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arXiv:cond-mat/9811172 · Physical Review Letters

Strong-Coupling Resistivity in the Kondo Model

Frédéric Lesage, Hubert Saleur

Published May 31, 1999• 18 citations
arXivHTMLPDF

Abstract

By applying methods of integrable quantum field theory to the Kondo problem, we develop a systematic perturbation expansion near the IR (strong coupling) fixed point. This requires knowledge of an infinity of irrelevant operators and their couplings, which we determine exactly. A low temperature expansion (i.e., all the corrections to Fermi liquid theory) of the resistivity follows, extending the well-known Nozi\`eres ${T}^{2}$ result in the exactly screened case to arbitrary order. The example of the ordinary Kondo model is worked out in detail: We determine $\ensuremath{\rho}$ up to order ${T}^{6}$ and compare the result with available numerical data.

DOI
10.1103/physrevlett.82.4540

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By applying methods of integrable quantum field theory to the Kondo problem, we develop a systematic perturbation expansion near the IR (strong coupling) fixed point. This requires knowledge of an infinity of irrelevant operators and their couplings, which we determine exactly. A low temperature expansion (i.e., all the corrections to Fermi liquid theory) of the resistivity follows, extending the well-known Nozi`eres T2{T}^{2}T2 result in the exactly screened case to arbitrary order. The example of the ordinary Kondo model is worked out in detail: We determine \ensuremathρ\ensuremath{\rho}\ensuremathρ up to order T6{T}^{6}T6 and compare the result with available numerical data.
We extend the Fermi liquid theory of Nozi\`eres by introducing the next-to-leading order corrections to the Fermi liquid fixed point. For a general SU(N)\text{SU}(N)SU(N) Kondo impurity away from half-filling, this extension is necessary to compute observables (resistivity, current or noise) at low energy. Three additional contributions are identified and their coupling constants are related using an original (and more complete) formulation of the Kondo resonance floating. In the conformal field theory language, a single cubic operator is proposed that produces the same three contributions with the same coupling constants. Comparison with an exact free-energy expansion further relates the leading and next-to-leading order corrections so that a single energy scale, the Kondo temperature, eventually governs the low-energy regime. We compare our results at large NNN with the approach of Read and Newns and find analytical agreement.
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 study the nonequilibrium transport properties of fully (exactly) screened Kondo quantum dots subject to a finite bias voltage or a finite temperature. First, we calculate the Fermi-liquid coefficients of the conductance for models with arbitrary spin, i.e., its leading behavior for small bias voltages or temperatures. Second, we determine the low-temperature behavior of the static susceptibility from the exactly known Bethe ansatz results for the magnetization. Third, we study the crossover from strong to weak coupling in the spin-1/2 and the spin-1 models coupled to one or two screening channels, respectively. Using a real-time renormalization group method we calculate the static and dynamical spin-spin correlation functions for the spin-1/2 model as well as the linear and differential conductance and the static susceptibility for the spin-1 model. We define various Kondo scales and discuss their relations. We assess the validity of the renormalization-group treatment by comparing with known results for the temperature dependence of the linear conductance and static susceptibility as well as the Fermi-liquid behavior at low energies.
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We revisit the physics of a Kondo impurity coupled to a fermionic host with a diverging power-law density of states near the Fermi level, $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}{|\ensuremath{\omega}|}^{r}$, with exponent $\ensuremath{-}1<r<0$. Using the analytical understanding of several fixed points, based partially on powerful mappings between models with bath exponents rrr and $(\ensuremath{-}r)$, combined with accurate numerical renormalization group calculations, we determine thermodynamic quantities within the stable phases and also near the various quantum phase transitions. Antiferromagnetic Kondo coupling leads to strong screening with a negative zero-temperature impurity entropy, while ferromagnetic Kondo coupling can induce a stable fractional spin moment. We formulate the quantum field theories for all critical fixed points of the problem, which are fermionic in nature and allow for a perturbative renormalization-group treatment.
We present here the details of a method [A. B. Culver and N. Andrei, Many-body wavefunctions for quantum impurities out of equilibrium, Phys. Rev. B 103, L201103 (2021)] for calculating the time-dependent many-body wavefunction that follows a local quench. We apply the method to the voltage-driven nonequilibrium Kondo model to find the exact time-evolving wavefunction following a quench where the dot is suddenly attached to the leads at t=0t=0t=0. The method, which does not use Bethe ansatz, also works in other quantum impurity models and may be of wider applicability. We show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction of the Kondo model is a current-carrying nonequilibrium steady state that satisfies the Lippmann-Schwinger equation. We show that the electric current in the time-evolving wavefunction is given by a series expression that can be expanded either in weak coupling or in strong coupling, converging to all orders in the steady-state limit in either case. The series agrees to leading order with known results in the well-studied regime of weak antiferromagnetic coupling and also reveals a universal regime of strong ferromagnetic coupling with Kondo temperature ${T}_{K}^{(F)}=D{e}^{\ensuremath{-}\frac{3{\ensuremath{\pi}}^{2}}{8}\ensuremath{\rho}|J|}$ ($J<0, \ensuremath{\rho}|J|\ensuremath{\rightarrow}\ensuremath{\infty}$). In this regime, the differential conductance dI/dVdI/dVdI/dV reaches the unitarity limit 2e2/h2{e}^{2}/h2e2/h asymptotically at large voltage or temperature.