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Using a real-time renormalization group method we study the minimal model of a quantum dot dominated by charge fluctuations, the two-lead interacting resonant level model, at finite bias voltage. We develop a set of RG equations to treat the case of weak and strong charge fluctuations, together with the determination of power-law exponents up to second order in the Coulomb interaction. We derive analytic expressions for the charge susceptibility, the steady-state current and the conductance in the situation of arbitrary system parameters, in particular away from the particle-hole symmetric point and for asymmetric Coulomb interactions. In the generic asymmetric situation we find that power laws can be observed for the current only as function of the level position (gate voltage) but not as function of the voltage. Furthermore, we study the quench dynamics after a sudden switch-on of the level-lead couplings. The time evolution of the dot occupation and current is governed by exponential relaxation accompanied by voltage-dependent oscillations and characteristic algebraic decay., Comment: 24 pages, 13 figures; revised version

We study the steady-state current in a minimal model for a quantum dot dominated by charge fluctuations and analytically describe the time evolution into this state. The current is driven by a finite bias voltage V across the dot, and two different renormalization group methods are used to treat small to intermediate local Coulomb interactions. The corresponding flow equations can be solved analytically which allows to identify all microscopic cutoff scales. Exploring the entire parameter space we find rich non-equilibrium physics which cannot be understood by simply considering the bias voltage as an infrared cutoff. For the experimentally relevant case of left-right asymmetric couplings, the current generically shows a power-law suppression for large V. The relaxation dynamics towards the steady state features characteristic oscillations as well as an interplay of exponential and power-law decay.

We investigate equilibrium and steady-state non-equilibrium transport properties of a spinless resonant level locally coupled to two conduction bands of width ~\Gamma via a Coulomb interaction U and a hybridization t'. In order to study the effects of finite bias voltages beyond linear response, a generalization of the functional renormalization group to Keldysh frequency space is employed. Being mostly unexplored in the context of quantum impurity systems out of equilibrium, we benchmark this method against recently-published time-dependent density matrix renormalization group data. We thoroughly investigate the scaling limit \Gamma\to\infty characterized by the appearance of power laws. Most importantly, at the particle-hole symmetric point the steady-state current decays like J ~ V^{-\alpha_J} as a function of the bias voltage V>>t', with an exponent \alpha_J(U) that we calculate to leading order in the Coulomb interaction strength. In contrast, we do not observe a pure power-law (but more complex) current-voltage-relation if the energy \epsilon of the resonant level is pinned close to either one of the chemical potentials \pm V/2.

We consider the steady state non-equilibrium physics of the multichannel interacting resonant level model in the weak coupling regime. By using the scattering state method we show in agreement with the rate equations that the negative differential conductance at large enough voltages is due to the renormalization of the hopping amplitude thus of the vertex., Comment: 3+ pages, 4 figures

Boulat, Saleur and Schmitteckert (BSS) reported results on the full $I-V$ characteristics of the interacting resonant level model (IRLM) exhibiting region with unexpected negative differential conductance (NDC). Using time-dependent density matrix renormalization group complemented with the exact solution performed at a special point (the self-dual point) in the parameter space BSS have shown that at nonzero Coulomb interaction $U$ the current flowing through the impurity level (IL) exhibits a power-law asymptotics as a function of large applied bias voltage. Similar conclusion was earlier reached by Doyon. Even though their results are solid and supported by both analytic and numeric arguments, BSS concluded that "the NDC at large voltage seems a truly nonperturbative behavior, with unclear physical origin". On the contrary, the remarkable physics of NDC can be explained by simple physical arguments and, as we shall show in this Comment, in certain circumstances can be calculated in the framework of perturbation theory., Comment: 1 page 1 figure

We consider the spatial spin correlations around a partially screened spin-1 magnetic moment in a metal exhibiting the underscreened Kondo effect. We find that the underscreening of the impurity spin results in spatial spin correlations that are more pronounced as compared to the fully screened Kondo effect; their power-law decay is weaker because of characteristic logarithmic corrections at large distances. The spin correlator also changes sign as a function of distance to the impurity allowing for ferromagnetic correlations between conduction electron spin density and the local moment. The numerical findings are shown to be in agreement with the predictions deriving from an effective ferromagnetic Kondo Hamiltonian., Comment: 4+ pages, 2 figures, submitted to Phys. Rev. B

