A Coulomb blockaded M-Majorana island coupled to normal metal leads realizes a novel type of Kondo effect where the effective impurity "spin" transforms under the orthogonal group SO(M). The impurity spin stems from the nonlocal topological ground state degeneracy of the island and thus the effect is known as the topological Kondo effect. We introduce a physically motivated N-channel generalization of the topological Kondo model. Starting from the simplest case N=2, we conjecture a stable intermediate coupling fixed point and evaluate the resulting low-temperature impurity entropy. The impurity entropy indicates that an emergent Fibonacci anyon can be realized in the N=2 model. We also map the case N=2, M=4 to the conventional four-channel Kondo model and find the conductance at the intermediate fixed point. By using the perturbative renormalization group, we also analyze the large-N limit, where the fixed point moves to weak coupling. In the isotropic limit, we find an intermediate stable fixed point, which is stable to "exchange" coupling anisotropies, but unstable to channel anisotropy. We evaluate the fixed point impurity entropy and conductance to obtain experimentally observable signatures of our results. In the large-N limit, we evaluate the full crossover function describing the temperature-dependent conductance.
In the multichannel Kondo effect, overscreening of a magnetic impurity by conduction electrons leads to a frustrated exotic ground state. It has been proposed that multichannel topological Kondo (MCTK) model involving topological Cooper pair boxes with M M Majorana modes [SO( M M ) “spin”] and N N spinless electron channels exhibits an exotic intermediate coupling fixed point. This intermediate fixed point has been analyzed through large- N N perturbative calculations, which gives a zero-temperature conductance decaying as 1/N^2 1 / N 2 in the large- N N limit. However, the conductance at this intermediate fixed point has not been calculated for generic N N . Using representation theory, we verify the existence of this intermediate-coupling fixed point and find the strong-coupling effective Hamiltonian for the case M=4 M = 4 . Using conformal field theory techniques for SO( M M ), we generalize the notion of overscreening and conclude that the MCTK model is an overscreened Kondo model. We find the fixed-point finite-size energy spectrum and the leading irrelevant operator (LIO). We express the fixed-point conductance in terms of the modular S-matrix of SO( M M ) for general N N , confirming the previous large- N N result. We describe the finite-temperature corrections to the conductance by the LIO and find that they are qualitatively different for the cases N=1 N = 1 and N≥2 N ≥ 2 due to the different fusion outcomes with the current operator. We also compare the multichannel topological Kondo model to the topological symplectic Kondo model.
Multiple conduction channels interacting with a quantum impurity---a spin in the conventional ``multichannel Kondo effect'' or a topological mesoscopic device (``topological Kondo effect'')---has been proposed as a platform to realize anyonic quasiparticles. However, the above implementations require either perfect channel symmetry or the use of Majorana fermions. Here we propose a Majorana-free mesoscopic setup which implements the Kondo effect of the symplectic Lie group and can harbor emergent anyons (including Majorana fermions, Fibonacci anyons, and
Z3 parafermions) even in the absence of perfect channel symmetry. In addition to the detailed prescription of the implementation, we present the strong coupling solution by mapping the model to the multichannel Kondo effect associated to an internal
SU(2) symmetry and exploit conformal field theory to predict the nontrivial scaling of a variety of observables, including conductance, as a function of temperature. This work does not only open the door for robust Kondo-based anyon platforms, but also sheds light on the physics of strongly correlated materials with competing order parameters.
