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arXiv:cond-mat/0009285 · Physical review. B, Condensed matter

Fractional quantum Hall junctions and two-channel Kondo models

Nancy Sandler, Eduardo Fradkin

Published May 8, 2001• 2 citations
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Abstract

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 $S=\frac{1}{2}\mathrm{ln}2.$ We construct the space of perturbations at the strong coupling fixed point and find that the dimension of the tunneling operator is $\frac{1}{2}.$ 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.

DOI
10.1103/physrevb.63.235301

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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 \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=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.
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.
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.
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 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 how the non-Fermi-liquid nature of the overscreened multichannel Kondo impurity model affects the response to a BCS pairing term that, in the absence of the impurity, opens a gap $\mathrm{\ensuremath{\Delta}}$. We find that the low-energy spectrum in the limit $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$ actually does not correspond to the spectrum strictly at $\mathrm{\ensuremath{\Delta}}=0$. In particular, in the two-channel Kondo model, the $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$ ground state is an orbitally degenerate spin singlet, while it is an orbital singlet with a residual spin degeneracy at $\mathrm{\ensuremath{\Delta}}=0$. In addition, there are fractionalized spin-1/2 subgap excitations whose energy in units of $\mathrm{\ensuremath{\Delta}}$ tends toward a finite and universal value when $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$, as if the universality of the anomalous power-law exponents that characterize the overscreened Kondo effect turned into universal energy ratios when the scale invariance is broken by $\mathrm{\ensuremath{\Delta}}\ensuremath{\ne}0$. This intriguing phenomenon can be explained by the renormalization flow toward the overscreened fixed point and the gap cutting off the orthogonality catastrophe singularities. We also find other non-Fermi-liquid features at finite $\mathrm{\ensuremath{\Delta}}$: the local density of states lacks coherence peaks, the states in the continuum above the gap are unconventional, and the boundary entropy is a nonmonotonic function of temperature. The persistent subgap excitations are characteristic of the non-Fermi-liquid fixed point of the model, and thus depend on the impurity spin and the number of screening channels.
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.