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arXiv:1112.0925 · Physical Review B

Magnetotransport in the Kondo model with ferromagnetic exchange interaction

Mathias Cabrera Cano, Serge Florens

Published Jul 3, 2013• 5 citations
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Abstract

We consider the transport properties in an applied magnetic field of the spin $S=1/2$ Kondo model with ferromagnetic exchange coupling to electronic reservoirs, a description relevant for the strong-coupling limit of underscreened spin $S=1$ 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 $\mathcal{T}$-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 $S=1$ molecular quantum dots.

DOI
10.1103/physrevb.88.035104

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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.
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.
Using different techniques, and Fermi-liquid relationships, we calculate the variation with the applied magnetic field (up to second order) of the zero-temperature equilibrium conductance through a quantum dot described by the impurity Anderson model. We focus on the strong-coupling limit $U\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$, where UUU is the Coulomb repulsion and $\mathrm{\ensuremath{\Delta}}$ is half the resonant-level width, and consider several values of the dot level energy Ed{E}_{d}Ed​, ranging from the Kondo regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$ to the intermediate-valence regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy. We have mainly used the density-matrix renormalization group (DMRG) and the numerical renormalization group (NRG) combined with renormalized perturbation theory (RPT). Results for the dot occupancy and magnetic susceptibility from the DMRG and NRG+RPT\mathrm{NRG}+\mathrm{RPT}NRG+RPT are compared with the corresponding Bethe ansatz results for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, showing an excellent agreement once Ed{E}_{d}Ed​ is renormalized by a constant Haldane shift. For $U<3\mathrm{\ensuremath{\Delta}}$ a simple perturbative approach in UUU agrees very well with the other methods. The conductance decreases with the applied magnetic field for dot occupancies ${n}_{d}\ensuremath{\sim}1$ and increases for ${n}_{d}\ensuremath{\sim}0.5$ or ${n}_{d}\ensuremath{\sim}1.5$ regardless of the value of UUU. We also relate the energy scale for the magnetic-field dependence of the conductance with the width of the low-energy peak in the spectral density of the dot.
We study a system consisting of a junction of NNN wires, where the junction is characterized by a scalar SSS-matrix, and an impurity spin is coupled to the electrons near the junction. The wires are modeled as weakly interacting Tomonaga-Luttinger liquids. We derive the renormalization group (RG) equations for the Kondo couplings of the spin to the electrons on different wires. We analyze the RG flows and fixed points for different values of the initial Kondo couplings and of the junction SSS-matrix, such as the decoupled SSS-matrix and the Griffiths (connected) SSS-matrix. We find that the Kondo couplings flow either towards a ferromagnetic (FM) fixed point or towards large and antiferromagnetic (AFM) values in one of two ways. For the Griffiths SSS-matrix, one of the strong coupling flows is towards a FM fixed point with decoupled wires; this is seen by a perturbative analysis. Thus if we start with a system of connected wires with an AFM coupling to the spin impurity, the flow at large distances is towards a system of disconnected wires at the FM fixed point. For the decoupled SSS-matrix, the flow is either to a FM fixed point or to one of two strong coupling fixed points in which all the channels are strongly coupled to each other through the impurity spin. Strong interactions between the electrons with ${K}_{\ensuremath{\rho}}<N∕(N+2)$ can stabilize a multichannel fixed point in which the coupling between different channels goes to zero. We also study the temperature dependence of the conductance at the decoupled and Griffiths SSS-matrices.
The Kondo effect in a one-dimensional spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model in the gapless XY\mathrm{XY}XY regime $(\ensuremath{-}1<\ensuremath{\Delta}<~1)$ is studied both analytically and numerically. In our model an impurity spin (S=1/2)(S=1/2)(S=1/2) is coupled to a single spin in the XXZ\mathrm{XXZ}XXZ spin chain. Perturbative renormalization-group (RG) analysis is performed for various limiting cases to deduce low-energy fixed points. It is shown that in the ground state the impurity spin is screened by forming a singlet with a spin in the host XXZ\mathrm{XXZ}XXZ chain. In the antiferromagnetic side $(0<\ensuremath{\Delta}<~1)$ the host chain is cut into two semi-infinite chains by the singlet. In the ferromagnetic side $(\ensuremath{-}1<\ensuremath{\Delta}<0),$ on the other hand, the host XXZ\mathrm{XXZ}XXZ chain remains as a single chain through ``healing'' of a weakened bond in the low-energy (long-distance) limit. The density-matrix renormalization-group method is used to study the size scaling of finite-size energy gaps and the power-law decay of correlation functions in the ground state. The numerical results are in good agreement with the predictions of the RG analysis. Low-temperature behaviors of specific heat and susceptibility are also discussed.
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.
It is well known that the free electron Kondo problem can be described by a one-dimensional (1D) model because only the s-wave part of the electronic wavefunction is affected by the Kondo coupling. Moreover, since only the spin degrees of freedom are involved in the Kondo interaction, and due to spin–charge separation in 1D, the universal low energy long distance physics of the Kondo model also arises when a magnetic impurity is coupled to the end of a gapless antiferromagnetic J 1 –J 2 spin- chain, where J 1 ( J 2 ) is the (next) nearest neighbor coupling. Experimental realizations of such spin chain models are possible and, using various analytical and numerical techniques, we present a detailed and quantitative comparison between the usual free electron Kondo model and such spin chain versions of the Kondo problem. For the gapless J 1 –J 2 spin chain two cases are studied, with zero next nearest neighbor coupling, J 2 = 0, and with a critical second-neighbor coupling, J 2 = J 2 c . We first focus on the spin chain impurity model with a critical second-neighbor antiferromagnetic exchange where a bulk marginal coupling present in the spin chain model for J 2 < J 2 c vanishes . There, the usual Kondo physics is recovered in the spin chain model in the low energy regime (up to negligible corrections, dropping as powers of inverse length or energy). At J 2 c the spin chain model is not exactly solvable and we demonstrate the equivalence to the Kondo problem by comparing density matrix renormalization group calculations on the frustrated spin chain model with exact Bethe ansatz calculations of the electronic Kondo problem. We then analyze the nearest neighbor model ( J 2 = 0) where a new kind of Kondo effect occurs due to the presence of the bulk marginal coupling. This marginal coupling slightly alters the β function for the Kondo coupling, leading to a slower variation of the Kondo temperature, T K , with the bare Kondo coupling. In their exact Bethe ansatz solution of this spin chain impurity model ( J 2 = 0), Frahm and Zvyagin noted this relation as well as the connection to the Kondo problem. Here, by numerically solving the Bethe ansatz equations we provide further evidence for the connection to Kondo physics and in addition we present low temperature quantum Monte Carlo results for the impurity susceptibility that further support this connection.