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arXiv:cond-mat/9506099 · Physical review. B, Condensed matter

Magnetic impurity in a Luttinger liquid: A conformal field theory approach

Per Fröjdh, Henrik Johannesson

Published Feb 1, 1996• 53 citations
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Abstract

We study the low-temperature properties of a spin-1/2 magnetic impurity coupled to a one-dimensional interacting electron system. Using a formalism by Affleck and Ludwig, with a scale invariant boundary condition replacing the impurity, we exploit boundary conformal field theory to deduce the impurity thermal and magnetic response. In the case of only forward electron scattering off the impurity, we predict the same critical scaling as for the two-channel Kondo effect for noninteracting electrons, but with a different Wilson ratio. Backward electron scattering off the impurity destabilizes this behavior and drives the system to a new fixed point. In the case of equal amplitudes for forward and backward scattering (Kondo interaction), we show that there are only two types of scaling behaviors consistent with the symmetries of the problem: either a local Fermi liquid or a critical theory with an anomalous specific heat. The latter case agrees with a recent poor-man's scaling result proposed by Furusaki and Nagaosa. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.3211

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We report on exact results for the low-temperature thermodynamics of a spin- 12\frac{1}{2}21​ magnetic impurity coupled to a one-dimensional interacting electron system. By using boundary conformal field theory, we show that there are only two types of critical behaviors consistent with the symmetries of the problem: either a local Fermi liquid or a theory with an anomalous response identical to that recently proposed by Furusaki and Nagaosa. Suppression of backscattering off the impurity leads to the same critical properties as for the two-channel Kondo effect.
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).
We present results for the Kondo effect in two crossed Luttinger liquids by using boundary conformal field theory. We predict two types of critical behaviors: either a two-channel Kondo fixed point with a nonuniversal Wilson ratio or a theory with an anomalous reponse that is identical to that found by Furusaki and Nagaosa for the Kondo effect in a single Luttinger liquid. Moreover, we discuss the relevance of perturbations like channel anisotropy in restoring a Fermi-liquid-like Kondo fixed point, and compare our results with the Kondo effect in a two-band Hubbard system modeled by a channel-dependent Luttinger Hamiltonian. The suppression of backscattering off the impurity produces a model similar to the four-channel Kondo theory. Consequences are discussed.
To study the impurity potential effect to the Kondo problem in a Luttinger liquid, we propose an integrable model of two interacting half-chains coupled with a single magnetic impurity ferromagnetically. It is shown that the scalar potential effectively reconciles the spin dynamics at low temperatures. Generally, there is a competition between the Kondo coupling J and the impurity potential V. When the ferromagnetic Kondo coupling dominates over the impurity potential (V<∣SJ∣),(V<|\mathrm{SJ}|),(V<∣SJ∣), the Furusaki-Nagaosa many-body singlet can be perfectly realized. However, when the impurity potential dominates over the Kondo coupling (V> ∣SJ∣),(V>~|\mathrm{SJ}|),(V> ∣SJ∣), the fixed point predicted by Furusaki and Nagaosa is unstable and the system must flow to a weak-coupling fixed point. It is also found that the effective moment of the impurity measured from the susceptibility is considerably enlarged by the impurity potential. In addition, some quantum phase transitions driven by the impurity potential are found and the anomaly residual entropy is discussed.
Using a coset Ising-Bose representation, we show how backscattering of electrons off a magnetic impurity destabilizes the two-channel Kondo fixed point and drives the system to a new fixed point, in agreement with previous results [P. Fr\"ojdh and H. Johannesson, Phys. Rev. Lett. 75, 300 (1995)]. In addition, we find that the presence of several leading correction-to-boundary operators in the spinon basis, for nonzero U, implies a not well-defined Wilson ratio. Finally, we show that a strong potential scattering at the impurity site stabilizes the two-channel Kondo fixed point in the neighborhood of half-filling.