The applicability of bosonization and the Anderson-Yuval (AY) approach at strong coupling is investigated by considering two generic impurity models: the multichannel interacting resonant-level and the anisotropic Kondo models. The two methods differ in the renormalization of the conduction-electron density of states (DoS) near the impurity site. Reduction of the DoS, absent in bosonization but accounted for in the AY approach, is shown to be vital in some models yet redundant in others. The criterion being the stability of the strong-coupling fixed point. Renormalization of the DoS is essential for an unstable fixed point, but redundant when a decoupled entity with local dynamics is formed. This rule can be used to boost the accuracy of both methods at strong coupling., Comment: 4+ pages, 3 figures

We analyze the phase diagram of a quantum particle confined to a finite chain, subject to a dissipative environment described by an Ohmic spectral function. Analytical and numerical techniques are employed to explore both the perturbative and non-perturbative regime of the model. For small dissipation the coupling to the environment leads to a narrowing of the density distribution, and to a displacement towards the center of the array of accessible sites. For large values of the dissipation, we find a phase transition to a doubly degenerate phase which reflects the formation of an inhomogeneous effective potential within the array., Comment: 9 pages, 7 figures. Typos corrected

We systematically study the influence of ferromagnetic leads on the Kondo resonance in a quantum dot tuned to the local moment regime. We employ Wilson's numerical renormalization group method, extended to handle leads with a spin asymmetric density of states, to identify the effects of (i) a finite spin polarization in the leads (at the Fermi-surface), (ii) a Stoner splitting in the bands (governed by the band edges) and (iii) an arbitrary shape of the leads density of states. For a generic lead density of states the quantum dot favors being occupied by a particular spin-species due to exchange interaction with ferromagnetic leads leading to a suppression and splitting of the Kondo resonance. The application of a magnetic field can compensate this asymmetry restoring the Kondo effect. We study both the gate-voltage dependence (for a fixed band structure in the leads) and the spin polarization dependence (for fixed gate voltage) of this compensation field for various types of bands. Interestingly, we find that the full recovery of the Kondo resonance of a quantum dot in presence of leads with an energy dependent density of states is not only possible by an appropriately tuned external magnetic field but also via an appropriately tuned gate voltage. For flat bands simple formulas for the splitting of the local level as a function of the spin polarization and gate voltage are given., Comment: 18 pages, 18 figures, accepted for publication in PRB

We study finite-frequency transport properties of the double-dot system recently constructed to observe the two-channel Kondo effect [R. M. Potok et al., Nature 446, 167 (2007)]. We derive an analytical expression for the frequency-dependent linear conductance of this device in the Kondo regime. We show how the features characteristic of the 2-channel Kondo quantum critical point emerge in this quantity, which we compute using the results of conformal field theory as well as numerical renormalization group methods. We determine the universal cross-over functions describing non-Fermi liquid vs. Fermi liquid cross-overs and also investigate the effects of a finite magnetic field., Comment: 11 pages in PRB format

We use the framework set up recently to compute non-perturbatively inelastic scattering from quantum impurities [G. Zar\'and {\it et al.}, Phys. Rev. Lett. {\bf 93}, 107204 (2004)] to study the the energy dependence of the single particle $S$-matrix and the inelastic scattering cross section for a number of quantum impurity models. We study the case of the spin $S=1/2$ two-channel Kondo model, the Anderson model, and the usual $S=1/2$ single-channel Kondo model. We discuss the difference between non-Fermi liquid and Fermi liquid models and study how a cross-over between the non-Fermi liquid and Fermi liquid regimes appears in case of channel anisotropy for the $S=1/2$ two-channel Kondo model. We show that for the most elementary non-Fermi liquid system, the two-channel Kondo model, half of the scattering remains inelastic even at the Fermi energy. Details of the derivation of the reduction formulas and a simple path integral approach to connect the $T$-matrix to local correlation functions are also presented., Comment: published version

We review the non-perturbative theoretical framework set up recently to compute the inelastic scattering cross section from quantum impurities [G. Zar\'and {\it et al.}, Phys. Rev. Lett. {\bf 93}, 107204 (2004)] and show how it can be applied to a number of quantum impurity models. We first use this method for the $S=1/2$ single-channel Kondo model and the Anderson model. In both cases, a large plateau is found in the inelastic scattering rate for incoming energies above $T_K$, and a quasi-linear regime appears in the energy range $0.05 T_K < \omega < 0.5 T_K$, in agreement with the experimental observations. We also present results for the 2-channel Kondo model, the prototype of all non-Fermi liquid models, and show that there half of the scattering remains inelastic even at the Fermi energy., Comment: Review submitted to the proceedings of the conference DECONS 2006

