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arXiv:1507.04307 · Physical Review B

Universal spatial correlations in the anisotropic Kondo screening cloud: Analytical insights and numerically exact results from a coherent state expansion

Serge Florens, Izak Snyman

Published Nov 4, 2015• 24 citations
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Abstract

We analyze the spatial correlation structure of the spin density of an electron gas in the vicinity of an antiferromagnetically coupled Kondo impurity. Our analysis extends to the regime of spin-anisotropic couplings, where there are no quantitative results for spatial correlations in the literature. We use an original and numerically exact method, based on a systematic coherent-state expansion of the ground state of the underlying spin-boson Hamiltonian. It has not yet been applied to the computation of observables that are specific to the fermionic Kondo model. We also present an important technical improvement to the method that obviates the need to discretize modes of the Fermi sea, and allows one to tackle the problem in the thermodynamic limit. As a result, one can obtain excellent spatial resolution over arbitrary length scales, for a relatively low computational cost, a feature that gives the method an advantage over popular techniques such as the numerical and density-matrix renormalization groups. We find that the anisotropic Kondo model shows rich universal scaling behavior in the spatial structure of the entanglement cloud. First, SU(2) spin-symmetry is dynamically restored in a finite domain in the parameter space in the vicinity of the isotropic line, as expected from poor man's scaling. More surprisingly, we are able to obtain in closed analytical form a set of different, yet universal, scaling curves for strong exchange asymmetry, which are parametrized by the longitudinal exchange coupling. Deep inside the cloud, i.e., for distances smaller than the Kondo length, the correlation between the electron spin density and the impurity spin oscillates between ferromagnetic and antiferromagnetic values at the scale of the Fermi wavelength, an effect that is drastically enhanced at strongly anisotropic couplings. Our results also provide further numerical checks and alternative analytical approximations for the Kondo overlaps that were recently computed by Lukyanov, Saleur, Jacobsen, and Vasseur [Phys. Rev. Lett. 114, 080601 (2015)].

DOI
10.1103/physrevb.92.195106

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We analyze the spatial correlation structure of the spin density of an electron gas in the vicinity of an antiferromagnetically coupled Kondo impurity. Our analysis extends to the regime of spin-anisotropic couplings, where there are no quantitative results for spatial correlations in the literature. We use an original and numerically exact method, based on a systematic coherent-state expansion of the ground state of the underlying spin-boson Hamiltonian. It has not yet been applied to the computation of observables that are specific to the fermionic Kondo model. We also present an important technical improvement to the method that obviates the need to discretize modes of the Fermi sea, and allows one to tackle the problem in the thermodynamic limit. As a result, one can obtain excellent spatial resolution over arbitrary length scales, for a relatively low computational cost, a feature that gives the method an advantage over popular techniques such as the numerical and density-matrix renormalization groups. We find that the anisotropic Kondo model shows rich universal scaling behavior in the spatial structure of the entanglement cloud. First, SU(2) spin-symmetry is dynamically restored in a finite domain in the parameter space in the vicinity of the isotropic line, as expected from poor man's scaling. More surprisingly, we are able to obtain in closed analytical form a set of different, yet universal, scaling curves for strong exchange asymmetry, which are parametrized by the longitudinal exchange coupling. Deep inside the cloud, i.e., for distances smaller than the Kondo length, the correlation between the electron spin density and the impurity spin oscillates between ferromagnetic and antiferromagnetic values at the scale of the Fermi wavelength, an effect that is drastically enhanced at strongly anisotropic couplings. Our results also provide further numerical checks and alternative analytical approximations for the Kondo overlaps that were recently computed by Lukyanov, Saleur, Jacobsen, and Vasseur [Phys. Rev. Lett. 114, 080601 (2015)].
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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.
Abstract The screening of an impurity spin in the Kondo model occurs over a characteristic length scale ξ K , that defines the size of the Kondo screening cloud or ‘mist’. The presence of such a length scale in experimental and numerical results is rather subtle. A consistent way to show the presence of the screening cloud is to demonstrate scaling in the spatial correlations in terms of the single variable, r / ξ K instead of the correlations depending on r and ξ K separately. Here we study the paradigmatic one channel Kondo model using a spin chain representation, with an impurity spin at one end of the chain coupled with a strength J K ′ . Using Fermi liquid theory combined with numerical results, we obtain new high precision estimates of the non-universal terms in the entanglement entropy which leads to a verification of the expected non-integer ground-state degeneracy, g. This then allows us to study the impurity contribution to the entanglement in detail. If the impurity coupling J K ′ is varied, a precise determination of ξ K can then be obtained. The length scale, ξ K , is then shown to characterize the scaling of both the uniform and alternating part of a measure of the magnetization of part of an odd length chain.
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 address the fundamental question how the spatial Kondo correlations are building up in time assuming an initially decoupled impurity spin ${\stackrel{P\vec}{S}}_{\mathrm{imp}}$. We investigate the time-dependent spin-correlation function $\ensuremath{\chi}(\stackrel{P\vec}{r},t)=\ensuremath{\langle}{\stackrel{P\vec}{S}}_{\mathrm{imp}}\stackrel{P\vec}{s}(\stackrel{P\vec}{r})\ensuremath{\rangle}(t)$ in the Kondo model with antiferromagnetic and ferromagnetic couplings, where $\stackrel{P\vec}{s}(\stackrel{P\vec}{r})$ denotes the spin density of the conduction electrons after switching on the Kondo coupling at time t=0t=0t=0. We present data obtained from a time-dependent numerical renormalisation group (TD-NRG) calculation. We gauge the accuracy of our two-band NRG by the spatial sum rules of the equilibrium correlation functions and the reproduction of the analytically exactly known spin-correlation function of the decoupled Fermi sea. We find a remarkable building up of Kondo correlation outside of the light cone defined by the Fermi velocity of the host metal. By employing a perturbative approach exact in second-order of the Kondo coupling, we connect these surprising correlations to the intrinsic spin-density entanglement of the Fermi sea. The thermal wavelength supplies a cutoff scale at finite temperatures beyond which correlations are exponentially suppressed. We present data for the frequency dependent retarded spin-spin susceptibility and use the results to calculate the real-time response of a weak perturbation in linear response: within the spatial resolution no response outside of the light cone is found.