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arXiv:1711.10835 · Physical review. B./Physical review. B

Width of the charge-transfer peak in the SU( N ) impurity Anderson model and its relevance to nonequilibrium transport

J. Fernández, Franco T. Lisandrini, P. Roura-Bas, C. J. Gazza, Armando A. Aligia

Published Jan 25, 2018• 10 citations
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Abstract

We calculate the width $2{\mathrm{\ensuremath{\Delta}}}_{\text{CT}}$ and intensity of the charge-transfer peak (the one lying at the on-site energy ${E}_{d}$) in the impurity spectral density of states as a function of ${E}_{d}$ in the SU($N$) impurity Anderson model (IAM). We use the dynamical density-matrix renormalization group (DDMRG) and the noncrossing approximation (NCA) for $N=4$ and a 1/$N$ variational approximation in the general case. In particular, while for ${E}_{d}\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$, where $\mathrm{\ensuremath{\Delta}}$ is the resonant level half-width, ${\mathrm{\ensuremath{\Delta}}}_{\text{CT}}=\mathrm{\ensuremath{\Delta}}$ as expected in the noninteracting case, for $\ensuremath{-}{E}_{d}\ensuremath{\gg}N\mathrm{\ensuremath{\Delta}}$ one has ${\mathrm{\ensuremath{\Delta}}}_{\text{CT}}=N\mathrm{\ensuremath{\Delta}}$. In the $N=2$ case, some effects of the variation of ${\mathrm{\ensuremath{\Delta}}}_{\text{CT}}$ with ${E}_{d}$ were observed in the conductance through a quantum dot connected asymmetrically to conducting leads at finite bias [J. K\"onemann et al., Phys. Rev. B 73, 033313 (2006)]. More dramatic effects are expected in similar experiments that can be carried out in systems of two quantum dots, carbon nanotubes or other, realizing the SU(4) IAM.

