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Journal of Applied Physics

Quantum critical points of an anisotropic multichannel Kondo impurity

P. Schlottmann

Published Jun 1, 2001• 5 citations
Publisher

Abstract

The low-temperature behavior of a magnetic impurity of spin S interacting with an electron gas via an anisotropic spin exchange is studied via Bethe’s ansatz. The multichannel Kondo model with U(1) invariance is integrable as a function of two continuous (the exchange and the anisotropy) and two discrete parameters, namely the impurity spin S and the number of channels n. As a function of S and n we distinguish: (i) the compensated case with n=2S, (ii) the overcompensated case if n>2S, and (iii) the undercompensated case (n<2S). While in case (i) the ground state is a singlet, the cases (ii), and (iii) yield quantum critical points. The undercompensated one is of a new type with the critical exponents depending on the anisotropy.

DOI
10.1063/1.1357846

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The low-temperature behavior of a magnetic impurity of spin S interacting with an electron gas via an anisotropic spin exchange is studied via Bethe’s ansatz. The multichannel Kondo model with U(1) invariance is integrable as a function of two continuous (the exchange and the anisotropy) and two discrete parameters, namely the impurity spin S and the number of channels n. As a function of S and n we distinguish: (i) the compensated case with n=2S, (ii) the overcompensated case if n>2S, and (iii) the undercompensated case (n<2S). While in case (i) the ground state is a singlet, the cases (ii), and (iii) yield quantum critical points. The undercompensated one is of a new type with the critical exponents depending on the anisotropy.
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.
The numerical solution of the Bethe ansatz equations of the n-channel Kondo problem is presented. The thermodynamics of the impurity is obtained as a function of temperature, external field, impurity spin S and the number of channels. 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 briefly discussed in the context of magnetic impurities, the quadrupolar Kondo effect, an impurity spin embedded in the Takhtajan-Babujian model and a two-level system interacting with conduction electrons.
A multichannel Kondo model, where two or more equivalent but independent channels of electrons compete to screen a spin-12\frac{1}{2}21​ impurity, shows overcompensation of the impurity spin, leading to the non-Fermi-liquid behavior in various thermodynamic and transport properties. However, when the channel symmetry is broken, an impurity quantum phase transition can occur at zero temperature. Identification of an order parameter describing the impurity quantum phase transition is very difficult since it is beyond the conventional Landau-Ginzburg-Wilson theory. By employing the natural orbitals renormalization group method, we study both two-channel and three-channel Kondo models, from the perspective of spin correlation between the impurity and electrons in electronic channels. Here we demonstrate that by introducing the spin-correlation ratio as an order parameter we can characterize impurity quantum phase transitions driven by channel asymmetry. In particular, the universal critical exponents $\ensuremath{\beta}$ of the spin-correlation ratio and $\ensuremath{\nu}$ of the correlation length are explicitly determined by finite-size-scaling analysis, namely, $\ensuremath{\beta}=0.10(1)$ and $\ensuremath{\nu}=2.0(1)$, and $\ensuremath{\beta}=0.10(1)$ and $\ensuremath{\nu}=2.5(1)$ for the two-channel and three-channel Kondo models, respectively.
We study the low-temperature thermodynamics of a spin-SSS magnetic impurity coupled to m> 2m>~2m> 2 degenerate bands of interacting electrons in one dimension. By exploiting boundary-conformal-field-theory techniques, we derive exact results for the possible impurity thermal and magnetic response. The leading behavior of the impurity magnetic susceptibility is shown to be insensitive to the electron-electron interaction. In contrast, there are two types of scaling behavior of the impurity specific heat consistent with the symmetries of the problem: Either it remains the same as for the ordinary multichannel Kondo problem for noninteracting electrons or it acquires a new leading term governed by a critical exponent ${\ensuremath{\alpha}}_{m}{=(g}_{m}^{\ensuremath{-}1}\ensuremath{-}1)/m,$ where gm< 1{g}_{m}<~1gm​< 1 is a generalized (mmm channel) Luttinger-liquid charge parameter measuring the strength of the repulsive electron-electron interaction. We conjecture that the latter behavior is indeed realized when the impurity is exactly screened (m=2S).(m=2S).(m=2S).
A detailed and comprehensive study of the one-impurity multichannel Kondo model is presented. In the limit of a large number of conduction electron channels k>>1, the low-energy fixed point is accessible to a renormalization group improved perturbative expansion in 1/k. This straightforward approach enables us to examine the scaling, thermodynamics and dynamical response functions in great detail and make clear the following features: (i) the criticality of the fixed point, (ii) the universal non-integer degeneracy, and (iii) that the compensating spin cloud has a spatial extent of the order of one lattice spacing.