We study the transport properties of a quantum dot (QD) with highly resistive gate electrodes, and show that the QD displays a quantum phase transition analogous to the famous dissipative phase transition first identified by S. Chakravarty [Phys. Rev. Lett. {\bf 49}, 681 (1982)]; for a review see [A. J. Leggett {\em et al.}, Rev. Mod. Phys. {\bf 59}, 1 (1987)]. At temperature T=0, the charge on the central island of a conventional QD changes smoothly as a function of gate voltage, due to quantum fluctuations. However, for sufficiently large gate resistance charge fluctuations on the island can freeze out even at the degeneracy point, causing the charge on the island to change in sharp steps as a function of gate voltage. For $R_g

We consider low temperature transport through a lateral quantum dot asymmetrically coupled to two conducting leads, and tuned to the mixed-valence region separating two adjacent Coulomb blockade valleys with spin S=1/2 and S=1 on the dot. We demonstrate that this system exhibits a quantum phase transition driven by the gate voltage. In the vicinity of the transition the spin on the dot is quantized, even though the fluctuations of charge are strong. The spin-charge separation leads to an unusual Fano-like dependence of the conductance on the gate voltage and to an almost perfect spin polarization of the current through the dot in the presence of a magnetic field.

We study inelastic scattering of energetic electrons off a Kondo impurity. If the energy E of the incoming electron (measured from the Fermi level) exceeds significantly the Kondo temperature T_K, then the differential inelastic cross-section \sigma (E,w), i.e., the cross-section characterizing scattering of an electron with a given energy transfer w, is well-defined. We show that \sigma (E,w) factorizes into two parts. The E-dependence of \sigma (E,w) is logarithmically weak and is due to the Kondo renormalization of the effective coupling. We are able to relate the w-dependence to the spin-spin correlation function of the magnetic impurity. Using this relation, we demonstrate that in the absence of magnetic field the dynamics of the impurity spin causes the electron scattering to be inelastic at any temperature. Quenching of the spin dynamics by an applied magnetic field results in a finite elastic component of the electron scattering cross-section. The differential scattering cross-section may be extracted from the measurements of relaxation of hot electrons injected in conductors containing localized spins., Comment: 15 pages, 9 figures; final version as published, minor changes, reference added

We study a model of frustration of decoherence in an open quantum system. Contrary to other dissipative ohmic impurity models, such as the Kondo model or the dissipative two-level system, the impurity model discussed here never presents overdamped dynamics even for strong coupling to the environment. We show that this unusual effect has its origins in the quantum mechanical nature of the coupling between the quantum impurity and the environment. We study the problem using analytic and numerical renormalization group methods and obtain expressions for the frequency and temperature dependence of the impurity susceptibility in different regimes., Comment: 14 pages, 5 figures

Recent experiments measuring the emission of exciton recombination in a self-organized single quantum dot (QD) have revealed that novel effects occur when the wetting layer surrounding the QD becomes filled with electrons, because the resulting Fermi sea can hybridize with the local electron levels on the dot. Motivated by these experiments, we study an extended Anderson model, which describes a local conduction band level coupled to a Fermi sea, but also includes a local valence band level. We are interested, in particular, on how many-body correlations resulting from the presence of the Fermi sea affect the absorption and emission spectra. Using Wilson's numerical renormalization group method, we calculate the zero-temperature absorption (emission) spectrum of a QD which starts from (ends up in) a strongly correlated Kondo ground state. We predict two features: Firstly, we find that the spectrum shows a power law divergence close to the threshold, with an exponent that can be understood by analogy to the well-known X-ray edge absorption problem. Secondly, the threshold energy $\omega_0$ - below which no photon is absorbed (above which no photon is emitted) - shows a marked, monotonic shift as a function of the exciton binding energy $U_{\rm exc}$, Comment: 10 pages, 9 figures

In a recent work, Le Hur has shown that dissipative coupling to gate electrodes may play an important role in a quantum box near its degeneracy point [K. Le Hur, Phys. Rev. Lett. {\bf 92}, 196804 (2004)]: While quantum fluctuations of the charge of the dot tend to round Coulomb blockade charging steps of the box, strong enough dissipation suppresses these fluctuations and leads to the reappearance of sharp charging steps. In the present paper we study this quantum phase transition in detail using bosonization and numerical renormalization group methods in the limit of vanishing level spacing., Comment: 10 pages, 9 figures

