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arXiv:1008.0150 · Physical Review B

Nonequilibrium frequency-dependent noise through a quantum dot: A real-time functional renormalization group approach

Cătălin Paşcu Moca, Pascal Simon, C. H. Chung, Gergely Zaránd

Published May 10, 2011• 33 citations
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Abstract

We construct a real time current-conserving functional renormalization group (RG) scheme on the Keldysh contour to study frequency-dependent transport and noise through a quantum dot in the local moment regime. We find that the current vertex develops a nontrivial nonlocal structure in time that is governed by a new set of RG equations. Solving these RG equations, we compute the complete frequency and temperature dependence of the noise spectrum. For voltages that are large compared to the Kondo temperature (i.e., $\mathit{eV}\ensuremath{\gg}{k}_{B}{T}_{K}$), two sharp antiresonances are found in the noise spectrum at frequencies $\ensuremath{\hbar}\ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}eV$ and, correspondingly, two Kondo-assisted peaks appear in the ac conductance through the dot.

DOI
10.1103/physrevb.83.201303

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We construct a real time current-conserving functional renormalization group (RG) scheme on the Keldysh contour to study frequency-dependent transport and noise through a quantum dot in the local moment regime. We find that the current vertex develops a nontrivial nonlocal structure in time that is governed by a new set of RG equations. Solving these RG equations, we compute the complete frequency and temperature dependence of the noise spectrum. For voltages that are large compared to the Kondo temperature (i.e., eV\ensuremath≫kBTK\mathit{eV}\ensuremath{\gg}{k}_{B}{T}_{K}eV\ensuremath≫kB​TK​), two sharp antiresonances are found in the noise spectrum at frequencies \ensuremathℏ\ensuremathω=\ifmmode±\else\textpm\fieV\ensuremath{\hbar}\ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}eV\ensuremathℏ\ensuremathω=\ifmmode±\else\textpm\fieV and, correspondingly, two Kondo-assisted peaks appear in the ac conductance through the dot.
We apply the real-time renormalization group (RG) in nonequilibrium to an arbitrary quantum dot in the Coulomb blockade regime. Within one-loop RG equations, we include self-consistently the kernel governing the dynamics of the reduced density matrix of the dot. As a result, we find that relaxation and dephasing rates generically cut off the RG flow. In addition, we include all other cutoff scales defined by temperature, energy excitations, frequency, and voltage. We apply the formalism to transport through single molecular magnets, realized by the fully anisotropic Kondo model (with three different exchange couplings Jx{J}_{x}Jx​, Jy{J}_{y}Jy​, and Jz{J}_{z}Jz​) in a magnetic field hz{h}_{z}hz​. We calculate the differential conductance as function of bias voltage VVV and discuss a quantum phase transition which can be tuned by changing the sign of JxJyJz{J}_{x}{J}_{y}{J}_{z}Jx​Jy​Jz​ via the anisotropy parameters. Finally, we calculate the noise $S(\ensuremath{\Omega})$ at finite frequency $\ensuremath{\Omega}$ for the isotropic Kondo model and find that the dephasing rate determines the height of the shoulders in $dS(\ensuremath{\Omega})∕d\ensuremath{\Omega}$ near $\ensuremath{\Omega}=V$.
We calculate the finite-frequency current noise of a nonequilibrium resonance-level quantum dot close to a dissipative quantum phase transition of the Kosterlitz-Thouless (KT) type between a de-localized phase for weak dissipation and a localized phase for strong dissipation. The resonance-level is coupled to two spinless fermionic baths with a finite bias voltage and an Ohmic boson bath representing the dissipative environment. The system is equivalent to an effective anisotropic Kondo model out of equilibrium. To compute the finite-frequency noise, we combine two recently developed Functional Renormalization Group (FRG) approaches in Refs.[17,22] and in Ref.[23]. The nonequilibrium current noise at zero-temperature and finite frequencies shows a singular dip in the de-localized phase for the magnitude of frequencies equal to the bias voltage; while the dip is smeared out as the system moves to the localized phase. The corresponding peak-to-dip crossover is found in the AC conductance for the magnitude of frequencies equal to the bias voltage. The relevance and applications of our results for the experiments and for tunnelings between Fractional Quantum Hall Edge (FQHE) states and chiral Luttinger liquids are discussed.
We present a detailed study for the finite-frequency current noise of a Kondo quantum dot in the presence of a magnetic field by using a recently developed real-time functional renormalization group approach [C. P. Moca, P. Simon, C. H. Chung, and G. Zar\'and, Phys. Rev. B 83, 201303(R) (2011)]. The scaling equations are modified in an external magnetic field; the couplings and nonlocal current vertices become strongly anisotropic, and develop new singularities. Consequently, in addition to the natural emission threshold frequency, $\ensuremath{\hbar}\ensuremath{\omega}=|eV|$, a corresponding singular behavior is found to emerge in the noise spectrum at frequencies $\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\approx}|eV\ifmmode\pm\else\textpm\fi{}B|$. The predicted singularities are measurable with present-day experimental techniques.
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 nonequilibrium properties of the single-impurity Anderson model by means of the functional renormalization group (fRG) within Keldysh formalism. We present how the level broadening $\ensuremath{\Gamma}/2$ can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi-liquid behavior that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions $U\ensuremath{\lesssim}3\ensuremath{\Gamma}$. In equilibrium we find excellent agreement with numerical RG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with time-dependent density-matrix RG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second-order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.
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