bias-driven quantum dots.

Hierarchical equations of motion approach for accurate characterization of spin excitations in quantum impurity systems

Recent technological advancement in scanning tunneling microscopes has enabled the measurement of spin-field and spin-spin interactions in single atomic or molecular junctions with an unprecedentedly high resolution. Theoretically, although the fermionic hierarchical equations of motion (HEOM) method has been widely applied to investigate the strongly correlated Kondo states in these junctions, the existence of low-energy spin excitations presents new challenges to numerical simulations. These include the quest for a more accurate and efficient decomposition for the non-Markovian memory of low-temperature environments and a more careful handling of errors caused by the truncation of the hierarchy. In this work, we propose several new algorithms, which significantly enhance the performance of the HEOM method, as exemplified by the calculations on systems involving various types of low-energy spin excitations. Being able to characterize both the Kondo effect and spin excitation accurately, the HEOM method offers a sophisticated and versatile theoretical tool, which is valuable for the understanding and even prediction of the fascinating quantum phenomena explored in cutting-edge experiments.

Numerical renormalization group study of the Loschmidt echo in Kondo systems

We study the dynamical properties of the one-channel and two-channel spin-1/2 Kondo models after quenching in Hamiltonian variables. Eigen spectrum of the initial and final Hamiltonians is calculated by using the numerical renormalization group method implemented within the matrix product states formalism. We consider multiple quench protocols in the considered Kondo systems, also in the presence of external magnetic field of different intensities. The main emphasis is put on the analysis of the behavior of the Loschmidt echo L(t), which measures the ability of the system's revival to its initial state after a quench. We show that the decay of the Loschmidt echo strongly depends on the type of quench and the ground state of the system. For the one-channel Kondo model, we show that L(t) decays as, [Formula: see text], where [Formula: see text] is the Kondo temperature, while for the two-channel Kondo model, we demonstrate that the decay is slower and given by [Formula: see text]. In addition, we also determine the dynamical behavior of the impurity's magnetization, which sheds light on identification of the relevant time scales in the system's dynamics.

Transient dynamics of a single molecular transistor in the presence of local electron-phonon and electron-electron interactions and quantum dissipation

We consider a single molecular transistor in which a quantum dot with local electron–electron and electron–phonon interactions is coupled to two metallic leads, one of which acts like a source and the other like a drain. The system is modeled by the Anderson-Holstein (AH) model. The quantum dot is mounted on a substrate that acts as a heat bath. Its phonons interact with the quantum dot phonons by the Caldeira–Leggett interaction giving rise to dissipation in the dynamics of the quantum dot system. A simple canonical transformation exactly treats the interaction of the quantum dot phonons with the substrate phonons. The electron–phonon interaction of the quantum dot is eliminated by the celebrated Lang-Firsov transformation. The time-dependent current is finally calculated by the Keldysh Green function technique with various types of bias. The transient-time phase diagram is analysed as a function of the system parameters to explore regions that can be used for fast switching in devices like nanomolecular switches.

Two-Channel Charge-Kondo Physics in Graphene Quantum Dots

Nanoelectronic quantum dot devices exploiting the charge-Kondo paradigm have been established as versatile and accurate analogue quantum simulators of fundamental quantum impurity models. In particular, hybrid metal-semiconductor dots connected to two metallic leads realize the two-channel Kondo (2CK) model, in which Kondo screening of the dot charge pseudospin is frustrated. In this article, a two-channel charge-Kondo device made instead from graphene components is considered, realizing a pseudogapped version of the 2CK model. The model is solved using Wilson's Numerical Renormalization Group method, uncovering a rich phase diagram as a function of dot-lead coupling strength, channel asymmetry, and potential scattering. The complex physics of this system is explored through its thermodynamic properties, scattering T-matrix, and experimentally measurable conductance. The strong coupling pseudogap Kondo phase is found to persist in the channel-asymmetric two-channel context, while in the channel-symmetric case, frustration results in a novel quantum phase transition. Remarkably, despite the vanishing density of states in the graphene leads at low energies, a finite linear conductance is found at zero temperature at the frustrated critical point, which is of a non-Fermi liquid type. Our results suggest that the graphene charge-Kondo platform offers a unique possibility to access multichannel pseudogap Kondo physics.

