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arXiv:cond-mat/0407624 · Journal of Physics Condensed Matter

Hamiltonian theory of the strongly coupled limit of the Kondo problem in the overscreened case

Domenico Giuliano, Arturo Tagliacozzo

Published Aug 14, 2004• 4 citations
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Abstract

By properly generalizing Nozie`res' Fermi liquid theory, we construct an Hamiltonian approach to the scattering of conduction electrons off a spin-1/2 impurity in the ovescreneed Kondo regime, as T -> 0. We derive the S-matrix at the interacting fixed point, and the corresponding phase shifts, together with leading energy corrections to the unitary limit. We apply our results to obtain the low-temperature dependence of the 2-channel Kondo conductance, and we relate it to possible transport experiments in a Quantum Dot

DOI
10.1088/0953-8984/16/34/008

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By properly generalizing Nozie`res' Fermi liquid theory, we construct an Hamiltonian approach to the scattering of conduction electrons off a spin-1/2 impurity in the ovescreneed Kondo regime, as T -> 0. We derive the S-matrix at the interacting fixed point, and the corresponding phase shifts, together with leading energy corrections to the unitary limit. We apply our results to obtain the low-temperature dependence of the 2-channel Kondo conductance, and we relate it to possible transport experiments in a Quantum Dot
We investigate quantum transport through the Kondo impurity assuming both a large number of orbital channels $\mathcal{K}\ensuremath{\gg}1$ for the itinerant electrons and a semiclassical spin $\mathcal{S}\phantom{\rule{4pt}{0ex}}\ensuremath{\gg}\phantom{\rule{4pt}{0ex}}1$ for the impurity. The non-Fermi-liquid regime of the Kondo problem is achieved in the overscreened sector K>2S\mathcal{K}>2\mathcal{S}K>2S. We show that there exist two distinct semiclassical regimes for the quantum transport through impurity: (i) $\mathcal{K}\phantom{\rule{4pt}{0ex}}\ensuremath{\gg}\phantom{\rule{4pt}{0ex}}\mathcal{S}\phantom{\rule{4pt}{0ex}}\ensuremath{\gg}\phantom{\rule{4pt}{0ex}}1$, differential conductance vanishes, and (ii) S/K=C\mathcal{S}/\mathcal{K}=\mathcal{C}S/K=C with 0<C<1/20<\mathcal{C}<1/20<C<1/2, differential conductance reaches some nonvanishing fraction of its unitary value. Using the conformal field theory approach we analyze the behavior of the quantum transport observables and residual entropy in both semiclassical regimes. We show that the semiclassical limit (ii) preserves the key features of resonance scattering and the most essential fingerprints of the non-Fermi-liquid behavior. We discuss possible realization of two semiclassical regimes in semiconductor quantum transport experiments.
The non-Fermi-liquid fixed point of the overscreened Kondo problem is studied in the limit of a large number of conduction electron channels, k\ensuremath{\gg}1, using perturbation theory in 1/k. By expanding the beta function and self-energy to sub-leading-order in 1/k, we examine the scaling, thermodynamics, and dynamic response functions in detail. Our results compliment and extend previous work based on conformal field theory. We present the dynamical spin susceptibility for the first time and show that the impurity spin is marginally quenched, with an inelastic linewidth of order T.
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.
We consider the transport properties in an applied magnetic field of the spin S=1/2S=1/2S=1/2 Kondo model with ferromagnetic exchange coupling to electronic reservoirs, a description relevant for the strong-coupling limit of underscreened spin S=1S=1S=1 Kondo impurities. Because the ferromagnetic Kondo interaction is marginally irrelevant, perturbative methods should prove accurate down to low energies. For the purpose of this study, we use a combination of Majorana diagrammatic theory with density matrix numerical renormalization group simulations. In the standard case of antiferromagnetic Kondo exchange, we first show that our simple analytical technique recovers results for the T\mathcal{T}T-matrix and spin relaxation at weak coupling (above the Kondo temperature), which were previously obtained using functional renormalization tools. Considering then the ferromagnetic case, we demonstrate how the low-energy Kondo anomaly splits for arbitrary small values of the Zeeman energy, in contrast to fully screened Kondo impurities near the strong-coupling Fermi liquid fixed point, and in agreement with recent experimental findings for spin S=1S=1S=1 molecular quantum dots.
The Kondo problem, which describes the interaction of a spin sss magnetic impurity with a free Fermi gas, is a classic example of strongly coupled physics. Historically, the problem has been solved by Wilson's numerical renormalization group and later by Bethe ansatz. In this paper, we present an alternate analytic solution of the Kondo problem that combines an expansion in 1/s1/s1/s with the renormalization group. We study both the case of an impurity interacting with a single channel K=1K=1K=1 of fermions in the s→∞s \rightarrow \inftys→∞ limit and the case with KKK channels in the double-scaling limit K→∞K \rightarrow \inftyK→∞, s→∞s \to \inftys→∞, K/sK/sK/s fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large sss and compute thermodynamic observables such as the impurity entropy and susceptibility. We find these observables to agree with the Bethe ansatz results. We also compute the impurity spectral function, finite temperature resistivity and the Kondo screening cloud profile, properties that are not easily accessible from Bethe ansatz. Notably, in the regime K>2sK > 2sK>2s we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.
We analyze the single-channel Kondo model using the recently developed unitary renormalization-group (URG) method and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in good agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both kkk-space number diagonal (Fermi liquid) and off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The URG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local-moment and strong-coupling ground states and is related to a change in the Luttinger volume of the conduction bath under the crossover to strong coupling. Our results offer fresh insight into the nature of the emergent many-particle entanglement within the Kondo cloud and pave the way for further investigations in more exotic contexts such as the fixed point of the overscreened multichannel Kondo problem.