A great variety of experiments, like heat release measurements, acoustic measurements, and transport measurements on mesoscopic samples have proved that two level systems (TLSs) have a crucial role in the low temperature thermal and electric properties of disordered systems. This paper is aimed at reviewing the role of slow TLSs in point contacts. First the theory of point contacts is summarized, concentrating on the discussion of different point contact models, and on the different regimes of electron flow in the contact, mainly focusing on the ballistic and diffusive limit. The Boltzmann equation is solved in both regimes, and the position dependence of the electrical potential is determined. Then the scattering processes in point contacts are investigated, particularly concentrating on the scattering on slow TLSs. If the the electron assisted transitions between the two states are negligible the electron-two level system interaction can be treated with a simplified Hamiltonian. The scattering on such slow TLSs causes nonlinearity in the current-voltage characteristics of the point contact, which can be determined using Fermi's golden role. These calculations are presented showing both the contribution of elastic and inelastic scattering, and including the dependence on the position of the TLS, and on the effect of high frequency irradiation. These results are used to discuss the differences between these slow TLSs and the fast centers which may be described by the two channel Kondo model. The available experimental results are analyzed, distinguishing between the effects due to the different types of TLSs., Comment: 53 pages, 40 figures

The effect of a gate voltage ($V_g$) on the spin-splitting of an electronic level in a quantum dot (QD) attached to ferromagnetic leads is studied in the Kondo regime using a generalized numerical renormalization group technique. We find that the $V_g$-dependence of the QD level spin-splitting strongly depends on the shape of the density of states (DOS). For one class of DOS shapes there is nearly no $V_g$-dependence, for another, $V_g$ can be used to control the magnitude and sign of the spin-splitting, which can be interpreted as a local exchange magnetic field. We find that the spin-splitting acquires a new type of logarithmic divergence. We give an analytical explanation for our numerical results and explain how they arise due to spin-dependent charge fluctuations., Comment: 4 pages, 2 figures

We study, using Numerical Renormalization Group methods, the generalization of the Anderson impurity model where the hopping depends on the filling of the impurity. We show that the model, for sufficiently large values of the assisted hopping term, shows a regime where local pairing correlations are enhanced. These correlations involve pairs fluctuating between on site and nearest neighbor positions.

We show that thermodynamics is insufficient to probe the nature of the low energy dynamics of quantum impurity models and a more subtle analysis based on scattering theory is required. Traditionally, quantum impurity models are classified into one of two categories: Fermi liquids and non-Fermi liquids, depending on the analytic properties of the various thermodynamic quntities. We show, however, that even when a quantum impurity model is a Fermi liquid (an incoming electron at the Fermi level scatters elastically off the impurity), one may find singular thermodynamic behavior if characteristics of quasiparticles are not analytic near the Fermi surface. Prompted by this observation, we divide Fermi liquids into two categories: regular Fermi liquids and singular Fermi Liquids. The difference between regular Fermi liquids, singular Fermi liquids, and non-Fermi liquids fixed points is explained using properties of the many-body S-matrix for impurity quasiparticle scattering. Using the Bethe-Ansatz and numerical RG, we show that whereas the ordinary Kondo Model is a regular Fermi liquid the underscreened Kondo model is a a singular Fermi liquid. This results in a breakdown of Nozieres' Fermi liquid picture for the underscreened and explains the singular thermodynamic behavior noticed in Bethe Ansatz and large-N calculations. Furthermore, we show that conventional regular Fermi liquid behavior is re-established in an external magnetic field H, but with a density of states which diverges as 1/H. Possible connections with the field-tuned quantum criticality recently observed in heavy electron materials are also discussed., Comment: 15 pages, 4 figures

We use the numerical renormalization group method tocalculate the single particle matrix elements $\cal T$ of the many body $T$-matrix of the conduction electrons scattered by a magnetic impurity at T=0 temperature. Since $\cal T$ determines both the total and the elastic, spin-diagonal scattering cross sections, we are able to compute the full energy-, spin- and magnetic field dependence of the inelastic scattering cross section, $\sigma_{\rm inel}(\omega)$. We find an almost linear frequency dependence of $\sigma_{\rm inel}(\omega)$ below the Kondo temperature, $T_K$, which crosses over to a $\sim \omega^2$ behavior only at extremely low energies. Our method can be generalized to other quantum impurity models., Comment: 4 pages

We develop a theory of electron transport in a double quantum dot device recently proposed for the observation of the two-channel Kondo effect. Our theory provides a strategy for tuning the device to the non-Fermi-liquid fixed point, which is a quantum critical point in the space of device parameters. We explore the corresponding quantum phase transition, and make explicit predictions for behavior of the differential conductance in the vicinity of the quantum critical point.

