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arXiv:1102.2187 · Physical Review B

Comment on “Scaling feature of magnetic field induced Kondo-peak splittings”

Sebastian Schmitt, Frithjof B. Anders

Published May 23, 2011• 4 citations
arXivHTMLPDF

Abstract

In a recent work, Zhang and coworkers [Phys. Rev. B 82, 075111 (2010)] studied the Zeeman splitting of the Kondo resonance for the single impurity Anderson model in a finite magnetic field $B$ with the numerical renormalization group (NRG) method. They report a discrepancy between the position of the Kondo resonance in the total spectral function and the position in the spin-resolved spectral function at large magnetic fields. Additionally, the position of the Kondo maximum exceeded the Zeeman energy for $B/{T}_{K}\ensuremath{\gtrsim}5\ensuremath{-}10$, where ${T}_{K}$ is the low-energy Kondo scale of the model. In this comment, we argue that both these findings are a result of the specific choice of NRG parameter values. However, we reproduce the crossover in the splitting from Kondo-like behavior to a nonuniversal splitting larger than the Zeeman energy, but this crossover occurs at much larger fields of the order of the charge scale.

DOI
10.1103/physrevb.83.197101

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In a recent work, Zhang and coworkers [Phys. Rev. B 82, 075111 (2010)] studied the Zeeman splitting of the Kondo resonance for the single impurity Anderson model in a finite magnetic field BBB with the numerical renormalization group (NRG) method. They report a discrepancy between the position of the Kondo resonance in the total spectral function and the position in the spin-resolved spectral function at large magnetic fields. Additionally, the position of the Kondo maximum exceeded the Zeeman energy for B/TK\ensuremath≳5\ensuremath−10B/{T}_{K}\ensuremath{\gtrsim}5\ensuremath{-}10B/TK​\ensuremath≳5\ensuremath−10, where TK{T}_{K}TK​ is the low-energy Kondo scale of the model. In this comment, we argue that both these findings are a result of the specific choice of NRG parameter values. However, we reproduce the crossover in the splitting from Kondo-like behavior to a nonuniversal splitting larger than the Zeeman energy, but this crossover occurs at much larger fields of the order of the charge scale.
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)].
Using a recently developed perturbative renormalization group (RG) scheme, we calculate analytically the spectral function of a Kondo impurity for either large frequencies $\ensuremath{\omega}$ or large magnetic field B and arbitrary frequencies. For large $\ensuremath{\omega}\ensuremath{\gg}\mathrm{max}[{B,T}_{K}]$ the spectral function decays as $1/{\mathrm{ln}}^{2}[\ensuremath{\omega}{/T}_{K}]$ with prefactors which depend on the magnetization. The spin-resolved spectral function displays a pronounced peak at $\ensuremath{\omega}\ensuremath{\sim}B$ with a characteristic asymmetry. In a detailed comparison with results from numerical renormalization group (NRG) and bare perturbation theory in next-to-leading logarithmic order, we show that our perturbative RG scheme is controlled by the small parameter $1/\mathrm{ln}[\mathrm{max}(\ensuremath{\omega}{,B)/T}_{K}].$ Furthermore, we assess the ability of the NRG to resolve structures at finite frequencies.
In a recent work [Merker, Kirchner, Mu\~noz, and Costi, Phys. Rev. B 87, 165132 (2013)], the authors compared the results of numerical renormalization group (NRG) and a perturbative approach for the dependence on temperature TTT and magnetic field BBB of the conductance through a quantum dot described by the impurity Anderson model, for small TTT and BBB. We show that the equation used to extract the dependence on BBB from NRG results is incorrect out of the particle-hole symmetric case. As a consequence, in the Kondo regime, the correct NRG results have a weaker dependence on BBB and the disagreement between both approaches increases.
This paper is a corrected version of Phys. Rev. B 95, 165404 (2017), which we have retracted because it contained a trivial but fatal sign error that lead to incorrect conclusions. We extend a recently developed Fermi liquid (FL) theory for the asymmetric single-impurity Anderson model [C. Mora et al., Phys. Rev. B 92, 075120 (2015)] to the case of an arbitrary local magnetic field. To describe the system's low-lying quasiparticle excitations for arbitrary values of the bare Hamiltonian's model parameters, we construct an effective low-energy FL Hamiltonian whose FL parameters are expressed in terms of the local level's spin-dependent ground-state occupations and their derivatives with respect to level energy and local magnetic field. These quantities are calculable with excellent accuracy from the Bethe ansatz solution of the Anderson model. Applying this effective model to a quantum dot in a nonequilibrium setting, we obtain exact results for the curvature of the spectral function, cA{c}_{A}cA​, describing its leading $\ensuremath{\sim}{\ensuremath{\varepsilon}}^{2}$ term, and the transport coefficients cV{c}_{V}cV​ and cT{c}_{T}cT​, describing the leading $\ensuremath{\sim}{V}^{2}$ and $\ensuremath{\sim}{T}^{2}$ terms in the nonlinear differential conductance. A sign change in cA{c}_{A}cA​ or cV{c}_{V}cV​ is indicative of a change from a local maximum to a local minimum in the spectral function or nonlinear conductance, respectively, as is expected to occur when an increasing magnetic field causes the Kondo resonance to split into two subpeaks. We find that the fields BA{B}_{A}BA​, BT{B}_{T}BT​, and BV{B}_{V}BV​ at which cA{c}_{A}cA​, cT{c}_{T}cT​, and cV{c}_{V}cV​ change sign, respectively, are all of order TK{T}_{K}TK​, as expected, with BA=BT=BV=0.75073TK{B}_{A}={B}_{T}={B}_{V}=0.75073{T}_{K}BA​=BT​=BV​=0.75073TK​ in the Kondo limit.
We study the single-impurity Anderson model out of equilibrium under the influence of a bias voltage ϕ and a magnetic field B . We investigate the interplay between the shift ( ) of the Kondo peak in the spin-resolved density of states (DOS) and the one ( ) of the conductance anomaly. In agreement with experiments and previous theoretical calculations we find that, while the latter displays a rather linear behavior with an almost constant slope as a function of B down to the Kondo scale, the DOS shift first features a slower increase reaching the same behavior as only for . Our auxiliary master equation approach yields highly accurate nonequilibrium results for the DOS and for the conductance all the way from within the Kondo up to the charge fluctuation regime, showing excellent agreement with a recently introduced scheme based on a combination of numerical renormalization group with time-dependent density matrix renormalization group.
Numerical renormalization group (NRG) is formulated for nonequilibrium steady-state by converting finite-lattice many-body eigenstates into scattering states. Extension of the full-density-matrix NRG for a biased Anderson impurity model, simplified by formulating with the original orbital basis as the Hamiltonian, enables detailed studies of the sub-Kondo spectral evolution in the zero-temperature limit, confirming the double-resonance structure at bias of the Kondo energy scale TKT_KTK​. The distribution shows distinct multi-scale spectral features at energy ωωω below the Kondo scale (ω≲TKω\lesssim T_Kω≲TK​) and near the bias (ω≳Vω\gtrsim Vω≳V), leading to the nonequilibrium temperature TlocT_{\rm loc}Tloc​ local to the Kondo dot scaling as kBTloc≈Vk_BT_{\rm loc}\approx VkB​Tloc​≈V for V≫TKV\gg T_KV≫TK​. The current-voltage relation in the low-temperature limit (T≪TKT\ll T_KT≪TK​) deviates from the unitary limit as the bias exceeds the Kondo scale (V/2≳TKV/2\gtrsim T_KV/2≳TK​) and reaches the current saturation regime.