Spontaneous Josephson Pi Junctions With Topological Superconductors

We examine the question of whether a pi junction can spontaneously form in a Josephson junction between two topological superconductors. We study a junction between two time-reversal-invariant topological superconductors and show this system goes through a series of multiple transitions between a 0-junction phase, where the free energy has its minimum for a superconducting phase difference of zero, and a pi-junction phase, where the free energy has its minimum for a superconducting phase difference of pi. These transitions occur in the absence of a Coulomb blockade or magnetic impurities. Rather, they are driven by the spin-orbit coupling in the junction, and can be probed, for example, by measuring the tunneling density of states or the critical current as a function of the junction's length or its Fermi velocity.