Abstract
Spin-orbit coupled superconductors are potentially interesting candidates for realizing topological and potentially non-Abelian states with Majorana Fermions. We argue that time-reversal broken spin-orbit coupled superconductors generically can be characterized as having sub-gap states that are bound to localized non-magnetic impurities. Such bound states, which are referred to as Shiba states, can be detected as sharp resonances in the tunneling spectrum of the spin-orbit coupled superconductors. The Shiba state resonance can be tuned using a gate-voltage or a magnetic field from being at the edge of the gap at zero magnetic fields to crossing zero energy when the Zeeman splitting is tuned into the topological superconducting regime. The zero-crossing signifies a Fermion parity changing first order quantum phase transition, which is characterized by a Pfaffian topological invariant. These zero-crossings of the impurity level can be used to locally characterize the topological superconducting state from tunneling experiments.
Highlights
Recent experimental results[17–20] on these systems have already shown promising evidence for the existence of Majorana fermions (MF). While experiments such as tunneling and the fractional Josephson effect to detect the presence of end MFs8,11,14,15,21,22 and edge MFs23–25 have been proposed and demonstrated,[17,26] local probes that characterize the topological superconducting (TS) state of the wire, away from the physical ends are still missing
We show that the semiconductor nanowire systems that have been proposed to realize MFs are ideal for generating subgap Shiba states bound to nonmagnetic impurities
We find that in the nanowire the Shiba bound states crossover from weakly bound states to deeply bound states as the magnetic field is increased, indicating a crossover from an s-wave-like state for VZ to a p-wave-like phase for VZ
Summary
We find that the energy associated with the Shiba state can be tuned to cross zero energy as the impurity strength is increased [see Fig. 1(c)], providing a realization of the fermion-parity changing transition.[32] Such a zero crossing will be shown to be associated with a local version of the Pfaffian topological invariant that may be used to characterize the TS phase. Consistent with Anderson’s theorem,[36,38] Eq (5) describing subgap states bound to nonmagnetic impurities, is found to have no solutions for VZ = 0 This continues to hold, even in the presence of a spatially uniform Zeeman potential VZ = 0, if the spin-orbit coupling α vanishes. Nonmagnetic impurities may lead to localized Shiba bound states in wires only in the presence of a combination of Zeeman splitting and spin-orbit coupling, which are the ingredients to realize a TS phase.
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