Universal Thermal Entanglement of Multichannel Kondo Effects

Quantum entanglement between an impurity and its environment is expected to be central in quantum impurity problems. We develop a method to compute the entanglement in spin-1/2 impurity problems, based on the entanglement negativity and the boundary conformal field theory (BCFT). Using the method, we study the thermal decay of the entanglement in the multichannel Kondo effects. At zero temperature, the entanglement has the maximal value independent of the number of the screening channels. At low temperature, the entanglement exhibits a power-law thermal decay. The power-law exponent equals two times of the scaling dimension of the BCFT boundary operator describing the impurity spin, and it is attributed to the energy-dependent scaling behavior of the entanglement in energy eigenstates. These agree with numerical renormalization group results, unveiling quantum coherence inside the Kondo screening length.

Spin Seebeck effect of correlated magnetic molecules

In this paper we investigate the spin-resolved thermoelectric properties of strongly correlated molecular junctions in the linear response regime. The magnetic molecule is modeled by a single orbital level to which the molecular core spin is attached by an exchange interaction. Using the numerical renormalization group method we analyze the behavior of the (spin) Seebeck effect, heat conductance and figure of merit for different model parameters of the molecule. We show that the thermopower strongly depends on the strength and type of the exchange interaction as well as the molecule's magnetic anisotropy. When the molecule is coupled to ferromagnetic leads, the thermoelectric properties reveal an interplay between the spin-resolved tunneling processes and intrinsic magnetic properties of the molecule. Moreover, in the case of finite spin accumulation in the leads, the system exhibits the spin Seebeck effect. We demonstrate that a considerable spin Seebeck effect can develop when the molecule exhibits an easy-plane magnetic anisotropy, while the sign of the spin thermopower depends on the type and magnitude of the molecule's exchange interaction.

Non-equilibrium electronic transport through a quantum dot with strong Coulomb repulsion in the presence of a magnetic field

The non-equilibrium electronic transport through a nanoscale device composed of a single quantum dot between two metallic contacts is studied theoretically within the framework of the Keldysh formalism. The quantum dot consists of a single energy level subject to an applied magnetic field. Correlations due to the Coulomb repulsion between electrons on the dot are treated with a Green's function decoupling scheme which, although similar to the Hubbard-I approximation, captures some of the dynamics beyond. The scheme is exact in the so-called atomic limit, defined by vanishing tunneling between contacts and dot, and in the non-interacting limit, where the on-dot Coulomb repulsion is zero. Explicit analytic solutions, valid for arbitrary magnetic fields, are obtained for two important setups: (i) the stationary regime, with constant voltage bias between the leads, and (ii) the time-dependent regime for metallic leads with constant density of states of infinite width. In these regimes, the current through the dot is evaluated numerically for various parameter sets and its main features interpreted in terms of the underlying physical processes. The results are compared to the non-crossing approximation (NCA) and diagrammatic non-equilibrium quantum Monte-Carlo (QMC) where available.

Signatures of Mottness and Hundness in archetypal correlated metals

Physical properties of multi-orbital materials depend not only on the strength of the effective interactions among the valence electrons but also on their type. Strong correlations are caused by either Mott physics that captures the Coulomb repulsion among charges, or Hund physics that aligns the spins in different orbitals. We identify four energy scales marking the onset and the completion of screening in orbital and spin channels. The differences in these scales, which are manifest in the temperature dependence of the local spectrum and of the charge, spin and orbital susceptibilities, provide clear signatures distinguishing Mott and Hund physics. We illustrate these concepts with realistic studies of two archetypal strongly correlated materials, and corroborate the generality of our conclusions with a model Hamiltonian study.

