Fermion-parity anomaly of the critical supercurrent in the quantum spin-Hall effect

TL;DR

This paper investigates how fermion parity affects the critical supercurrent in a quantum spin-Hall insulator's edge state, revealing a parity-dependent doubling of the current in long junctions, which signals the 4pi-periodic Josephson effect.

Contribution

It extends the analysis of supercurrent in quantum spin-Hall edges beyond the short-junction limit, highlighting fermion parity's role in long junctions.

Findings

Fermion parity doubles the critical current in long junctions at low temperature.

In short junctions, parity constraints do not affect the critical current.

The results provide a phase-insensitive signature of the 4pi-periodic Josephson effect.

Abstract

The helical edge state of a quantum spin-Hall insulator can carry a supercurrent in equilibrium between two superconducting electrodes (separation L, coherence length xi). We calculate the maximum (critical) current I_c that can flow without dissipation along a single edge, going beyond the short-junction restriction L<<xi of earlier work, and find a dependence on the fermion parity of the ground state when L becomes larger than xi. Fermion-parity conservation doubles the critical current in the low-temperature, long-junction limit, while for a short junction I_c is the same with or without parity constraints. This provides a phase-insensitive, DC signature of the 4pi-periodic Josephson effect.