Thermoelectric effects in correlated quantum dots and molecules

We investigate thermoelectric properties of correlated quantum dots and molecules, described by a single level Anderson model coupled to conduction electron leads, by using Wilson's numerical renormalization group method. In the Kondo regime, the thermopower, $S(T)$, exhibits two sign changes, at temperatures $T=T_{1}$ and $T=T_{2}>T_{1}$. We find that $T_{2}$ is of order the level width $\Gamma$ and $T_{1}> T_{p}\approx T_{K}$, where $T_{p}$ is the position of the Kondo induced peak in the thermopower and $T_{K}$ is the Kondo scale. No sign change is found outside the Kondo regime, or, for weak correlations, making a sign change in $S(T)$ a particularly sensitive signature of strong correlations and Kondo physics. For molecules, we investigate the effect of screening by conduction electrons on the thermoelectric transport. We find that a large screening interaction enhances the figure of merit in the Kondo and mixed valence regimes.Comment: 4 pages, 3 figures; to appear in the Proceedings of the International Conference on Strongly Correlated Electron Systems, Santa Fe 2010; revised version: typos corrected and references update

Numerical renormalization group calculation of impurity internal energy and specific heat of quantum impurity models

We introduce a method to obtain the specific heat of quantum impurity models via a direct calculation of the impurity internal energy requiring only the evaluation of local quantities within a single numerical renormalization group (NRG) calculation for the total system. For the Anderson impurity model, we show that the impurity internal energy can be expressed as a sum of purely local static correlation functions and a term that involves also the impurity Green function. The temperature dependence of the latter can be neglected in many cases, thereby allowing the impurity specific heat, $C_{\rm imp}$, to be calculated accurately from local static correlation functions; specifically via $C_{\rm imp}=\frac{\partial E_{\rm ionic}}{\partial T} + 1/2\frac{\partial E_{\rm hyb}}{\partial T}$, where $E_{\rm ionic}$ and $E_{\rm hyb}$ are the energies of the (embedded) impurity and the hybridization energy, respectively. The term involving the Green function can also be evaluated in cases where its temperature dependence is non-negligible, adding an extra term to $C_{\rm imp}$. For the non-degenerate Anderson impurity model, we show by comparison with exact Bethe ansatz calculations that the results recover accurately both the Kondo induced peak in the specific heat at low temperatures as well as the high temperature peak due to the resonant level. The approach applies to multiorbital and multichannel Anderson impurity models with arbitrary local Coulomb interactions. An application to the Ohmic two state system and the anisotropic Kondo model is also given, with comparisons to Bethe ansatz calculations. The new approach could also be of interest within other impurity solvers, e.g., within quantum Monte Carlo techniques.Comment: 16 pages, 15 figures, published versio

Real-Time-RG Analysis of the Dynamics of the Spin-Boson Model

Using a real-time renormalization group method we determine the complete dynamics of the spin-boson model with ohmic dissipation for coupling strengths $\alpha\lesssim 0.1-0.2$. We calculate the relaxation and dephasing time, the static susceptibility and correlation functions. Our results are consistent with quantum Monte Carlo simulations and the Shiba relation. We present for the first time reliable results for finite cutoff and finite bias in a regime where perturbation theory in $\alpha$ or in tunneling breaks down. Furthermore, an unambigious comparism to results from the Kondo model is achieved.Comment: 4 pages, 5 figures, 1 tabl

Transport Coefficients of the Anderson Model via the Numerical Renormalization Group

The transport coefficients of the Anderson model are calculated by extending Wilson's NRG method to finite temperature Green's functions. Accurate results for the frequency and temperature dependence of the single--particle spectral densities and transport time $\tau(\omega,T)$ are obtained and used to extract the temperature dependence of the transport coefficients in the strong correlation limit. The low temperature anomalies in the resistivity, $\rho(T)$, thermopower, $S(T)$, thermal conductivity $\kappa(T)$ and Hall coefficient, $R_{H}(T)$, are discussed. All quantities exhibit the expected Fermi liquid behaviour at low temperature with power law dependecies on $T/T_{K}$ in very good agreement with analytic results based on Fermi liquid theory. Scattering of conduction electrons in higher, $l>0$, angular momentum channels is also considered and an expression is derived for the corresponding transport time and used to discuss the influence of non--resonant scattering on the transport properties.Comment: 45 pages, RevTeX, 28 figures, available on reques

