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arXiv:1606.08634 · Physical review. B./Physical review. B

Non-Kondo many-body physics in a Majorana-based Kondo type system

Ian J. van Beek, Bernd Braunecker

Published Sep 12, 2016• 13 citations
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Abstract

We carry out a theoretical analysis of a prototypical Majorana system, which demonstrates the existence of a Majorana-mediated many-body state and an associated intermediate low-energy fixed point. Starting from two Majorana bound states, hosted by a Coulomb-blockaded topological superconductor and each coupled to a separate lead, we derive an effective low-energy Hamiltonian, which displays a Kondo-like character. However, in contrast to the Kondo model which tends to a strong- or weak-coupling limit under renormalization, we show that this effective Hamiltonian scales to an intermediate fixed point, whose existence is contingent upon teleportation via the Majorana modes. We conclude by determining experimental signatures of this fixed point, as well as the exotic many-body state associated with it.

DOI
10.1103/physrevb.94.115416

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We carry out a theoretical analysis of a prototypical Majorana system, which demonstrates the existence of a Majorana-mediated many-body state and an associated intermediate low-energy fixed point. Starting from two Majorana bound states, hosted by a Coulomb-blockaded topological superconductor and each coupled to a separate lead, we derive an effective low-energy Hamiltonian, which displays a Kondo-like character. However, in contrast to the Kondo model which tends to a strong- or weak-coupling limit under renormalization, we show that this effective Hamiltonian scales to an intermediate fixed point, whose existence is contingent upon teleportation via the Majorana modes. We conclude by determining experimental signatures of this fixed point, as well as the exotic many-body state associated with it.
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.
Conduction electrons coupled to a mesoscopic superconducting island hosting Majorana bound states have been shown to display a topological Kondo effect with robust non-Fermi liquid correlations. With MMM bound states coupled to MMM leads, this is an SO(MMM) Kondo problem, with the asymptotic high and low energy theories known from bosonization and conformal field theory studies. Here we complement these approaches by analyzing the Bethe ansatz equations describing the exact solution of these models at all energy scales. We apply our findings to obtain nonperturbative results on the thermodynamics of M→M−2M\rightarrow M-2M→M−2 crossovers induced by tunnel couplings between adjacent Majorana bound states.
Using the thermodynamic Bethe ansatz, we investigate the topological Kondo model, which describes a set of one-dimensional external wires, pertinently coupled to a central region hosting a set of Majorana bound states. After a short review of the Bethe ansatz solution, we study the system at finite temperature and derive its free energy for arbitrary (even and odd) number of external wires. We then analyse the ground state energy as a function of the number of external wires and of their couplings to the Majorana bound states. Then, we compute, both for small and large temperatures, the entropy of the Majorana degrees of freedom localized within the central region and connected to the external wires. Our exact computation of the impurity entropy provides evidence of the importance of fermion parity symmetry in the realization of the topological Kondo model. Finally, we also obtain the low-temperature behaviour of the specific heat of the Majorana bound states, which provides a signature of the non-Fermi-liquid nature of the strongly coupled fixed point.
Abstract. Conduction electrons coupled to a mesoscopic superconducting island hosting Majorana bound states have been shown to display a topological Kondo effect with robust non-Fermi liquid correlations. With M bound states coupled to M leads, this is an SO(M) Kondo problem, with the asymptotic high and low energy theories known from bosonization and conformal field theory studies. Here we complement these approaches by analyzing the Bethe ansatz equations describing the exact solution of these models at all energy scales. We apply our findings to obtain nonperturbative results on the thermodynamics of M → M − 2 crossovers induced by tunnel couplings between adjacent Majorana bound states. PACS numbers: 71.10.Pm, 74.50.+r, 74.90.+n 1.
This thesis aims to investigate the novel physics around Majorana bound states (MBSs). MBSs are quasiparticle excitations predicted to exist in topological superconductors. Due to the inherent protection they obtain from their topology, they have long been regarded as a promising setup to build a topologically protected quantum computer. Apart from this very practical application, they continue to fascinate as platforms built with MBS exhibit unique effects like nonabelian braiding and non-Fermi liquid behaviour in the topological Kondo effect.The thesis starts with a general introduction to quantum computing, highlighting both the promises and challenges of that field. Afterwards follows an introduction to systems that are candidates to host MBSs. It focuses on the theoretical proposals and experimental realizations that are most relevant to the presented work. We continue with a summary of the most fundamental effects and the setups that make the physics of MBSs accessible. The author's contributions in this field are theoretical. However, each publication comes with the explicit goal of investigating setups that make the unique physics of MBSs directly accessible in experiments. The first three papers have been peer-reviewed and were published in scientific journals. The fourth one is currently under peer review. The first two papers investigate DC transport setups connecting to a qubit based on MBSs called the MBS tetron. We show that via an effect called parity blockade, it is possible to initialize the qubit, quantify its lifetime, and determine whether it holds MBSs or trivial (normal fermionic) states. Also employing parity blockade, the second paper establishes a transport setup that enables access to MBS fusion rules, an effect directly related to nonabelian braiding. The third paper translates the concept of a MBS tetron into the newly experimentally established approach of MBSs in quantum dot-based Kitaev chains. We denote this system the poor man's Majorana tetron. We find that the poor man's Majorana tetron in a transport setup with appropriate tuning exhibits the famed topological Kondo effect. Paper~IV focuses on nonabelian braiding. It investigates the hybridization-based braiding protocol and accounts for imperfections in MBS localization, breaking the ground state degeneracy. This detrimental effect is corrected by the introduction of a correction term, followed by a full analytical solution of the transformation induced by this generalized braiding protocol.
In this paper, we study a set-up in which quantum dot is tunnel-coupled to Majorana-Kramers pair(MKP) which are the edge modes of time reversal invariant topological superconductor (TRITOPS), which is an experimental realization of time reversal invariant Kitaev chain model. Our motivation is to study the effect of Majorana-Kramers pair on Kondo fixed point and conversely the effect of Kondo interaction on Majorana-Kramers pair. This way we can capture the signatures of Majorana fermions on Kondo effect in quantum dots which can be probed feasibly in experiments. Employing the renormalization group method, we report the first study of the signatures of Majorana-Kramers pair in the dynamical spin susceptibility of the quantum dot electrons. We not only calculate the flow equations for coupling constants but also study the renormalization of observables and especially calculate dynamical spin susceptibility. We propose that since these dynamical quantities have signatures of Majorana-Kramers pair, so they provide an unambiguous experimental probe for detection of Majorana fermions in TRITOPS.