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arXiv:cond-mat/9304031 · Physical Review Letters

Kinks in the Kondo problem

Paul Fendley

Published Oct 11, 1993• 64 citations
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Abstract

We find the exact quasiparticle spectrum, elastic S matrix, and free energy for the continuum Kondo problem of k species of electrons coupled to an impurity of spin S. Here, the impurity becomes an immobile quasiparticle sitting on the boundary. The particles are ``kinks,'' which can be thought of as field configurations interpolating between adjacent wells of a potential with k+1 degenerate minima. For the overscreened case k>2S, the impurity in the continuum is a kink as well, which explains the noninteger number of boundary states.

DOI
10.1103/physrevlett.71.2485

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We find the exact quasiparticle spectrum, elastic S matrix, and free energy for the continuum Kondo problem of k species of electrons coupled to an impurity of spin S. Here, the impurity becomes an immobile quasiparticle sitting on the boundary. The particles are ``kinks,'' which can be thought of as field configurations interpolating between adjacent wells of a potential with k+1 degenerate minima. For the overscreened case k>2S, the impurity in the continuum is a kink as well, which explains the noninteger number of boundary states.
We consider the single-channel Kondo problem with the Kondo coupling between a spin SSS impurity and conduction electrons with spin jjj. These problems arise as multicritical points in the parameter spaces of two-and higher-level tunneling systems and some impurity models of heavy fermion compounds. In contrast to the previous Bethe-ansatz conjectures, it turns out that the dynamics of the spin sector is the same as that of a spin-SSS impurity coupled to k(j)k(j)k(j) channels of spin-\textonehalf{} electrons with k(j)=2j(j+1)(2j+1)3k(j)=2\frac{j(j+1)(2j+1)}{3}k(j)=23j(j+1)(2j+1)​. As a result, for 2S<k(j)2S<k(j)2S<k(j), the system shows non-Fermi-liquid behavior with the same exponents for the thermodynamic quantities as those of the k(j)k(j)k(j)-channel Kondo problem. However, both the finite-size spectrum and the operator content are different due to the presence of the other sectors and can be obtained by conformal field theory techniques.
We review analytical and numerical results derived from the Bethe ansatz solution of the n-channel Kondo model of arbitrary spin S as a function of temperature, external field, impurity spin S and the number of channels n. Three situations have to be distinguished: (i) If n = 2S the conduction electrons exactly compensate the impurity spin into a singlet at low temperatures, (ii) if n < 2S the impurity spin is only partially compensated (undercompensated), and (iii) if n > 2S the impurity spin is said to be overcompensated giving rise to critical behaviour. The results are discussed in the context of magnetic impurities, e.g. Fe, Cr and Tm in simple metals, the quadrupolar Kondo effect, an impurity spin embedded in the Takhtajan-Babujian Heisenberg model and electron assisted-tunnelling of an atom in a double-well potential.
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We investigate the boundary effects that arise when spin-12\frac{1}{2}21​ impurities interact with the edges of the antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg chain through spin exchange interactions. We consider both cases when the couplings are ferromagnetic or antiferromagnetic. We find that in the case of antiferromagnetic interaction, when the impurity coupling strength is much weaker than that in the bulk, the impurity is screened in the ground state via the Kondo effect. The Kondo phase is characterized by the Lorentzian density of states and a dynamical scale, the Kondo temperature TK{T}_{K}TK​, is generated. As the impurity coupling strength increases, TK{T}_{K}TK​ increases until it reaches its maximum value ${T}_{0}=2\ensuremath{\pi}J$ which is the maximum energy carried by a single spinon. When the impurity coupling strength is increased further, we enter another phase, the bound mode phase, where the impurity is screened in the ground state by a single particle bound mode exponentially localized at the edge to which the impurity is coupled. We find that, in contrast to the Kondo phase, the impurity can be unscreened by unoccupying the single particle bound mode. This costs an energy Eb{E}_{b}Eb​ that is greater than T0{T}_{0}T0​. There exists a boundary eigenstate phase transition between the Kondo and the bound-mode phases, a transition which is characterized by the change in the number of towers of the Hilbert space. The transition also manifests itself in local thermodynamic quantities---local impurity density of states and the local impurity magnetization in the ground state. When the impurity coupling is ferromagnetic, the impurity is unscreened in the ground state; however, when the absolute value of the ratio of the impurity and bulk coupling strengths is greater than 45\frac{4}{5}54​, the impurity can be screened by adding a bound mode that costs energy greater than T0{T}_{0}T0​. When two impurities are considered, the phases exhibited by each impurity remain unchanged in the thermodynamic limit, but nevertheless the system exhibits a rich phase diagram.
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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.