Coulomb stability of the 4\pi-periodic Josephson effect of Majorana fermions

TL;DR

This paper investigates the stability of the 4π-periodic Josephson effect in Majorana fermions, demonstrating that topological conditions preserve this periodicity despite charging effects, while trivial segments restore conventional periodicity.

Contribution

It provides a detailed calculation of the ground state energy in a ring geometry, showing the conditions under which 4π-periodicity is maintained or restored.

Findings

4π-periodicity persists in topologically nontrivial rings regardless of energy ratios

Charging energy can induce phase slips in trivial segments, restoring 2π-periodicity

Topological state of the entire ring is crucial for the stability of the 4π-periodic Josephson effect

Abstract

The Josephson energy of two superconducting islands containing Majorana fermions is a 4\pi-periodic function of the superconducting phase difference. If the islands have a small capacitance, their ground state energy is governed by the competition of Josephson and charging energies. We calculate this ground state energy in a ring geometry, as a function of the flux -\Phi- enclosed by the ring, and show that the dependence on the Aharonov-Bohm phase 2e\Phi/\hbar remains 4\pi-periodic regardless of the ratio of charging and Josephson energies - provided that the entire ring is in a topologically nontrivial state. If part of the ring is topologically trivial, then the charging energy induces quantum phase slips that restore the usual 2\pi-periodicity.