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arXiv:1312.4686 · Physical Review B

Finite-frequency-dependent noise of a quantum dot in a magnetic field

Cătălin Paşcu Moca, Pascal Simon, Chung‐Hou Chung, Gergely Zaránd

Published Apr 29, 2014• 15 citations
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Abstract

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.

DOI
10.1103/physrevb.89.155138

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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, \ensuremathℏ\ensuremathω=∣eV∣\ensuremath{\hbar}\ensuremath{\omega}=|eV|\ensuremathℏ\ensuremathω=∣eV∣, a corresponding singular behavior is found to emerge in the noise spectrum at frequencies \ensuremathℏ\ensuremathω\ensuremath≈∣eV\ifmmode±\else\textpm\fiB∣\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\approx}|eV\ifmmode\pm\else\textpm\fi{}B|\ensuremathℏ\ensuremathω\ensuremath≈∣eV\ifmmode±\else\textpm\fiB∣. The predicted singularities are measurable with present-day experimental techniques.
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.
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.
Single-particle dynamics of the Anderson impurity model in the presence of a magnetic field H are considered, using a recently developed local moment approach that encompasses all energy scales, field and interaction strengths. For strong coupling in particular, the Kondo scaling regime is recovered. Here the frequency (ω/ω K ) and field ( H /ω K ) dependence of the resultant universal scaling spectrum is obtained in large part analytically, and the field-induced destruction of the Kondo resonance investigated. The scaling spectrum is found to exhibit the slow logarithmic tails recently shown to dominate the zero-field scaling spectrum. At the opposite extreme of the Fermi level, it gives asymptotically exact agreement with results for statics known from the Bethe ansatz. Good agreement is also found with the frequency and field-dependence of recent numerical renormalization group calculations. Differential conductance experiments on quantum dots in the presence of a magnetic field are likewise considered, and appear to be well accounted for by the theory. Some new exact results for the problem are also established.
By using the full density matrix approach to spectral functions within the numerical renormalization group method, we present a detailed study of the magnetic field induced splittings in the spin-resolved and the total spectral densities of a Kondo correlated quantum dot described by the single level Anderson impurity model. The universal scaling of the splittings with magnetic field is examined by varying the Kondo scale either by a change of local level position at a fixed tunnel coupling or by a change of the tunnel coupling at a fixed level position. We find that the Kondo-peak splitting $\ensuremath{\Delta}/{T}_{K}$ in the spin-resolved spectral function always scales perfectly for magnetic fields B<8TKB<8{T}_{K}B<8TK​ in either of the two TK{T}_{K}TK​-adjusted paths. Scaling is destroyed for fields B>10TKB>10{T}_{K}B>10TK​. On the other hand, the Kondo peak splitting $\ensuremath{\delta}/{T}_{K}$ in the total spectral function does slightly deviate from the conventional scaling theory in whole magnetic field window along the coupling-varying path. Furthermore, we show the scaling analysis suitable for all field windows within the Kondo regime and two specific fitting scaling curves are given from which certain detailed features at low field are derived. In addition, the scaling dimensionless quantity $\ensuremath{\Delta}/2B$ and $\ensuremath{\delta}/2B$ are also studied and they can reach and exceed 1 in the large magnetic field region, in agreement with a recent experiment [T. M. Liu et al., Phys. Rev. Lett. 103, 026803 (2009)].
We calculate the dynamical spin-spin correlation functions of a Kondo dot coupled to two noninteracting leads held at different chemical potentials. To this end we generalize a recently developed real-time renormalization group (RG) method in frequency space [H. Schoeller, Eur. Phys. J. Spec. Top. 168, 179 (2009)] to allow the calculation of dynamical correlation functions of arbitrary dot operators in systems describing spin and/or orbital fluctuations. The resulting two-loop RG equations are analytically solved in the weak-coupling regime. This implies that the method can be applied provided either the voltage VVV through the dot or the external magnetic field h0{h}_{0}h0​ are sufficiently large, $\mathrm{max}{V,{h}_{0}}\ensuremath{\gg}{T}_{K}$, where the Kondo temperature TK{T}_{K}TK​ is the scale where the system enters the strong-coupling regime. Explicitly, we calculate the longitudinal and transverse spin-spin correlation and response functions as well as the resulting fluctuation-dissipation ratios. The correlation functions in real-frequency space can be calculated in Matsubara space without the need of any analytical continuation. We obtain analytic results for the line shape, the small- and large-frequency limits, and several other features like the height and width of the peak in the transverse susceptibility at $\ensuremath{\Omega}\ensuremath{\approx}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$ denotes the renormalized magnetic field. Furthermore, we discuss how the developed method can be generalized to calculate dynamical correlation functions of other operators involving reservoir degrees of freedom as well.
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 two-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 crossover functions describing non-Fermi-liquid vs Fermi-liquid crossovers and also investigate the effects of a finite magnetic field.