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arXiv:1609.06165 · Physical review. B./Physical review. B

At which magnetic field, exactly, does the Kondo resonance begin to split? A Fermi liquid description of the low-energy properties of the Anderson model

Michele Filippone, Cătălin Paşcu Moca, Andreas Weichselbaum, Jan von Delft, Christophe Mora

Published Aug 6, 2018• 40 citations
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Abstract

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, ${c}_{A}$, describing its leading $\ensuremath{\sim}{\ensuremath{\varepsilon}}^{2}$ term, and the transport coefficients ${c}_{V}$ and ${c}_{T}$, describing the leading $\ensuremath{\sim}{V}^{2}$ and $\ensuremath{\sim}{T}^{2}$ terms in the nonlinear differential conductance. A sign change in ${c}_{A}$ or ${c}_{V}$ 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 ${B}_{A}$, ${B}_{T}$, and ${B}_{V}$ at which ${c}_{A}$, ${c}_{T}$, and ${c}_{V}$ change sign, respectively, are all of order ${T}_{K}$, as expected, with ${B}_{A}={B}_{T}={B}_{V}=0.75073{T}_{K}$ in the Kondo limit.

DOI
10.1103/physrevb.98.075404

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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∼\ensuremathε2\ensuremath{\sim}{\ensuremath{\varepsilon}}^{2}\ensuremath∼\ensuremathε2 term, and the transport coefficients cV{c}_{V}cV​ and cT{c}_{T}cT​, describing the leading \ensuremath∼V2\ensuremath{\sim}{V}^{2}\ensuremath∼V2 and \ensuremath∼T2\ensuremath{\sim}{T}^{2}\ensuremath∼T2 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 generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozi\`eres that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives with respect to the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe ansatz and numerical renormalization group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature, and small bias voltage, and compute the coefficients of the $\ensuremath{\sim}{B}^{2}, \ensuremath{\sim}{T}^{2}$, and $\ensuremath{\sim}{V}^{2}$ terms exactly in terms of the Fermi-liquid parameters. The coefficients of T2,V2{T}^{2}, {V}^{2}T2,V2, and B2{B}^{2}B2 are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.
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In this report, we describe a recent development in a Fermi liquid theory for the Kondo effect in quantum dots under a finite bias voltage VVV. Applying the microscopic theory of Yamada and Yosida to a nonequilibrium steady state, we derive the Ward identities for the Keldysh Green's function, and determine the low-energy behavior of the differential conductance dI/dVdI/dVdI/dV exactly up to terms of order (eV)2(eV)^2(eV)2 for the symmetric Anderson model. These results are deduced from the fact that the Green's function at the impurity site is a functional of a nonequilibrium distribution feff(ω)f_{\text{eff}}(ω)feff​(ω), which at eV=0eV=0eV=0 coincides with the Fermi function. Furthermore, we provide an alternative description of the low-energy properties using a renormalized perturbation theory (RPT). In the nonequilibrium state the unperturbed part of the RPT is determined by the renormalized free quasiparticles, the distribution function of which is given by feff(ω)f_{\text{eff}}(ω)feff​(ω). The residual interaction between the quasiparticles U~\widetilde{U}U, which is defined by the full vertex part at zero frequencies, is taken into account by an expansion in the power series of U~\widetilde{U}U. We also discuss the application of the RPT to a high-bias region beyond the Fermi-liquid regime.