Tunable Josephson $pi$ junctions without Coulomb blockade

We study a Josephson junction between two time-reversal-invariant topological superconductors. We 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 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.