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arXiv:2210.16614 · Physical review. B./Physical review. B

Topological symplectic Kondo effect

Guangjie Li, Elio J. König, Jukka I. Väyrynen

Published May 5, 2023• 11 citations
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Abstract

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 ${\mathbb{Z}}_{3}$ 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 $\text{SU}(2)$ 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.

DOI
10.1103/physrevb.107.l201401

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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{\mathbb{Z}}_{3}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)\text{SU}(2)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.
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.
Fractionalized quasiparticles---anyons---bear a special role in present-day physics. At the same time, they display properties of interest both foundational, with quantum numbers that transcend the spin-statistics laws, and applied, providing a cornerstone for decoherence-free quantum computation. The development of platforms for realization and manipulation of these objects, however, remains a challenge. Typically these entail the zero-temperature ground state of incompressible, gapped fluids. Here we establish a strikingly different approach: The development and probing of anyon physics in a gapless fluid. The platform of choice is a chiral, multichannel, multi-impurity realization of the Kondo effect. We discuss how, in the proper limit, anyons appear at magnetic impurities, protected by an asymptotic decoupling from the fluid and by the emerging Kondo length scale. We discuss possible experimental realization schemes using integer quantum Hall edges. The gapless and charged degrees of freedom that coexist with the anyons suggest the possibility of extracting quantum information data by transport and simple correlation functions. To show that this is the case, we generalize the fusion ansatz of Cardy's boundary conformal field theory, now in the presence of multiple localized perturbations. The generalized fusion ansatz captures the idea that multiple impurities share quantum information nonlocally, in a way formally identical to anyonic zero modes. We display several examples supporting and illustrating this generalization and the extraction of quantum information data out of two-point correlation functions. With the recent advances in mesoscopic realizations of multichannel Kondo devices, our results imply that exotic anyon physics may be closer to reach than presently imagined.
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.
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.
Quantum gates for the manipulation of topological qubits rely on interactions between non-Abelian anyonic quasiparticles. We study the collective behavior of systems of anyons arising from such interactions. In particular, we study the effect of favoring different fusion channels of the screened Majorana spins appearing in the recently proposed topological Kondo effect. Based on the numerical solution of a chain of SO(5)2SO{(5)}_{2}SO(5)2​ anyons we identify two critical phases whose low-energy behavior is characterized by conformal field theories with central charges c=1c=1c=1 and c=8/7c=8/7c=8/7, respectively. Our results are complemented by exact results for special values of the coupling constants which provide additional information about the corresponding phase transitions.
Boundary conformal field theory is brought to bear on the study of topological insulating phases of non-Abelian anyonic chains. These phases display protected anyonic end modes. We consider spin-1/2 su(2)k{}_{k}k​ chains at any level kkk, focusing on the most prominent examples: the case k=2k=2k=2 describes Ising anyons (equivalent to Majorana fermions) and k=3k=3k=3 corresponds to Fibonacci anyons. The method we develop is quite general and rests on a deep connection between boundary conformal field theory and topological symmetry. This method tightly constrains the nature of the topological insulating phases of these chains for general kkk. Emergent anyons which arise at domain walls are shown to have the same braiding properties as the physical quasiparticles. This suggests a ``solid-state'' topological quantum computation scheme in which emergent anyons are braided by tuning the couplings of non-Abelian quasiparticles in a fixed network.