Auxiliary master equation approach within stochastic wave functions: Application to the interacting resonant level model

We present further developments of the auxiliary master equation approach (AMEA), a numerical method to simulate many-body quantum systems in as well as out of equilibrium and apply it to the interacting resonant level model to benchmark the new developments. In particular, our results are obtained by employing the stochastic wave functions method to solve the auxiliary open quantum system arising within AMEA. This development allows us to reach extremely low wall times for the calculation of correlation functions with respect to previous implementations of AMEA. An additional significant improvement is obtained by extrapolating a series of results obtained by increasing the number of auxiliary bath sites, N_{B}, used within the auxiliary open quantum system formally to the limit of N_{B}→∞. Results for the current-voltage characteristics and for equilibrium correlation functions are compared with the one obtained by exact and matrix-product states-based approaches. Further, we complement this benchmark by the presentation of spectral functions for higher temperatures where we find different behaviors around zero frequency depending on the hybridization strength.

Long-Range Entanglement near a Kondo-Destruction Quantum Critical Point

The numerical renormalization group is used to study quantum entanglement in the Kondo impurity model with a density of states ρ(ϵ)∝|ϵ|^{r} (0<r<1/2) that vanishes at the Fermi energy ϵ=0. This nonintegrable model features a Kondo-destruction quantum critical point (QCP) separating a partially screened phase from a local-moment phase. The impurity contribution S_{e}^{imp} to the entanglement entropy between a region of radius R around the magnetic impurity and the rest of the system reveals a length scale R^{*} that distinguishes a region R≪R^{*} of strong critical entanglement from one R≫R^{*} of weak entanglement. Within each phase, S_{e}^{imp} is a universal function of R/R^{*} with a power-law decay for R/R^{*}≫1. The entanglement length R^{*} diverges on approach to the interacting QCP, showing that the critical Kondo screening cloud subsumes the entire system as the impurity becomes maximally entangled with the conduction band. This work has implications for entanglement calculations in other models and for the nature of heavy-fermion quantum criticality.

Solving Quantum Impurity Problems in and out of Equilibrium with the Variational Approach

A versatile and efficient variational approach is developed to solve in- and out-of-equilibrium problems of generic quantum spin-impurity systems. Employing the discrete symmetry hidden in spin-impurity models, we present a new canonical transformation that completely decouples the impurity and bath degrees of freedom. Combining it with Gaussian states, we present a family of many-body states to efficiently encode nontrivial impurity-bath correlations. We demonstrate its successful application to the anisotropic and two-lead Kondo models by studying their spatiotemporal dynamics and universal behavior in the correlations, relaxation times, and the differential conductance. We compare them to previous analytical and numerical results. In particular, we apply our method to study new types of nonequilibrium phenomena that have not been studied by other methods, such as long-time crossover in the ferromagnetic easy-plane Kondo model. The present approach will be applicable to a variety of unsolved problems in solid-state and ultracold-atomic systems.

Doublon-Holon Origin of the Subpeaks at the Hubbard Band Edges

Dynamical mean-field theory (DMFT) studies frequently observe a fine structure in the local spectral function of the SU(2) Fermi-Hubbard model at half filling: In the metallic phase close to the Mott transition, subpeaks emerge at the inner edges of the Hubbard bands. Here we demonstrate that these subpeaks originate from the low-energy effective interaction of doublon-holon pairs, by investigating how the correlation functions of doublon and holon operators contribute to the subpeaks. A mean-field analysis of the low-energy effective Hamiltonian provides results consistent with our DMFT calculation using the numerical renormalization group as an impurity solver. In the SU(3) and SU(4) Hubbard models, the subpeaks become more pronounced due to the increased degeneracy of doublon-holon pair excitations.

