We use the method of infinitesimal unitary transformations to calculate zero-temperature correlation functions in the strong-coupling phase of the anisotropic Kondo model. We find the dynamics on all energy scales including the crossover behavior from weak to strong coupling. The integrable structure of the Hamiltonian is not used in our approach. Our method should also be useful in other strong-coupling models since few other analytical methods allow the evaluation of their correlation functions on all energy scales.
We use the method of infinitesimal unitary transformations to calculate zero-temperature correlation functions in the strong-coupling phase of the anisotropic Kondo model. We find the dynamics on all energy scales including the crossover behavior from weak to strong coupling. The integrable structure of the Hamiltonian is not used in our approach. Our method should also be useful in other strong-coupling models since few other analytical methods allow the evaluation of their correlation functions on all energy scales.
A path integral formulation of the SU(N) Kondo model is explored as a possible theoretical tool for studying generalizations of the model, such as the Kondo lattice, which are difficult to study by other means. Saddle point approximations to the path integral lead to an expansion in powers of 1/N. After developing the basic path integral formalism, the method is applied to three related problems. The first of these is an exactly soluble toy model which, like the Kondo model, shows smooth crossover between weakly mixed configurations at high energy scales and strongly mixed singlet configurations at low energy scales. Simple approximations are found for describing the high and low temperature limits of the model, and the failure of these approximations in the crossover region is interpreted in terms of differences between the high and low energy scales. In chapter 4 the low temperature approximation is used to study the Kondo crossover as a function of magnetic field at zero temperature. Smooth and plausible crossover is found provided the magnetic field is introduced in a way which preserves degeneracy proportional toN, indicating that the failure to describe temperature crossover is not a general limitation of the method. Finally, the same zero temperature equations are generalized in chapter 5 to a model of a magnetic impurity interacting with a gapless electronic band. Using a simple renormalization group argument this model is shown to have a transition as a function of coupling constant at zero temperature. This result is confirmed and extended using the zero temperature path integral equations, and certain special features of the magnetic field crossover are noted.
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.
We calculate the dynamical spin-spin correlation functions of a Kondo dot coupled to two noninteracting leads held at different chemical potentials. To this end we generalize a recently developed real-time renormalization group (RG) method in frequency space [H. Schoeller, Eur. Phys. J. Spec. Top. 168, 179 (2009)] to allow the calculation of dynamical correlation functions of arbitrary dot operators in systems describing spin and/or orbital fluctuations. The resulting two-loop RG equations are analytically solved in the weak-coupling regime. This implies that the method can be applied provided either the voltage V through the dot or the external magnetic field h0 are sufficiently large, $\mathrm{max}{V,{h}_{0}}\ensuremath{\gg}{T}_{K}$, where the Kondo temperature TK is the scale where the system enters the strong-coupling regime. Explicitly, we calculate the longitudinal and transverse spin-spin correlation and response functions as well as the resulting fluctuation-dissipation ratios. The correlation functions in real-frequency space can be calculated in Matsubara space without the need of any analytical continuation. We obtain analytic results for the line shape, the small- and large-frequency limits, and several other features like the height and width of the peak in the transverse susceptibility at $\ensuremath{\Omega}\ensuremath{\approx}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$ denotes the renormalized magnetic field. Furthermore, we discuss how the developed method can be generalized to calculate dynamical correlation functions of other operators involving reservoir degrees of freedom as well.
We reinvestigate the large degeneracy solution of the multichannel Kondo problem, and show how in the universal regime the complicated integral equations simplifying the problem can be mapped onto a first-order differential equation. This leads to an explicit expression for the full zero-temperature scaling functions at---and away from---the intermediate non-Fermi-liquid fixed point, providing complete analytic information on the universal low---and intermediate---energy properties of the model. These results also apply to the widely used noncrossing approximation of the Anderson model, taken in the Kondo regime. An extension of this formalism for studying finite-temperature effects is also proposed and offers a simple approach to solve models of strongly correlated electrons with relevance to the physics of heavy fermion compounds.
We analyze the spatial correlation structure of the spin density of an electron gas in the vicinity of an antiferromagnetically coupled Kondo impurity. Our analysis extends to the regime of spin-anisotropic couplings, where there are no quantitative results for spatial correlations in the literature. We use an original and numerically exact method, based on a systematic coherent-state expansion of the ground state of the underlying spin-boson Hamiltonian. It has not yet been applied to the computation of observables that are specific to the fermionic Kondo model. We also present an important technical improvement to the method that obviates the need to discretize modes of the Fermi sea, and allows one to tackle the problem in the thermodynamic limit. As a result, one can obtain excellent spatial resolution over arbitrary length scales, for a relatively low computational cost, a feature that gives the method an advantage over popular techniques such as the numerical and density-matrix renormalization groups. We find that the anisotropic Kondo model shows rich universal scaling behavior in the spatial structure of the entanglement cloud. First, SU(2) spin-symmetry is dynamically restored in a finite domain in the parameter space in the vicinity of the isotropic line, as expected from poor man's scaling. More surprisingly, we are able to obtain in closed analytical form a set of different, yet universal, scaling curves for strong exchange asymmetry, which are parametrized by the longitudinal exchange coupling. Deep inside the cloud, i.e., for distances smaller than the Kondo length, the correlation between the electron spin density and the impurity spin oscillates between ferromagnetic and antiferromagnetic values at the scale of the Fermi wavelength, an effect that is drastically enhanced at strongly anisotropic couplings. Our results also provide further numerical checks and alternative analytical approximations for the Kondo overlaps that were recently computed by Lukyanov, Saleur, Jacobsen, and Vasseur [Phys. Rev. Lett. 114, 080601 (2015)].
Wilson's momentum shell renormalization group method is used to solve for the dynamics of the dissipative two-state system. We utilize the mapping of the spin-boson model onto the anisotropic Kondo model (AKM) and solve for the dynamics of the latter. We find that the AKM captures the physics of the dissipative two-state system for dissipation strength $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}4$ corresponding to $\ensuremath{\infty}\ensuremath{\ge}{J}_{\ensuremath{\Vert}}\ensuremath{\ge}\ensuremath{-}\ensuremath{\infty}$ in the AKM. The dynamics of the AKM shows a smooth crossover between two strong coupling regimes corresponding to weak and strong dissipation in the spin-boson model.