# 4$\pi$ and 8$\pi$ dual Josephson effects induced by symmetry defects

## TL;DR

This paper demonstrates that dual Josephson effects with 4π and 8π periodicity can be induced by symmetry defects in topological insulator edges, relying on topological quantum pumps rather than Majorana modes.

## Contribution

It introduces a novel interpretation of the 4π Josephson effect via duality with quantum pumps and extends the concept to arbitrary periodicities using symmetry defects.

## Key findings

- The 4π Josephson effect can occur without Majorana modes.
- Symmetry defects enable systematic construction of dual Josephson effects.
- Phenomenology does not have a counterpart in conventional 2D systems.

## 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.