The traditional multichannel Kondo effect takes place when several gapless metallic electronic channels interact with a localized spin-
S impurity, with the number of channels
n exceeding the size of the impurity spin,
n>2S, leading to the emergence of non-Fermi liquid impurity behavior at low temperatures. Here, we show that the effect can be realized even when the electronic degrees of freedom are strongly correlated and gapped. The system under consideration consists of a single spin-
21 impurity coupled isotropically to
n spin singlet superconducting channels realized by one-dimensional leads with quasi-long-range superconducting order. The competition between the Kondo and superconducting fluctuations induces multiple distinct ground states and boundary phases depending on the relative strengths of the bulk and boundary interactions. Using the Bethe Ansatz technique, we identify four regimes: an overscreened Kondo phase, a zero-mode phase, a Yu-Shiba-Rusinov (YSR) phase, and a local-moment phase with an unscreened impurity, each with its own experimental characteristic. We describe the renormalization-group flow, the excitation spectrum, and the full impurity thermodynamics in each phase. Remarkably, even in the presence of a bulk mass gap, the boundary critical behavior in the Kondo phase is governed by the same exponents as in the gapless theory with the low-energy impurity sector flowing to the
SU(2)n Wess-Zumino-Witten (WZW) fixed point, and the impurity entropy monotonically decreasing as a function of temperature. In both the overscreened Kondo and zero-mode phases, the residual impurity entropy is
Simp(T→0)=ln[2cos(π/(n+2))]. In the YSR and unscreened phases on the other hand the impurity entropy exhibits non-monotonic temperature dependence and is effectively free at low temperatures with
Simp(T→0)=ln2.
The phase diagram of the multichannel Kondo Hamiltonian with an
XXZ spin-exchange anisotropy is revisited, revealing a far richer fixed-point structure than previously appreciated. For a spin-
21 impurity and
k>2 conduction-electron channels, a second ferromagneticlike domain is found deep inside the antiferromagnetic regime. The new domain extends above a (typically large) critical longitudinal coupling
Jz∗>0 and is separated from the antiferromagnetic domain by a second Kosterlitz-Thouless line. A similar line of stable ferromagneticlike fixed points with a residual isospin-
21 local moment is shown to exist for large ${J}_{z}⪢\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}>0$ and arbitrary
k and
s obeying $\ensuremath{\mid}k\ensuremath{-}2s\ensuremath{\mid}>1$. Here,
Jz is the longitudinal spin-exchange coupling, ${J}_{\ensuremath{\perp}}$ is the transverse coupling, and
s is the impurity spin. Near the free-impurity fixed point, spin-exchange anisotropy generates a highly relevant term for
s>1∕2 and arbitrary
k. Depending on the sign of ${J}_{z}^{2}\ensuremath{-}{J}_{\ensuremath{\perp}}^{2}$ and the parity of
2s, the system flows either to a conventional Fermi liquid with no residual degeneracy or to a
k-channel, spin-
21 Kondo effect or to a line of ferromagneticlike fixed points with a residual isospin-
21 local moment. These results are obtained through a combination of perturbative renormalization-group techniques, Abelian bosonization, a strong-coupling expansion in
1∕Jz, and explicit numerical renormalization-group calculations.
We propose to use residual anyons of the overscreened Kondo effect for topological quantum computation. A superconducting proximity gap of $\mathrm{\ensuremath{\Delta}}<{T}_{K}$ can be utilized to isolate the anyon from the continuum of excitations and stabilize the nontrivial fixed point. We use weak-coupling renormalization group, dynamical large-N technique, and bosonization to show that the residual entropy of multichannel Kondo impurities survives in a superconductor. We find that while (in agreement with recent numerical studies) the nontrivial fixed point is unstable against intrachannel pairing, it is robust in the presence of a finite interchannel pairing. Based on this observation, we suggest a superconducting charge Kondo setup for isolating and detecting the Majorana fermion in the two-channel Kondo system.
Non-Abelian anyons are fractional excitations of gapped topological models believed to describe certain topological superconductors or quantum Hall states. Here, we provide the first numerical evidence that they emerge as independent entities also in gapless electronic models. Starting from a multi-impurity multichannel chiral Kondo model, we introduce a novel mapping to a single-impurity model, amenable to Wilson's numerical renormalization group. We extract its spectral degeneracy structure and fractional entropy, and calculate the F matrices, which encode the topological information regarding braiding of anyons, directly from impurity spin-spin correlations. Impressive recent advances on realizing multichannel Kondo systems with chiral edges may thus bring anyons into reality sooner than expected.