Correlations between Majorana Fermions Through a Superconductor

Abstract

We consider a model of ballistic quasi-one dimensional semiconducting wire with intrinsic spinorbit interaction placed on the surface of a bulk s-wave superconductor (SC), in the presence of an external magnetic field. This setup has been shown to give rise to a topological superconducting state in the wire, characterized by a pair of Majorana-fermion (MF) bound states formed at the two ends of the wire. Here we demonstrate that, besides the well-known direct overlap-induced energy splitting, the two MF bound states may hybridize via elastic tunneling processes through virtual quasiparticles states in the SC, giving rise to an additional energy splitting between MF states from the same as well as from different wires. Here we consider a model of a wire with Rashba SOI brought into contact with a bulk s-wave SC. Due to the interplay of proximity-induced superconductivity, SOI, and magnetic field, the wire is expected to enter a topological superconducting (TSC) phase for strong enough magnetic fields, and to host MF bound states localized at its two ends 21-23 . The energy of an isolated MF is pinned to the Fermi level inside the mini gap, due to its topological nature. In realistic finite-size wires the two end MF wave functions overlap, and such coupling leads to the splitting in energy of the otherwise doubly degenerate level. In most theoretical approaches, after one has calculated the proximity-induced gap in the wire, one usually forgets about the bulk SC and works with an effective model for the wire. The smallness of the energy splitting of the MF state, relevant for quantum computing purposes, is then determined by the relation between the wire length L and the MF localization length ξ w . Namely, in order to have an exponentially small splitting, it is necessary to require L ξ w . However, we show here that coupling between MFs can be established also through the SC, on a relevant length scale dictated by the coherence length ξ s in the SC (modified by inverse power-law corrections in L). In such cases, the energy splitting is exponentially suppressed in the regime L ξ s . This SC-mediated effect becomes significant if ξ s > ξ w , and, together with other decoherence mechanisms 27-30 , it could become an important issue. On the other hand, this effect also provides useful signatures that can help to identify MFs experimentally.Generally speaking, tunneling between normal leads via an s-wave SC can occur via elastic single-electron cotunneling processes or via local or crossed Andreev reflection 31-33 , which is a two-particle tunneling process. Tunneling in systems with MFs are different in that sense. Two MF states form a single complex fermionic state and coupling via the anomalous propagator of the SC is thus not possible. Still, we show here that hybridization between two MFs can be induced by coherent tunneling of electrons via virtual quasiparticle states of the SC.Model.-The setups considered here are schematically arXiv:1305.4187v2 [cond-mat.mes-hall]

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