We study the effect of an external magnetic field on the transport properties of a quantum dot using a recently developed extension of the functional renormalization group approach to non-equilibrium situations. We discuss in particular the interplay and competition of the different energy scales of the dot and the magnetic field on the stationary non-equilibrium current and conductance. As rather interesting behavior we find a switching behavior of the magnetic field for intermediate correlations and bias voltage.
We study the effect of an external magnetic field on the transport properties of a quantum dot using a recently developed extension of the functional renormalization group approach to non-equilibrium situations. We discuss in particular the interplay and competition of the different energy scales of the dot and the magnetic field on the stationary non-equilibrium current and conductance. As rather interesting behavior we find a switching behavior of the magnetic field for intermediate correlations and bias voltage.
We study quantum many-body effects on the real-time evolution of the current through quantum dots. By using a non-equilibrium renormalization group approach, we provide analytic results for the relaxation dynamics into the stationary state and identify the microscopic cutoff scales that determine the transport rates. We find rich non-equilibrium physics induced by the interplay of the different energy scales. While the short-time limit is governed by universal dynamics, the long-time behavior features characteristic oscillations as well as an interplay of exponential and power-law decay.
Correlations are a central ingredient of modern condensed matter physics. With the advent of mesoscopic physics, it became possible to study correlated systems in non-equilibrium situations, which can be reached by e.g. driving a constant current through a quantum dot. A basic question is how correlations on the one hand and a non-equilibrium current on the other hand influence each other. Here this interplay is studied for the model system of a quantum dot in the Kondo regime using the real time renormalization group (RTRG) method. This method is a functional renormalization group (RG) method allowing a non-perturbative description of correlated systems in non-equilibrium. For the Kondo quantum dot the method is generalized to the case of double vertices. The RTRG method not only provides equations for the calculation of the renormalized coupling between the quantum dot spin and the reservoir electrons, but also equations for the calculation of the non-equilibrium decoherence rate and the non-equilibrium current.The RG equations are derived to second order. This approximation is expected to be valid as long as the quantum dot does not enter the strong coupling regime, i.e. the effective coupling strength between the quantum dot spin and the electrons does not become too large. It is found that this is the case if the bias voltage applied to the dot or the temperature is much larger than the Kondo temperature.It is shown that the RG equations lead to coupling functions depending on the energy of the incident and the scattered electron. This dependence is simplified to the dependence on the average energy of the incident and the scattered electron in the weak coupling regime.A closer inspection of the RG equations for the couplings shows that they not only contain terms leading to a logarithmic increase of the coupling strength but also subleading terms which are unimportant in the weak coupling regime. The RG equations are solved analytically for the case where the subleading terms are neglected. It is found that the coupling functions develop resonances at energies associated with the chemical potentials of the reservoirs. The height of these resonances is determined by the decoherence rate. However, for bias voltages much larger than the Kondo temperature these resonances do not affect the conductance for a two terminal setup, so the conductance does not provide a measure for the decoherence rate in this case. It is found that the conductance for the two terminal setup is a universal function of voltage/Kondo temperature and temperature/Kondo temperature in the weak coupling regime. This function is calculated analytically in the limits of vanishing voltage or temperature. Furthermore, it is found that for zero temperature the decoherence rate is proportional to the non-equilibrium current flowing through the system. This supports the physical picture that a non-equilibrium current is associated with a decoherence rate, which cuts the logarithmic scaling of the coupling strength.Finally, the RG equations are solved numerically for zero temperature including the subleading corrections found before. As expected, the subleading corrections have no impact on the decoherence rate and the conductance in the weak coupling regime. On the other hand, it is found that they lead to strong deviations from the results of the analytic leading logarithmic approximation if the bias voltage approaches the Kondo temperature. Generally, it is found that the subleading corrections lead to a saturation of the decoherence rate and the conductance to some finite value. However, for the numerical solution a numerical regularization parameter has to be introduced. It turns out that the saturation value found depends on its value, so no statements on the precise value of the decoherence rate and the current can be made in this regime.
