Chiral current-phase relation of topological Josephson junctions: A signature of the 4π-periodic Josephson effect

Abstract

The 4π-periodic Josephson effect is an indicator of Majorana zero modes and a ground-state degeneracy which are central to topological quantum computation. However, the observability of a 4π-periodic Josephson current-phase relation (CPR) is hindered by the necessity to fix the fermionic parity. As an alternative to a 4π-periodic CPR, this paper proposes a chiral CPR for the 4π-periodic Josephson effect. This is a CPR of the form J(ϕ)∝C|sin(ϕ/2)|, describing a unidirectional supercurrent with the chirality C=±1. Its nonanalytic dependence on the Josephson phase difference ϕ translates into the 4π-periodic CPR J(ϕ)∝sin(ϕ/2). The proposal requires a spin-polarized topological Josephson junction which is modeled here as a short link between spin-split superconducting channels at the edge of a two-dimensional topological insulator. In this case, C coincides with the Chern number of the occupied spin band of the topological insulator. This paper details three scenarios of achieving a chiral CPR: By only Zeeman-like splitting, by Zeeman splitting combined with bias currents, and by an external out-of-plane magnetic field.

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