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arXiv:1101.1326 · Physical Review B

Non-Fermi-liquid quantum impurity physics from non-Abelian quantum Hall states

Stuart A. Sevier, Gregory A. Fiete

Published Jul 11, 2011• 2 citations
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Abstract

We study the physics of electron tunneling between multiple quantum dots and the edge of a quantum Hall state. Our results generalize earlier work [Fiete, Bishara, and Nayak, Phys. Rev. Lett. 101, 176801 (2008)] in which it was shown that a single quantum dot tunnel coupled to a non-Abelian quantum Hall state can realize a stable multichannel Kondo fixed point at low energy. In this work, we investigate the physics of multiple dots and find that a rich set of possible low-energy fixed points arises, including those with non-Fermi-liquid properties. Previously unidentified fixed points may also be among the possibilities. We examine both the situation where the dots are spatially separated and where they are in close proximity. We discuss the relation to previous work on two-impurity Kondo models in Fermi liquids and highlight new research directions in multiple quantum impurity problems.

DOI
10.1103/physrevb.84.035101

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We study the physics of electron tunneling between multiple quantum dots and the edge of a quantum Hall state. Our results generalize earlier work [Fiete, Bishara, and Nayak, Phys. Rev. Lett. 101, 176801 (2008)] in which it was shown that a single quantum dot tunnel coupled to a non-Abelian quantum Hall state can realize a stable multichannel Kondo fixed point at low energy. In this work, we investigate the physics of multiple dots and find that a rich set of possible low-energy fixed points arises, including those with non-Fermi-liquid properties. Previously unidentified fixed points may also be among the possibilities. We examine both the situation where the dots are spatially separated and where they are in close proximity. We discuss the relation to previous work on two-impurity Kondo models in Fermi liquids and highlight new research directions in multiple quantum impurity problems.
Tunneling impurities in metals have been known for some time to have the potential for exhibiting Kondo-like physics. However, previous models based on an impurity hopping between two equivalent positions have run into trouble due to the existence of relevant operators that drive the system away from the non-Fermi-liquid Kondo fixed point. In the case of an impurity hopping among positions with higher symmetry, such as triangular symmetry, it is shown here that the non-Fermi-liquid behavior at low temperatures can be generic. Using various bosonization techniques, the fixed point is shown to be stable. However, unlike the conventional two-channel Kondo (2CK) model, it has four leading irrelevant operators, implying that while the form of the singular temperature dependence of physical quantities is similar to the 2CK model, there will not be simple universal amplitude ratios. The phase diagram of this system is analyzed and a critical manifold is found to separate the non-Fermi-liquid from a conventional Fermi liquid fixed point. Generalization to higher symmetries, such as cubic, and the possibility of physical realizations with dynamic Jahn-Teller impurities are discussed.
A quantum dot coupled to two helical edge states of a two-dimensional topological insulator through electron tunnelings is studied. We show that if the electron interactions on the edge states are repulsive, with Luttinger liquid parameter K<1K<1K<1, the system reaches a stable two-channel Kondo fixed point at low temperatures. This is in contrast to the Luttinger liquid leads case in which K<1/2K<1/2K<1/2 is needed. This two-channel fixed point is described by a boundary sine-Gordon Hamiltonian with a KKK dependent boundary term. The impurity entropy, the impurity specific heat and the conductance are calculated.
A mapping between fractional quantum Hall (FQH) junctions and the two-channel Kondo model is presented. We discuss this relation in detail for the particular case of a junction of a FQH state at $\ensuremath{\nu}=1/3$ and a normal metal. We show that in the strong coupling regime this junction has a non-Fermi-liquid fixed point. At this fixed point the electron Green's function has a branch cut and the impurity entropy is equal to S=12ln2.S=\frac{1}{2}\mathrm{ln}2.S=21​ln2. We construct the space of perturbations at the strong coupling fixed point and find that the dimension of the tunneling operator is 12.\frac{1}{2}.21​. These properties are strongly reminiscent of the non-Fermi-liquid fixed points of a number of quantum impurity models, particularly the two-channel Kondo model. However we have found that, in spite of these similarities, the Hilbert spaces of these two systems are quite different. In particular, although in a special limit the Hamiltonians of both systems are the same, their Hilbert spaces are not since they are determined by physically distinct boundary conditions. As a consequence the spectrum of operators in the two problems is different.
A complete Bethe ansatz solution of the $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(f)$ Coqblin-Schrieffer model and a detailed analysis of some physical applications of the model are given. As in the usual multichannel Kondo model, a variety of Fermi-liquid and non-Fermi-liquid (NFL) fixed points is found, whose nature depends on the impurity representation \ensuremath{\mu}. For $\ensuremath{\mu}=f,$ we find a Fermi-liquid fixed point, with the impurity spin completely screened. For $f>\ensuremath{\mu},$ the impurity is overscreened and the model has NFL properties. The form the NFL behavior takes depends on the NNN and f:f:f: for N< f,N<~f,N< f, the specific heat and the susceptibility are dominated by the NFL contributions; for N>fN>fN>f the leading contributions are Fermi-liquid-like, and the NFL behavior can be seen only to subleading order; and for N=fN=fN=f the behavior is marginal. We also analyze the possibility of physical realizations. We show by a detailed renormalization-group and 1/f1/f1/f analysis that the tunneling NNN-state problem can be mapped into the $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(f)$ exchange model, and discuss the subtle differences between the two models. As another physical realization we suggest a double quantum dot structure that can be described by means of an $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ model if the parameters of the dots are tuned appropriately.
We study triangular clusters of three spin-1/2 Kondo or Anderson impurities that are coupled to two conduction leads. In the case of Kondo impurities, the model takes the form of an antiferromagnetic Heisenberg ring with Kondo-type exchange coupling to continuum electrons. We show that this model exhibits many types of the behavior found in various simpler one- and two-impurity models, thereby enabling the study of crossovers between a number of Fermi-liquid (FL) and non-Fermi-liquid (NFL) fixed points. In particular, we explore a direct crossover between the two-impurity Kondo-model NFL fixed point and the two-channel Kondo-model NFL fixed point. We show that the concept of the two-stage Kondo effect applies even in the case when the first-stage Kondo state is of NFL type. In the case of Anderson impurities, we consider the transport properties of three coupled quantum dots. This class of models includes, as limiting cases, the familiar serial double quantum dot and triple quantum dot nanostructures. By extracting the quasiparticle scattering phase shifts, we compute the low-temperature conductance as a function of the interimpurity tunneling coupling. We point out that due to the existence of exponentially low-temperature scales, there is a parameter range where the stable ``zero-temperature'' fixed point is essentially never reached (not even in numerical renormalization group calculations). The zero-temperature conductance is then of no interest and it may only be meaningful to compute the conductance at finite temperature. This illustrates the perils of studying the conductance in the ground state and considering thermal fluctuations only as a small correction.
The article reviews recent results for the low energy physics of fast tunneling centers in metallic environments. For strong enough couplings to the environment these tunneling centers display an orbital Kondo effect and give rise to a non-Fermi-liquid behavior. This latter property is explained by establishing a mapping of the tunneling center model to the multichannel Kondo model via the renormalization group transformation combined with a 1/Nf expansion. The case of M-state systems, the role of the splittings and the present experimental situation are also discussed.