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arXiv:1108.6142 · Physical Review B

Quantum impurity coupled to Majorana edge fermions

Rok Žitko, Pascal Simon

Published Nov 8, 2011• 17 citations
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Abstract

We study a quantum impurity coupled to the edge states of a two-dimensional helical topological superconductor, that is, to a pair of counterpropagating Majorana fermion edge channels with opposite spin polarizations. For an impurity described by the Anderson impurity model, we show that the problem maps onto a variant of the interacting resonant two-level model which, in turn, maps onto the ferromagnetic Kondo model. Both magnetic and nonmagnetic impurities are considered. For magnetic impurities, an analysis relying on bosonization and the numerical renormalization group shows that the system flows to a fixed point characterized by a residual $\mathrm{ln}2$ entropy and anisotropic static and dynamical impurity magnetic susceptibilities. For nonmagnetic impurities, the system flows instead to a fixed point with no residual entropy and we find diamagnetic impurity response at low temperatures. We comment on the Schrieffer-Wolff transformation for problems with nonstandard conduction band continua and on the differences which arise when we describe the impurities by either Anderson or Kondo impurity models.

DOI
10.1103/physrevb.84.195310

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A quantum dot coupled to two helical edge states of a two-dimensional topological insulator through electron tunnelings is studied. We show that if the electron interactions on the edge states are repulsive, with Luttinger liquid parameter K<1K<1K<1, the system reaches a stable two-channel Kondo fixed point at low temperatures. This is in contrast to the Luttinger liquid leads case in which K<1/2K<1/2K<1/2 is needed. This two-channel fixed point is described by a boundary sine-Gordon Hamiltonian with a KKK dependent boundary term. The impurity entropy, the impurity specific heat and the conductance are calculated.
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We consider strongly correlated regimes which emerge at low temperature in coupled quantum dot (or magnetic impurity) systems. In strongly correlated systems a single particle description fails to explain the observed behaviour, so we resort to many body methods. We describe our system using a 2-impurity Anderson model and develop a numerical renormalisation group procedure which provides non-perturbative insight into the low energy behaviour, through calculation of dynamic quantities. We combine this approach with renormalised perturbation theory, thus acquiring a picture of how the Hamiltonian and interactions change at low energies. These approaches are first used to study the emergence of a Kondo effect with an SU(4) symmetry in capacitively-coupled double quantum dot systems. We classify the 'types' of SU(4) symmetry which can emerge and show how an experimentalist might achieve such emergence through tuning their system. We provide a way of distinguishing between the SU(2) and SU(4) Kondo regimes by considering the conductance. We also study a quantum critical point which occurs in the Heisenberg coupled quantum dot/impurity model. There is an anomalous entropy contributed by the impurities in this regime which is indicative of an uncoupled Majorana Fermion. We calculate dynamic quantities in regimes with different symmetries and establish correspondence with the 2-channel Kondo model. We formulate possible pictures of the underlying mechanisms of the critical point and construct a Majorana fermion model for the case with particle-hole symmetry, which explains the non-Fermi liquid energy levels and degeneracies obtained. We conjecture that a Majorana zero mode is present, and that this is responsible for the anomalous entropy.