The Kondo effect in quantum dots (QDs) - artificial magnetic impurities - attached to ferromagnetic leads is studied with the numerical renormalization group (NRG) method. It is shown that the QD level is spin-split due to presence of ferromagnetic electrodes, leading to a suppression of the Kondo effect. We find that the Kondo effect can be restored by compensating this splitting with a magnetic field. Although the resulting Kondo resonance then has an unusual spin asymmetry with a reduced Kondo temperature, the ground state is still a locally-screened state, describable by Fermi liquid theory and a generalized Friedel sum rule, and transport in the unitary limit is not spin dependent., Comment: 4 pages, 4 figures

We discuss the problem of a spin 1/2 impurity immersed in a spin S magnetically ordered background. We show that the problem maps onto a generalization of the dissipative two level system (DTLS) with two independent heat baths, associated with the Goldstone modes of the magnet, that couple to different components of the impurity spin operator. Using analytical perturbative renormalization group (RG) methods and accurate numerical renormalization group (NRG) we show that contrary to other dissipative models there is quantum frustration of decoherence and quasi-scaling even in the strong coupling regime. We make predictions for the behavior of the impurity magnetic susceptibility that can be measured in nuclear magnetic resonance (NMR) experiments. Our results may also have relevance to quantum computation., Comment: 4 pages, 3 figures

The interaction between an atom moving in a model double-well potential and the conduction electrons is treated using renormalization group methods in next-to-leading logarithmic order. A large number of excited states is taken into account and the Kondo temperature $T_K$ is computed as a function of barrier parameters. We find that for special parameters $T_K$ can be close to $1 {\rm K}$ and it can be of the same order of magnitude as the renormalized splitting $\Delta$. However, in the perturbative regime we always find that $T_K \alt \Delta $ with a $T_K \alt 1 {\rm K}$ [Aleiner {\em et al.}, Phys. Rev. Lett. {\bf 86}, 2629 (2001)]. We also find that $\Delta$ remains unrenormalized at energies above the Debye frequency, $\omega_{\rm Debye}$., Comment: 9 pages, 9 figures, RevTeX

We study a symmetrical double quantum dot (DD) system with strong capacitive inter-dot coupling using renormalization group methods. The dots are attached to separate leads, and there can be a weak tunneling between them. In the regime where there is a single electron on the DD the low-energy behavior is characterized by an SU(4)-symmetric Fermi liquid theory with entangled spin and charge Kondo correlations and a phase shift $\pi/4$. Application of an external magnetic field gives rise to a large magneto-conductance and a crossover to a purely charge Kondo state in the charge sector with SU(2) symmetry. In a four lead setup we find perfectly spin polarized transmission., Comment: 4 pages, 4 figures, ReVTeX

This paper was withdrawn by the authors., Comment: This paper has been withdrawn

We study how the formation of the Kondo compensation cloud influences the dynamical properties of a magnetic impurity that tunnels between two positions in a metal. The Kondo effect dynamically generates a strong tunneling impurity-conduction electron coupling, changes the temperature dependence of the tunneling rate, and may ultimately result in the destruction of the coherent motion of the particle at zero temperature. We find an interesting two-channel Kondo fixed point as well for a vanishing overlap between the electronic states that screen the magnetic impurity. We propose a number of systems where the predicted features could be observed., Comment: 4 pages, 3 figures, ReVTeX

Earlier a magnetic anisotropy for magnetic impurities nearby the surface of non-magnetic host was proposed in order to explain the size dependence of the Kondo effect in dilute magnetic alloys. Recently Giordano has measured the magnetoresistance of dilute Au(Fe) films for different thicknesses well above the Kondo temperature $T_K$. In this way he verified the existence of that anisotropy even for such a case where the Kondo effect is not dominating. For detailed comparison of that suggestion with experiments, the magnetic field dependence of the magnetoresistance is calculated in the lowest approximation, thus in the second order of the exchange coupling. The strength of the anisotropy is very close to earlier estimates deduced from the size dependence of the Kondo resistivity amplitude., Comment: (11 pages, 8 figures, essential changes compared to the old version)