Time Evolution of the Kondo Resonance in Response to a Quench

We investigate the time evolution of the Kondo resonance in response to a quench by applying the time-dependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within the TDNRG approach a numerically tractable expression for the retarded two-time nonequilibrium Green function G(t+t^{'},t), and its associated time-dependent spectral function, A(ω,t), for times t both before and after the quench. Quenches from both mixed valence and Kondo correlated initial states to Kondo correlated final states are considered. For both cases, we find that the Kondo resonance in the zero temperature spectral function, a preformed version of which is evident at very short times t→0^{+}, only fully develops at very long times t≳1/T_{K}, where T_{K} is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale 1/Γ during the time interval -1/Γ≲t≲+1/Γ, where Γ is the hybridization strength. Initial and final state spectral functions are recovered in the limits t→-∞ and t→+∞, respectively. Our formulation of two-time nonequilibrium Green functions within the TDNRG approach provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.

Quantum entanglement and temperature fluctuations

In this paper, we consider entanglement in a system out of equilibrium, adopting the viewpoint given by the formalism of superstatistics. Such an approach yields a good effective description for a system in a slowly fluctuating environment within a weak interaction between the system and the environment. For this purpose, we introduce an alternative version of the formalism within a quantum mechanical picture and use it to study entanglement in the Heisenberg XY model, subject to temperature fluctuations. We consider both isotropic and anisotropic cases and explore the effect of different temperature fluctuations (χ^{2}, log-normal, and F distributions). Our results suggest that particular fluctuations may enhance entanglement and prevent it from vanishing at higher temperatures than those predicted for the same system at thermal equilibrium.

Kondo blockade due to quantum interference in single-molecule junctions

Molecular electronics offers unique scientific and technological possibilities, resulting from both the nanometre scale of the devices and their reproducible chemical complexity. Two fundamental yet different effects, with no classical analogue, have been demonstrated experimentally in single-molecule junctions: quantum interference due to competing electron transport pathways, and the Kondo effect due to entanglement from strong electronic interactions. Here we unify these phenomena, showing that transport through a spin-degenerate molecule can be either enhanced or blocked by Kondo correlations, depending on molecular structure, contacting geometry and applied gate voltages. An exact framework is developed, in terms of which the quantum interference properties of interacting molecular junctions can be systematically studied and understood. We prove that an exact Kondo-mediated conductance node results from destructive interference in exchange-cotunneling. Nonstandard temperature dependences and gate-tunable conductance peaks/nodes are demonstrated for prototypical molecular junctions, illustrating the intricate interplay of quantum effects beyond the single-orbital paradigm.

Spin Seebeck effect in quantum dot side-coupled to topological superconductor

The spin-resolved thermoelectric transport properties of a quantum dot coupled to ferromagnetic leads and side-coupled to a topological superconductor wire hosting Majorana zero-energy modes are studied theoretically. The calculations are performed in the linear response regime by using the numerical renormalization group method. It is shown that transport characteristics are determined by the interplay of Kondo correlations, exchange field due to the presence of ferromagnets and the strength of coupling to Majorana wire. These different energy scales are revealed in the behavior of the Seebeck and spin Seebeck coefficients, which exhibit an enhancement for temperatures of the order of the coupling strength to topological wire. Moreover, it is demonstrated that additional sign changes of the thermopower can occur due to the presence of Majorana zero-energy modes. These findings may provide additional fingerprints of the presence of Majorana fermions.

Strong spin Seebeck effect in Kondo T-shaped double quantum dots

We investigate, taking a theoretical approach, the thermoelectric and spin thermoelectric properties of a T-shaped double quantum dot strongly coupled to two ferromagnetic leads, focusing on the transport regime in which the system exhibits the two-stage Kondo effect. We study the dependence of the (spin) Seebeck coefficient, the corresponding power factor and the figure of merit on temperature, leads' spin polarization and dot level position. We show that the thermal conductance fulfills a modified Wiedemann-Franz law, also in the regime of suppression of subsequent stages of the Kondo effect by the exchange field resulting from the presence of ferromagnets. Moreover, we demonstrate that the spin thermopower is enhanced at temperatures corresponding to the second stage of Kondo screening. Very interestingly, the spin-thermoelectric response of the system is found to be highly sensitive to the spin polarization of the leads. In some cases spin polarization of the order of 1% is sufficient for a strong spin Seebeck effect to occur. This is explained as a consequence of the interplay between the two-stage Kondo effect and the exchange field induced in the double quantum dot. Due to the possibility of tuning the exchange field by the choice of gate voltage, the spin thermopower may also be tuned to be maximal for desired spin polarization of the leads. All calculations are performed with the aid of the numerical renormalization group technique.

