Anomalous topological pumps and fractional Josephson effects

We discover novel topological pumps in the Josephson effects for superconductors with time-reversal symmetry. 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. In contrast to topological pumps with conventional parameters, these pumps are characterized by $\mathbb{Z}\times \mathbb{Z}$ or ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ strong invariants. We determine the general classifications in class AIII, and those in class DIII with a single anomalous parameter. For the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ topological pumps in class DIII, the first ${\mathbb{Z}}_2$ invariant describes the coincidence of fermion parity and spin pumps whereas the second ${\mathbb{Z}}_2$ invariant reflects the non-Abelian statistics of Majorana Kramers pairs, leading to three distinct fractional Josephson effects.

Figure 1

The three ${\mathbb{Z}}_2$ topological and one trivial pumps in the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ Josephson effects for class DIII SCs. Green and red (solid and dashed) states are related by TRS (PHS). The shaded areas are the bulk continuum.

Figure 2

(a) The extended BZ of $\mathcal{H}(k,\varphi )$ in class DIII. (b)–(e) Schematics of the contraction and deformation of the special lines and points on the torus (a) in our homotopy theory.