The resonant-level model represents a paradigmatic quantum system which serves as a basis for many other quantum impurity models. We provide a comprehensive analysis of the nonequilibrium transport near a quantum phase transition in a spinless dissipative resonant-level model, extending earlier work [Phys. Rev. Lett 102, 216803 (2009)]. A detailed derivation of a rigorous mapping of our system onto an effective Kondo model is presented. A controlled energy-dependent renormalization-group approach is applied to compute the nonequilibrium current in the presence of a finite bias voltage V. In the linear-response regime $V\ensuremath{\rightarrow}0$, the system exhibits as a function of the dissipative strength a localized-delocalized quantum transition of the Kosterlitz-Thouless (KT) type. We address fundamental issues of the nonequilibrium transport near the quantum phase transition: Does the bias voltage play the same role as temperature to smear out the transition? What is the scaling of the nonequilibrium conductance near the transition? At finite temperatures, we show that the conductance follows the equilibrium scaling for V<T, while it obeys a distinct nonequilibrium profile for V>T. We furthermore provide different signatures of the transition in the finite-frequency current noise and ac conductance via a recently developed functional renormalization group (FRG) approach. The generalization of our analysis to nonequilibrium transport through a resonant level coupled to two chiral Luttinger liquid leads, generated by fractional quantum Hall edge states, is discussed. Our work on the dissipative resonant level has direct relevance to experiments on a quantum dot coupled to a resistive environment, such as H. Mebrahtu et al., [Nature (London) 488, 61 (2012)].
Using different techniques, and Fermi-liquid relationships, we calculate the variation with the applied magnetic field (up to second order) of the zero-temperature equilibrium conductance through a quantum dot described by the impurity Anderson model. We focus on the strong-coupling limit $U\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$, where U is the Coulomb repulsion and $\mathrm{\ensuremath{\Delta}}$ is half the resonant-level width, and consider several values of the dot level energy Ed, ranging from the Kondo regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$ to the intermediate-valence regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy. We have mainly used the density-matrix renormalization group (DMRG) and the numerical renormalization group (NRG) combined with renormalized perturbation theory (RPT). Results for the dot occupancy and magnetic susceptibility from the DMRG and NRG+RPT are compared with the corresponding Bethe ansatz results for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, showing an excellent agreement once Ed is renormalized by a constant Haldane shift. For $U<3\mathrm{\ensuremath{\Delta}}$ a simple perturbative approach in U agrees very well with the other methods. The conductance decreases with the applied magnetic field for dot occupancies ${n}_{d}\ensuremath{\sim}1$ and increases for ${n}_{d}\ensuremath{\sim}0.5$ or ${n}_{d}\ensuremath{\sim}1.5$ regardless of the value of U. We also relate the energy scale for the magnetic-field dependence of the conductance with the width of the low-energy peak in the spectral density of the dot.
The accurate quantitative description of the nonequilibrium transport in interacting nanoscale junctions presents a formidable challenge. Tensor network methods have entered the arena as a highly competitive, versatile, and well-controlled approach. Here, in particular, the authors use a recently developed hybrid renormalization group thermofield quench approach. With this, they obtain reliable predictions for spintronic transport through a Kondo-correlated quantum dot coupled to magnetic leads in far-from-equilibrium conditions. The work focuses on the effects of the induced exchange field. As a key result, they demonstrate that an additional external magnetic field can be utilized to achieve a full Kondo revival in an out-of-equilibrium setting.
The aim of this work is the extension of the functional renormalization group (FRG) formalism to treat non-equilibrium situations. To this end, we reformulate the FRG equations in terms of the Keldysh method which is the standard technique to treat systems out of equilibrium. As simplest non-trivial application to test the potential and weakness of the non-equilibrium FRG we choose the single impurity Anderson model (SIAM). This model represents the paradigm for correlation effects in condensed matter physics and it is at the heart of a large range of experimental and theoretical investigations. In particular, the SIAM can be considered as the standard model for describing the physical properties of certain nanostructures and mesoscopic systems, such as quantum dots. These devices are interesting because they show fancy physical effects such as single electron tunneling and they could open new perspectives for future generations of electronic devices. After a general introduction to mesoscopic systems and the basics of the Keldysh technique, we derive the FRG equations for treating non-equilibrium situations and we point out the differences between equilibrium and non-equilibrium FRG schemes. The FRG can be adopted to describe both fermionic and bosonic systems and, at least formally, time-dependent situations as well as the stationary case. In the second part of this work we analyze the transport properties of quantum dots in several physical situations. First, we study the ""easiest"" case, namely we apply the non-equilibrium FRG formalism to the SIAM with finite bias voltage V_B in the stationary state at T=0. Here, we can test the technique by comparing it to available exact results for the linear response regime V_B = 0. As next step we switch on an external magnetic field B and the temperature T in order to observe the effect of these two parameters on the transport properties of a quantum dot. The comparison to known results for V_B = 0 allows us to critically survey the range of applicability of the Non-equilibrium FRG and the accuracy of the results.