Slowdown of the Electronic Relaxation Close to the Mott Transition

We investigate the time-dependent reformation of the quasiparticle peak in a correlated metal near the Mott transition, after the system is quenched into a hot electron state and equilibrates with an environment which is colder than the Fermi-liquid crossover temperature. Close to the transition, we identify a purely electronic bottleneck time scale, which depends on the spectral weight around the Fermi energy in the bad metallic phase in a nonlinear way. This time scale can be orders of magnitude larger than the bare and renormalized electronic hopping time, so that a separation of electronic and lattice time scales may break down. The results are obtained using nonequilibrium dynamical mean-field theory and a slave-rotor representation of the Anderson impurity model.

Dynamics of the two-spin spin-boson model with a common bath

Dynamics of the two-spin spin-boson model in the presence of Ohmic and sub-Ohmic baths is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Its accuracy in dynamics simulations of the two-spin SBM is improved significantly over the single D1 Ansatz, especially in the weak to moderately strong coupling regime. Validity of the multi-D1 Ansatz for various coupling strengths is also systematically examined by making use of the deviation vector which quantifies how faithfully the trial state obeys the Schrödinger equation. The time evolution of population difference and entanglement has been studied for various initial conditions and coupling strengths. Careful comparisons are carried out between our approach and three other methods, i.e., the time-dependent numerical renormalization group (TD-NRG) approach, the Bloch-Redfield theory, and a method based on a variational master equation. For strong coupling, the multi-D1 trial state yields consistent results as the TD-NRG approach in the Ohmic regime while the two disagree in the sub-Ohmic regime, where the multi-D1 trial state is shown to be more accurate. For weak coupling, the multi-D1 trial state agrees with the two master-equation methods in the presence of both Ohmic and sub-Ohmic baths, but shows considerable differences with the TD-NRG approach in the presence of a sub-Ohmic bath, calling into question the validity of the TD-NRG results at long times in the literature.

Transient currents of a single molecular junction with a vibrational mode

By using a propagation scheme for current matrices and an auxiliary mode expansion method, we investigate the transient dynamics of a single molecular junction coupled with a vibrational mode. Our approach is based on the spinless Anderson-Holstein model and the dressed tunnelling approximation for the electronic self-energy in the polaronic regime. The time-dependent currents after a sudden switching on the tunnelling to leads, an abrupt upward step bias pulse and a step potential on the quantum dot are calculated. We show that the strong electron-phonon interaction greatly influences the nonlinear response properties of the system, and gives rise to interesting characteristics on the time traces of transient currents.

Voltage Quench Dynamics of a Kondo System

We examine the dynamics of a correlated quantum dot in the mixed valence regime. We perform numerically exact calculations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibration times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. The time-dependent current saturation temperature connects the equilibrium Kondo temperature to a substantially increased value at voltages outside of the linear response. These signatures are directly observable by experiments in the time domain.

Taming the Dynamical Sign Problem in Real-Time Evolution of Quantum Many-Body Problems

Current nonequilibrium Monte Carlo methods suffer from a dynamical sign problem that makes simulating real-time dynamics for long times exponentially hard. We propose a new "inchworm algorithm," based on iteratively reusing information obtained in previous steps to extend the propagation to longer times. The algorithm largely overcomes the dynamical sign problem, changing the scaling from exponential to quadratic. We use the method to solve the Anderson impurity model in the Kondo and mixed valence regimes, obtaining results both for quenches and for spin dynamics in the presence of an oscillatory magnetic field.

