We discover novel topological pumps in the Josephson effects for superconductors. The phase difference, which is odd under the chiral symmetry defined by the product of time-reversal and particle-hole symmetries, acts as an anomalous adiabatic parameter. These pumping cycles are different from those in the "periodic table", and are characterized by Z×Z or Z2×Z2 strong invariants. We determine the general classifications in class AIII, and those in class DIII with a single anomalous parameter. For the Z2×Z2 topological pump in class DIII, one Z2 invariant describes the coincidence of fermion parity and spin pumps whereas the other one reflects the non-Abelian statistics of Majorana Kramers pairs, leading to three distinct fractional Josephson effects.
@article{arxiv.1310.5281,
title = {Anomalous Topological Pumps and Fractional Josephson Effects},
author = {Fan Zhang and C. L. Kane},
journal= {arXiv preprint arXiv:1310.5281},
year = {2014}
}