Coulomb stability of the $\mathbf{4\pi}$-periodic Josephson effect of Majorana fermions

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 [B.\ van Heck, F.\ Hassler, A.\ R. Akhmerov, and C.\ W.\ J. Beenakker, Phys. Rev. B {\bf 84}, 180502(R) (2011)].