Geometric Josephson effects in chiral topological nanowires

One of the salient signatures of Majorana zero modes and topological superconductivity is a 4 pi-periodic Josephson effect due to the combination of fermion parity conservation and the presence of a topologically protected odd number of zero-energy crossings in the Andreev spectrum. In this paper, we study this effect in Josephson junctions composed of two semiconducting nanowires with Rashba spin-orbit coupling and induced superconductivity from the proximity effect. For certain orientations of the externalmagnetic field, such junctions possess a chiral symmetry, and we show how this symmetry allows the Andreev spectrum and the protected crossings to be shifted by introducing a relative angle between the two wires. The junction then displays a geometrically induced anomalous Josephson effect, the flow of a supercurrent in the absence of external phase bias. Furthermore, we derive a proportionality relation between the local current density and the local curvature for a single curvedwire. This result can be viewed as a one-dimensional analog of the recently proposed geo-Josephson effect [T. Kvorning et al., Phys. Rev. Lett. 120, 217002 (2018)]. Our two proposed effects can, in principle, be used as signatures of topological superconductivity in one dimension.