Dynamical Mean-Field Theory Plus Numerical Renormalization-Group Study of Spin-Orbital Separation in a Three-Band Hund Metal

We show that the numerical renormalization group is a viable multi-band impurity solver for dynamical mean-field theory (DMFT), offering unprecedented real-frequency spectral resolution at arbitrarily low energies and temperatures. We use it to obtain a numerically exact DMFT solution to the Hund metal problem for a three-band model on a Bethe lattice at 1/3 filling. The ground state is a Fermi liquid. The one-particle spectral function undergoes a coherence-incoherence crossover with increasing temperature, with spectral weight being transferred from low to high energies. Further, it exhibits a strong particle-hole asymmetry. In the incoherent regime, the self-energy displays approximate power-law behavior for positive frequencies only. The spin and orbital spectral functions show "spin-orbital separation": spin screening occurs at much lower energies than orbital screening. The renormalization group flows clearly reveal the relevant physics at all energy scales.

Modeling the metastable dynamics of correlated structures

Metastable quantum dynamics of an asymmetric triangular cluster that is coupled to a reservoir is investigated. The dynamics is governed by bath-mediated transitions, which in part require a thermal activation process. The decay rate is controlled by tuning the excitation spectrum of the frustrated cluster. We use the master equation approach and construct transition operators in terms of many-body states. We analyze dynamics of observables and reveal metastability of an excited state and of a magnetically polarized ground state.

Nonequilibrium and nonperturbative dynamics of ultrastrong coupling in open lines

The time and space resolved dynamics of a qubit with an Ohmic coupling to propagating 1D photons is studied, from weak coupling to the ultrastrong coupling regime. A nonperturbative study based on matrix product states shows the following results, (i) The ground state of the combined systems contains excitations of both the qubit and the surrounding bosonic field. (ii) An initially excited qubit equilibrates through spontaneous emission to a state, which under certain conditions is locally close to that ground state, both in the qubit and the field. (iii) The resonances of the combined qubit-photon system match those of the spontaneous emission process and also the predictions of the adiabatic renormalization [A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987)]. Finally, nonperturbative ab initio calculations show that this physics can be studied using a flux qubit galvanically coupled to a superconducting transmission line.

Kondo memory in driven strongly correlated quantum dots

We investigate the real-time current response of strongly correlated quantum dot systems under sinusoidal driving voltages. By means of an accurate hierarchical equations of motion approach, we demonstrate the presence of prominent memory effects induced by the Kondo resonance on the real-time current response. These memory effects appear as distinctive hysteresis line shapes and self-crossing features in the dynamic current-voltage characteristics, with concomitant excitation of odd-number overtones. They emerge as a cooperative effect of quantum coherence-due to inductive behavior-and electron correlations-due to the Kondo resonance. We also show the suppression of memory effects and the transition to classical behavior as a function of temperature. All these phenomena can be observed in experiments and may lead to novel quantum memory applications.

Symmetry breaking and restoration using the equation-of-motion technique for nonequilibrium quantum impurity models

The dynamics of correlated electrons in quantum impurity models is typically described within the nonequilibrium Green function formalism combined with a suitable approximation. One common approach is based on the equation-of-motion technique often used to describe different regimes of the dynamic response. Here, we show that this approach may violate certain symmetry relations that must be fulfilled by the definition of the Green functions. These broken symmetries can lead to unphysical behaviour. To circumvent this pathological shortcoming of the equation-of-motion approach we provide a scheme to restore basic symmetry relations. Illustrations are given for the Anderson and double Anderson impurity models.