Kondo effect in a magnetic field and the magnetoresistivity of Kondo alloys

The effect of a magnetic field on the spectral density of a $\rm{S=1/2}$ Kondo impurity is investigated at zero and finite temperatures by using Wilson's numerical renormalization group method. A splitting of the total spectral density is found for fields larger than a critical value $H_{c}(T=0)\approx 0.5 T_{K}$, where $T_{K}$ is the Kondo scale. The splitting correlates with a peak in the magnetoresistivity of dilute magnetic alloys which we calculate and compare with the experiments on $\rm{Ce_{x}La_{1-x}Al_{2}}, x=0.0063$. The linear magnetoconductance of quantum dots exhibiting the Kondo effect is also calculated.Comment: 4 pages, 4 eps figure

The numerical renormalization group method for quantum impurity systems

In the beginning of the 1970's, Wilson developed the concept of a fully non-perturbative renormalization group transformation. Applied to the Kondo problem, this numerical renormalization group method (NRG) gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG has been later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method including some guidelines of how to calculate physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean field theory.Comment: 55 pages, 27 figures, submitted to Rev. Mod. Phy

Scaling and universality in the anisotropic Kondo model and the dissipative two-state system

Scaling and universality in the Ohmic two-state system is investigated by exploiting the equivalence of this model to the anisotropic Kondo model. For the Ohmic two-state system, we find universal scaling functions for the specific heat, $C_{\alpha}(T)$, static susceptibility, $\chi_{\alpha}(T)$, and spin relaxation function $S_{\alpha}(\omega)$ depending on the reduced temperature $T/\Delta_{r}$ (frequency $\omega/\Delta_{r}$), with $\Delta_{r}$ the renormalized tunneling frequency, and uniquely specified by the dissipation strength $\alpha$ ($0<\alpha<1$). The scaling functions can be used to extract $\alpha$ and $\Delta_{r}$ in experimental realizations.Comment: 5 pages (LaTeX), 4 EPS figures. Minor changes, typos corrected, journal reference adde

Transient thermoelectricity in a vibrating quantum dot in Kondo regime

We investigate the time evolution of the thermopower in a vibrating quantum dot suddenly shifted into the Kondo regime via a gate voltage by adopting the time-dependent non-crossing approximation and linear response Onsager relations. Behaviour of the instantaneous thermopower is studied for a range of temperatures both in zero and strong electron-phonon coupling. We argue that inverse of the saturation value of decay time of thermopower to its steady state value might be an alternative tool in determination of the Kondo temperature and the value of the electron-phonon coupling strength.Comment: 5 pages, 4 figures, to appear in Physics Letters

An Enhanced Perturbational Study on Spectral Properties of the Anderson Model

The infinite-$U$ single impurity Anderson model for rare earth alloys is examined with a new set of self-consistent coupled integral equations, which can be embedded in the large $N$ expansion scheme ($N$ is the local spin degeneracy). The finite temperature impurity density of states (DOS) and the spin-fluctuation spectra are calculated exactly up to the order $O(1/N^2)$. The presented conserving approximation goes well beyond the $1/N$-approximation ({\em NCA}) and maintains local Fermi-liquid properties down to very low temperatures. The position of the low lying Abrikosov-Suhl resonance (ASR) in the impurity DOS is in accordance with Friedel's sum rule. For $N=2$ its shift toward the chemical potential, compared to the {\em NCA}, can be traced back to the influence of the vertex corrections. The width and height of the ASR is governed by the universal low temperature energy scale $T_K$. Temperature and degeneracy $N$-dependence of the static magnetic susceptibility is found in excellent agreement with the Bethe-Ansatz results. Threshold exponents of the local propagators are discussed. Resonant level regime ($N=1$) and intermediate valence regime ($|\epsilon_f| <\Delta$) of the model are thoroughly investigated as a critical test of the quality of the approximation. Some applications to the Anderson lattice model are pointed out.Comment: 19 pages, ReVTeX, no figures. 17 Postscript figures available on the WWW at http://spy.fkp.physik.th-darmstadt.de/~frithjof

Renormalization Group Approach to Non-equilibrium Green Functions in Correlated Impurity Systems

We present a technique for calculating non-equilibrium Green functions for impurity systems with local interactions. We use an analogy to the calculation of response functions in the x-ray problem.The initial state and the final state problems, which correspond to the situations before and after the disturbance (an electric or magnetic field, for example) is suddenly switched on, are solved with the aid of Wilson's momentum shell renormalization group. The method is illustrated by calculating the non-equilibrium dynamics of the ohmic two-state problem.Comment: 7 pages, 2 figure