DOI
10.1103/physrevb.97.045144

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We calculate the width 2\ensuremathΔCT2{\mathrm{\ensuremath{\Delta}}}_{\text{CT}}2\ensuremathΔCT​ and intensity of the charge-transfer peak (the one lying at the on-site energy Ed{E}_{d}Ed​) in the impurity spectral density of states as a function of Ed{E}_{d}Ed​ in the SU(NNN) impurity Anderson model (IAM). We use the dynamical density-matrix renormalization group (DDMRG) and the noncrossing approximation (NCA) for N=4N=4N=4 and a 1/NNN variational approximation in the general case. In particular, while for Ed\ensuremath≫\ensuremathΔ{E}_{d}\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}Ed​\ensuremath≫\ensuremathΔ, where \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ is the resonant level half-width, \ensuremathΔCT=\ensuremathΔ{\mathrm{\ensuremath{\Delta}}}_{\text{CT}}=\mathrm{\ensuremath{\Delta}}\ensuremathΔCT​=\ensuremathΔ as expected in the noninteracting case, for \ensuremath−Ed\ensuremath≫N\ensuremathΔ\ensuremath{-}{E}_{d}\ensuremath{\gg}N\mathrm{\ensuremath{\Delta}}\ensuremath−Ed​\ensuremath≫N\ensuremathΔ one has \ensuremathΔCT=N\ensuremathΔ{\mathrm{\ensuremath{\Delta}}}_{\text{CT}}=N\mathrm{\ensuremath{\Delta}}\ensuremathΔCT​=N\ensuremathΔ. In the N=2N=2N=2 case, some effects of the variation of \ensuremathΔCT{\mathrm{\ensuremath{\Delta}}}_{\text{CT}}\ensuremathΔCT​ with Ed{E}_{d}Ed​ were observed in the conductance through a quantum dot connected asymmetrically to conducting leads at finite bias [J. K"onemann et al., Phys. Rev. B 73, 033313 (2006)]. More dramatic effects are expected in similar experiments that can be carried out in systems of two quantum dots, carbon nanotubes or other, realizing the SU(4) IAM.
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We study finite-temperature properties of the Kondo effect in a carbon nanotube (CNT) quantum dot using the Wilson numerical renormalization group (NRG). In the absence of magnetic fields, four degenerate energy levels of the CNT consisting of spin and orbital degrees of freedom give rise to the SU(4) Kondo effect. We revisit the universal scaling behavior of the SU(4) conductance for quarter- and half-filling in a wide temperature range. We find that the filling dependence of the universal scaling behavior at low temperatures TTT can be explained clearly with an extended Fermi-liquid theory. This theory clarifies that a T2{T}^{2}T2 coefficient of conductance becomes zero at quarter-filling, whereas the coefficient at half-filling is finite. We also study a field-induced crossover from the SU(4) to SU(2) Kondo state observed at the half-filled CNT dot. The crossover is caused by the matching of the spin and orbital Zeeman splittings, which lock two levels among the four at the Fermi level even in magnetic fields BBB. We find that the conductance shows the SU(4) scaling behavior at ${\ensuremath{\mu}}_{B}B<{k}_{B}{T}_{K}^{\mathrm{SU}(4)}$ and it exhibits the SU(2) universality at ${\ensuremath{\mu}}_{B}B\ensuremath{\gg}{k}_{B}{T}_{K}^{\mathrm{SU}(4)}$, where TKSU(4){T}_{K}^{\mathrm{SU}(4)}TKSU(4)​ is the SU(4) Kondo temperature. To clarify how the excited states evolve along the SU(4) to SU(2) crossover, we also calculate the spectral function. The results show that the Kondo resonance width of the two states locked at the Fermi level becomes sharper with increasing fields. The spectral peaks of the other two levels moving away from the Fermi level merge with atomic limit peaks for ${\ensuremath{\mu}}_{B}B\ensuremath{\gtrsim}{k}_{B}{T}_{K}^{\mathrm{SU}(4)}$.
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In this thesis we deal with the description of quantum impurity systems. In order to achieve such an aim, we exploit in the first part of this work the Reduced Density Matrix (RDM) formalism, in the second one the Density Matrix-Numerical Renormalization Group (DM-NRG) method. The first half of the thesis deals with the derivation and the application of out-of-equilibrium and non-perturbative approximations for the dynamics of the Single Impurity Anderson Model (SIAM) in the perturbative, intermediate coupling and Kondo cross-over regimes. In this part of the thesis, the main aim is to develop new technical tools useful to extend the RDM formalism from the well established perturbative regime to the intermediate and, in the future, to the strong coupling regimes. In this part, we analyse the advantages and drawbacks of the so-called Dressed Second Order (DSO), Dressed DSO (DDSO) and the Resonant Tunnelling Approximation (RTA). In the DSO we include processes capable to describe the broadening of the Coulomb blockade peaks but the DSO itself, as the RTA, is not capable to describe the Kondo resonance width properly. In the DDSO we propose a prescription to solve this issue and to describe properly the exponential dependence of the Kondo resonance width. However, in the DDSO the vertex corrections are not included and this yields a spurious conductance peak in the empty orbital regime (Coulomb valley with zero electrons). For this reason we turn our attention to the inclusion of the vertex corrections within the RTA framework. In the second part, we describe transport and thermodynamic properties of carbon-nanotubes (CNTs) based quantum dots using the DM-NRG method. We present results on the spectral function and the linear conductance in cooperation with Catalin Pascu Moca and Gergely Zarand from the Universities of Oradea and Budapest, respectively. In order to model the CNTs we consider an extended Anderson Hamiltonian. Exploiting such an Hamiltonian we study the crossover from the SU(4) to the SU(2) Kondo effect. We demonstrate numerically and analytically that the Kondo temperature scales inversely with respect to the SU(4)-symmetry breaking parameter Δ\DeltaΔ, tending to the Kondo temperature of the SU(2) Kondo effect in the large Δ\DeltaΔ limit. Furthermore we study the CNT quantum dot in the presence of a parallel and perpendicular (with respect to the CNT axis) magnetic field. We find necessary conditions to observe the so-called ``Kondo revivals'' for specific values of the magnetic field. In the final part of this thesis, we model a real world device realised by Jean-Pierre Cleuziou, Ngoc-Viet Nguyen and Wolfgang Wernsdorfer in Grenoble by means of the previously studied Hamiltonian. For this system we search for the best set of parameters to reproduce the experimentally measured linear conductance and we study the underlying Kondo state evaluating the CNT added entropy and specific heat.
We study the three-body Fermi liquid effects in the SU(NNN) Anderson impurity model within the strong interaction limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$, where the occupation number Nd{N}_{d}^{}Nd​ of the impurity levels varies over the range of 0<Nd<10<{N}_{d}^{}<10<Nd​<1. The three-body correlations arise among the impurity electrons when the electron-hole symmetry, the time-reversal symmetry, or both are broken by external fields or potentials. They play a key role in low-energy transport, particularly in the next-to-leading-order terms of power expansion with respect to temperature TTT and bias voltage eVeVeV. Using the numerical renormalization group approach, we calculate the differential conductance and nonlinear current noise through quantum dots, as well as the thermal conductivity of both quantum dots and magnetic alloys. Specifically, we focus on the SU(2) and SU(4) cases, demonstrating how the three-body correlations evolve in the strong interaction limit across the 1/N1/N1/N-filling Kondo regime at ${N}_{d}^{}\ensuremath{\simeq}1.0$ and the valence fluctuation regime where the occupation number Nd{N}_{d}^{}Nd​ varies rapidly with the position ${\ensuremath{\epsilon}}_{d}^{}$ of the impurity levels. We also show that the three-body correlations strongly couple with asymmetries in the tunnel couplings between quantum dots and reservoirs. For N=4N=4N=4, this coupling significantly affects the order (eV)3{(eV)}^{3}(eV)3 nonlinear current through the quarter-filling Kondo state, where the electron-hole symmetry is broken. In particular, in the $U\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, the expansion coefficients exhibit clear plateau structures associated with the Kondo effect at ${N}_{d}^{}\ensuremath{\simeq}1.0$. Our formulation and analysis pave the way for deducing the three-body correlations from the low-energy experiments on the next-to-leading-order transport.