Closed and open system dynamics in a fermionic chain with a microscopically specified bath: relaxation and thermalization

We study thermalization in a one-dimensional quantum system consisting of a noninteracting fermionic chain with each site of the chain coupled to an additional bath site. Using a density matrix renormalization group algorithm we investigate the time evolution of the observables in the chain after a quantum quench. For low densities we show that the intermediate time dynamics can be quantitatively described by a system of coupled equations of motion. For higher densities our numerical results show a prethermalization for the local observables at intermediate times and a full thermalization to the grand canonical ensemble at long times. For the case of a weak bath-chain coupling we find, in particular, a Fermi momentum distribution in the chain in equilibrium in spite of the seemingly oversimplified bath in our model.

Nonequilibrium dynamical mean-field theory: an auxiliary quantum master equation approach

We introduce a versatile method to compute electronic steady-state properties of strongly correlated extended quantum systems out of equilibrium. The approach is based on dynamical mean-field theory (DMFT), in which the original system is mapped onto an auxiliary nonequilibrium impurity problem imbedded in a Markovian environment. The steady-state Green's function of the auxiliary system is solved by full diagonalization of the corresponding Lindblad equation. The approach can be regarded as the nontrivial extension of the exact-diagonalization-based DMFT to the nonequilibrium case. As a first application, we consider an interacting Hubbard layer attached to two metallic leads and present results for the steady-state current and the nonequilibrium density of states.

Nonequilibrium thermodynamics of interacting tunneling transport: variational grand potential, density functional formulation and nature of steady-state forces

The standard formulation of tunneling transport rests on an open-boundary modeling. There, conserving approximations to nonequilibrium Green function or quantum statistical mechanics provide consistent but computational costly approaches; alternatively, the use of density-dependent ballistic-transport calculations (e.g., Lang 1995 Phys. Rev. B 52 5335), here denoted 'DBT', provides computationally efficient (approximate) atomistic characterizations of the electron behavior but has until now lacked a formal justification. This paper presents an exact, variational nonequilibrium thermodynamic theory for fully interacting tunneling and provides a rigorous foundation for frozen-nuclei DBT calculations as a lowest-order approximation to an exact nonequilibrium thermodynamic density functional evaluation. The theory starts from the complete electron nonequilibrium quantum statistical mechanics and I identify the operator for the nonequilibrium Gibbs free energy which, generally, must be treated as an implicit solution of the fully interacting many-body dynamics. I demonstrate a minimal property of a functional for the nonequilibrium thermodynamic grand potential which thus uniquely identifies the solution as the exact nonequilibrium density matrix. I also show that the uniqueness-of-density proof from a closely related Lippmann-Schwinger collision density functional theory (Hyldgaard 2008 Phys. Rev. B 78 165109) makes it possible to express the variational nonequilibrium thermodynamic description as a single-particle formulation based on universal electron-density functionals; the full nonequilibrium single-particle formulation improves the DBT method, for example, by a more refined account of Gibbs free energy effects. I illustrate a formal evaluation of the zero-temperature thermodynamic grand potential value which I find is closely related to the variation in the scattering phase shifts and hence to Friedel density oscillations. This paper also discusses the difference between the here-presented exact thermodynamic forces and the often-used electrostatic forces. Finally the paper documents an inherent adiabatic nature of the thermodynamic forces and observes that these are suited for a nonequilibrium implementation of the Born-Oppenheimer approximation.

Nonequilibrium Kondo model: crossover from weak to strong coupling

We analyze the nonequilibrium Kondo model at finite voltage and temperature by using a new formulation of the real-time renormalization group method with the Laplace variable as the flow parameter. We evaluate the energy-dependent spin relaxation rate and nonlinear conductance, and derive an approximate form for the universal line shape for the latter in the whole crossover regime from weak to strong coupling. The results are shown to agree well with exact methods in equilibrium, Fermi-liquid theory, weak-coupling expansions, and recent experiments. For the transient spin dynamics we find a universal exponential decay in the long-time limit along with a truncation-dependent pre-exponential power law. For multichannel models a pure power-law decay typical for non-Fermi-liquid behavior is predicted.

