We develop a general approach to the nonequilibrium dynamics of quantum-impurity systems for arbitrary coupling strength. The numerical renormalization group is used to generate a complete basis set necessary for the correct description of the time evolution. We benchmark our method with the exact analytical solution for the resonant-level model. As a first application, we investigate the equilibration of an ultrasmall quantum dot subject to a sudden change of gate voltage and external magnetic field. Two distinct relaxation times are identified for the spin and charge dynamics.

A detailed derivation of the recently proposed time-dependent numerical renormalization-group (TD-NRG) approach to nonequilibrium dynamics in quantum impurity systems is presented. We demonstrate that the method is suitable for fermionic as well as bosonic baths. A comparison with exact analytical results for the charge relaxation in the resonant-level model and for dephasing in the spin-boson model establishes the accuracy of the method. The real-time dynamics of a single spin coupled to both types of baths is investigated. We use the TD-NRG to calculate the spin relaxation and spin precession of a single Kondo impurity. The short-and long-time dynamics is studied as a function of temperature in the ferromagnetic and antiferromagnetic regimes. The short-time dynamics agrees very well with analytical results obtained at second order in the exchange coupling J. In the ferromagnetic regime, the long-time spin decay is described by the scaling variable x = 2ρF J(T )T t. In the antiferromagnetic regime it is governed for T < TK by the Kondo time scale 1/TK . Here ρF is the conduction-electron density of states and TK is the Kondo temperature. Results for spin precession are obtained by rotating the external magnetic field from the x axis to the z axis.

We present an exact solution to the nonequilibrium Kondo problem, based on a special point in the parameter space of the model where both the Hamiltonian and the operator describing the nonequilibrium distribution can be diagonalized simultaneously. Through this solution we are able to compute the differential conductance, spin current, charge-current noise, and magnetization, for arbitrary voltage bias. The differential conductance shows the standard zero-bias anomaly and its splitting under an applied magnetic field. A detailed analysis of the scaling properties at low temperature and voltage is presented. The spin current is independent of the sign of the voltage. Its direction depends solely on the sign of the magnetic field and the asymmetry in the transverse coupling to the left and right leads. The charge-current noise can exceed 2eI c for a large magnetic field, where I c is the charge current. This is not seen in noninteracting quantum problems, but occurs here because of the tunneling of pairs of electrons. The finite-frequency noise spectrum has singularities at ប⍀ϭ Ϯ2 eV, which cannot be explained in terms of noninteracting electrons. These singularities are traced to a different type of pair process involving the simultaneous creation or annihilation of two scattering states. The impurity susceptibility has three characteristic peaks as a function of magnetic field, two of which are due to interlead processes and one is due to intralead processes. Although the solvable point is only one point in the parameter space of the nonequilibrium Kondo problem, we expect it to correctly describe the strong-coupling regime of the model for arbitrary antiferromagnetic coupling constants and to be qualitatively correct as one leaves the strong-coupling regime. ͓S0163-1829͑98͒02442-4͔

A comprehensive theory is presented for the voltage, temperature, and spatial dependence of the tunneling current between a scanning tunneling microscope (STM) tip and a metallic surface with an individual magnetic adatom. Modeling the adatom by a nondegenerate Anderson impurity, a general expression is derived for a weak tunneling current in terms of the dressed impurity Green function, the impurity-free surface Green function, and the tunneling matrix elements. This generalizes Fano's analysis to the interacting case. The differential-conductance lineshapes seen in recent STM experiments with the tip directly over the magnetic adatom are reproduced within our model, as is the rapid decay, ∼ 10Å, of the low-bias structure as one moves the tip away from the adatom. With our simple model for the electronic structure of the surface, there is no dip in the differential conductance at approximately one lattice spacing from the magnetic adatom, but rather we see a resonant enhancement. The formalism for tunneling into small clusters of magnetic adatoms is developed.

We solve exactly a version of the nonequilibrium Kondo problem, which consists of two leads at different chemical potentials connected by a magnetic impurity. Using the nonequilibrium steady-state solution, a host of properties are computed: charge and spin current, current noise, magnetization, and susceptibility. Scaling of the differential conductance with V/T is shown to only be valid to leading order.In the last few years there have been several exciting experiments in strongly interacting mesoscopic systems, such as scattering from a single Kondo impurity ' and tunneling in a Luttinger liquid. In these strongly correlated systems the properties of the interacting system are markedly different from those of the noninteracting ones. Theoretically they are a challenge to describe because of the importance of interactions and because of the added challenge of doing transport calculations. In particular, transport quantities like the differential conductance away from linear response are not simple equilibrium correlation functions, but actually an infinite sum of correlation functions. In essence the finite applied bias leads to a new steady state of the system.In this paper we provide an exact solution for the nonequilibrium steady state of tunneling through a magnetic impurity. This solution is an extension of exact solutions in the equilibrium Kondo problem to the nonequilibriurn case, with the bias being included exactly to all orders. Although this model has been studied by a number of perturbative approaches, ' this is the first exact result on it and to the best of our knowledge the first exact solution of a nonequilibrium strongly correlated quantum problem.What does it mean to have an exact solution of a nonequilibrium quantum problem~Normally an exact solution means that one can diagonalize the Hamiltonian. For example, in the Toulouse limit of the Kondo problem one can perform a canonical transformation to reduce the Hamiltonian to a quadratic form, which can easily be diagonalized.For a nonequilibrium problem, in addition to the Hamiltonian one must define how one drives the system out of equilibrium, e.g. , by a chemical potential difference.In conventional perturbation theory for a tunneling problem one starts with two systems at different chemical potentials and connects them at some initial time. The system then evolves in time to a nonequilibriurn steady state. Recently, this same process has been written in an operator form. For this case one introduces an operator Yo which contains information about the initial distribution of the system. The operator Yo evolves in time into another operator Y, which satisfies t Y,~= i r/( Yo Y)Here, y is a positive infinitesimal included to ensure the correct boundary condition. It does not enter into any physical quantities. Thus, an exact solution means finding Y and diagonalizing, W~-Y. Typically this is more difficult than diagonalizing M alone, since one must find an explicit basis which diagonalizes both M~and Y. In general Y is a complicated many-body...