4π and 8π dual Josephson effects induced by symmetry defects

Abstract

In topological insulator edges, the duality between the Zeeman field orientation and the proximitized superconducting phase has been recently exploited to predict a magneto-Josephson effect with a 4$\pi$ periodicity. We revisit this latter Josephson effect in the light of this duality and show that the same 4$\pi$ quantum anomaly occurs when bridging two spinless Thouless pumps to a p-wave superconducting region that could be as small as a single and experimentally-relevant superconducting quantum dot - a point-like defect. This interpretation as a dual Josephson effect never requires the presence of Majorana modes but rather builds on the topological properties of adiabatic quantum pumps with Z topological invariants. It allows for the systematic construction of dual Josephson effects of arbitrary periodicity, such as 4$\pi$ and 8$\pi$, by using point-like defects whose symmetry differs from that of the pump, dubbed symmetry defects. Although adiabatic quantum pumps are typically discussed via mappings to two-dimensional geometries, we show that this phenomenology does not have any counterpart in conventional two-dimensional systems.