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physica status solidi (b)

Non‐equilibrium properties of mesoscopic systems – functional renormalization group approach

Thomas Pruschke, Riccardo Gezzi, V. Meden

Published Apr 26, 2007• 1 citation
Publisher

Abstract

Abstract Using the Keldysh technique for non‐equilibrium interacting many‐particle systems, the concept of the functional renormalization is extended to non‐equilibrium situations. The approach is completely general and thus allows calculations for both stationary and time‐dependent situations. As a specific example we present results for the stationary state transport through a quantum dot with finite bias voltage. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssb.200674621

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Abstract Using the Keldysh technique for non‐equilibrium interacting many‐particle systems, the concept of the functional renormalization is extended to non‐equilibrium situations. The approach is completely general and thus allows calculations for both stationary and time‐dependent situations. As a specific example we present results for the stationary state transport through a quantum dot with finite bias voltage. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We extend the concept of the functional renormalization for quantum many-body problems to nonequilibrium situations. Using a suitable generating functional based on the Keldysh approach, we derive a system of coupled differential equations for the mmm-particle vertex functions. The approach is completely general and allows calculations for both stationary and time-dependent situations. As a specific example we study the stationary state transport through a quantum dot with local Coulomb correlations at finite bias voltage employing two different truncation schemes for the infinite hierarchy of equations arising in the functional renormalization group scheme.
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.
Abstract We show that the N.N. Bogolubov generating functional method is a very effective tool for studying distribution functions of both equilibrium and non equilibrium states of classical many‐particle dynamical systems. In some cases the Bogolubov generating functionals can be represented by means of infinite Ursell‐Mayer diagram expansions, whose convergence holds under some additional constraints on statistical system. The classical Bogolubov idea [1] to use the Wigner density operator transformation for studying the non equilibrium distribution functions is developed and new analytic non‐stationary solution to the classical N.N. Bogolubov evolution functional equation is constructed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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.
We investigate equilibrium and steady-state nonequilibrium transport properties of a spinless resonant level locally coupled to two conduction bands of width $\ensuremath{\sim}\ensuremath{\Gamma}$ via a Coulomb interaction UUU and a hybridization ${t}^{\ensuremath{'}}$. 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 $\ensuremath{\Gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$ characterized by the appearance of power laws. Most importantly, at the particle-hole symmetric point the steady-state current decays like $J\ensuremath{\sim}{V}^{\ensuremath{-}{\ensuremath{\alpha}}_{J}}$ as a function of the bias voltage $V⪢{t}^{\ensuremath{'}}$, with an exponent ${\ensuremath{\alpha}}_{J}(U)$ 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 ϵϵϵ of the resonant level is pinned close to either one of the chemical potentials $\ifmmode\pm\else\textpm\fi{}V/2$.
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.