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arXiv:cond-mat/9607208 · Physical review. B, Condensed matter

Two-channel Kondo physics from tunneling impurities with triangular symmetry

Aris L. Moustakas, Daniel S. Fisher

Published Mar 15, 1997• 19 citations
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Abstract

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.

DOI
10.1103/physrevb.55.6832

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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.
It has been claimed that an impurity hopping between two sites in a metal can lead to non-Fermi-liquid behavior if the bias between the two sites is tuned to zero. Here it is shown that several extra tunneling processes are allowed; those involving bare impurity tunneling as well as two-electron-assisted tunneling play important roles. These terms generally drive the system away from the intermediate-coupling non-Fermi-liquid fixed point, which has the character of the two-channel Kondo problem. In order to find the non-Fermi-liquid behavior a combination of these terms must be small over a range of energy scales. The prospects for obtaining these conditions are studied by analyzing both the weak-coupling and the intermediate-coupling behavior, together with the connecting crossover region. Unfortunately, the conclusions are rather pessimistic. The phase diagram and the important crossover are analyzed via bosonization and refermionization and some of the results compared to those obtained from conformal field theory. Finally, an additional non-Fermi-liquid fixed point is found, a multicritical point with two relevant directions even in the unbiased case. \textcopyright{} 1996 The American Physical Society.
We analyze in detail all the possible fixed points of the effective Hamiltonian of a nonmagnetic impurity hopping between two sites in a metal obtained by Moustakas and Fisher (MF). We find a line of non-Fermi liquid fixed points which continuously interpolates between the two-channel Kondo fixed point (2CK) and the one-channel, two-impurity Kondo (2IK) fixed point. There is one relevant direction with scaling dimension 12\frac{1}{2}21​ and one leading irrelevant operator with dimension 32\frac{3}{2}23​. There is also one marginal operator in the spin sector moving along this line. The marginal operator, combined with the leading irrelevant operator, will generate the relevant operator. For the general position on this line, the leading low-temperature exponents of the specific heat, the hopping susceptibility and the electron conductivity ${C}_{\mathrm{imp}},{\ensuremath{\chi}}_{\mathrm{imp}}^{h},\ensuremath{\sigma}(T)$ are the same as those of the 2CK, but the finite-size spectrum depends on the position on the line. No universal ratios can be formed from the amplitudes of the three quantities except at the 2CK point on this line where the universal ratios can be formed. At the 2IK point on this line, $\ensuremath{\sigma}(T)\ensuremath{\sim}2{\ensuremath{\sigma}}_{u}{(1+aT}^{3/2})$, no universal ratio can be formed either. The additional non-Fermi-liquid fixed point found by MF has the same symmetry as the 2IK, it has two relevant directions with scaling dimension 12\frac{1}{2}21​, and is therefore also unstable. The leading low-temperature behaviors are ${C}_{\mathrm{imp}}\ensuremath{\sim}T,{\ensuremath{\chi}}_{\mathrm{imp}}^{h}\ensuremath{\sim}\mathrm{ln}T,\ensuremath{\sigma}(T)\ensuremath{\sim}2{\ensuremath{\sigma}}_{u}{(1+aT}^{3/2})$; no universal ratios can be formed. The system is shown to flow to a line of Fermi-liquid fixed points which continuously interpolates between the noninteracting fixed point and the two-channel spin-flavor Kondo fixed point discussed by the author previously. The effect of particle-hole symmetry breaking is discussed. The effective Hamiltonian in the external magnetic field is analyzed. The scaling functions for the physical measurable quantities are derived in the different regimes; their predictions for the experiments are given. Finally the implications are given for a nonmagnetic impurity hopping around three sites with triangular symmetry discussed by MF.
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.
We discuss the two-channel Kondo problem with a pseudogap density of states $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{r}$ of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic r=0r=0r=0 case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For r>rmaxr>{r}_{\mathrm{max}}r>rmax​, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions r=0r=0r=0 and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.
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.
A multipolar Kondo model of an impurity spin S_I=3/2 interacting with conduction electrons with spin s_c=3/2 is investigated using boundary conformal field theory. A two-channel Kondo (2CK) -like non-Fermi liquid (NFL) under the particle-hole symmetry is derived explicitly using a ``superspin absorption'' in the sector of a hidden symmetry, SO(5). We discuss the difference between the usual spin-1/2 2CK NFL fixed point and the present one. In particular, we find that, unlike the usual 2CK model, the low temperature impurity specific heat is proportional to temperature.