Variational numerical renormalization group: bridging the gap between NRG and density matrix renormalization group

The numerical renormalization group (NRG) is rephrased as a variational method with the cost function given by the sum of all the energies of the effective low-energy Hamiltonian. This allows us to systematically improve the spectrum obtained by NRG through sweeping. The ensuing algorithm has a lot of similarities to the density matrix renormalization group (DMRG) when targeting many states, and this synergy of NRG and DMRG combines the best of both worlds and extends their applicability. We illustrate this approach with simulations of a quantum spin chain and a single impurity Anderson model where the accuracy of the effective eigenstates is greatly enhanced as compared to the NRG, especially in the transition to the continuum limit.

Transient electron dynamics in a vibrating quantum dot in the Kondo regime

We employ the time-dependent non-crossing approximation to investigate the joint effect of strong electron-electron and electron-phonon interaction on the instantaneous conductance of a single-molecule transistor which is abruptly moved into the Kondo regime by means of a gate voltage. We find that the instantaneous conductance exhibits decaying sinusoidal oscillations on the long timescale for infinitesimal bias. The ambient temperature and electron-phonon coupling strength influence the amplitude of these oscillations. The frequency of the oscillations is found to be equal to the phonon frequency. We argue that the origin of these oscillations can be attributed to the interference between the emerging Kondo resonance and its phonon sidebands. We discuss the effect of finite bias on these oscillations.

Many-body dynamics of exciton creation in a quantum dot by optical absorption: a quantum quench towards Kondo correlations

We study a quantum quench for a semiconductor quantum dot coupled to a fermionic reservoir, induced by the sudden creation of an exciton via optical absorption. The subsequent emergence of correlations between spin degrees of freedom of dot and reservoir, culminating in the Kondo effect, can be read off from the absorption line shape and understood in terms of the three fixed points of the single-impurity Anderson model. At low temperatures the line shape is dominated by a power-law singularity, with an exponent that depends on gate voltage and, in a universal, asymmetric fashion, on magnetic field, indicative of a tunable Anderson orthogonality catastrophe.

Interaction quench dynamics in the Kondo model in the presence of a local magnetic field

In this work we investigate the quench dynamics in the Kondo model on the Toulouse line in the presence of a local magnetic field. It is shown that this setup can be realized by either applying the local magnetic field directly or by preparing the system in a macroscopically spin-polarized initial state. In the latter case, the magnetic field results from a subtlety in applying the bosonization technique where terms that are usually referred to as finite-size corrections become important in the present non-equilibrium setting. The transient dynamics are studied by analyzing exact analytical results for the local spin dynamics. The timescale for the relaxation of the local dynamical quantities turns out to be exclusively determined by the Kondo scale. In the transient regime, one observes damped oscillations in the local correlation functions with a frequency set by the magnetic field.

Charge transport through single molecules, quantum dots and quantum wires

We review recent progress in the theoretical description of correlation and quantum fluctuation phenomena in charge transport through single molecules, quantum dots and quantum wires. Various physical phenomena are addressed, relating to cotunneling, pair-tunneling, adiabatic quantum pumping, charge and spin fluctuations, and inhomogeneous Luttinger liquids. We review theoretical many-body methods to treat correlation effects, quantum fluctuations, non-equilibrium physics, and the time evolution into the stationary state of complex nanoelectronic systems.

Relaxation versus decoherence: spin and current dynamics in the anisotropic Kondo model at finite bias and magnetic field

Using a nonequilibrium renormalization group method we study the real-time evolution of spin and current in the anisotropic Kondo model (both antiferromagnetic and ferromagnetic) at a finite magnetic field h(0) and bias voltage V. We derive analytic expressions for all times in the weak-coupling regime max{V, h(0),1/t} >> T(c) (T(c) is the strong-coupling scale). We find that all observables decay both with the spin relaxation and decoherence rates Gamma(1/2). Various V-dependent logarithmic, oscillatory, and power-law contributions are predicted. The low-energy cutoff of logarithmic terms is generically identified by the difference of transport decay rates. For small times t << max{V, h(0)}(-1), we obtain universal dynamics for spin and current.