Josephson junctions of two-dimensional time-reversal invariant superconductors: Signatures of the topological phase

We determine the current-phase relation (CPR) of two-terminal configurations of Josephson junctions containing two-dimensional (2D) time-reversal invariant topological superconductors (TRITOPS), including TRITOPS-TRITOPS, as well as junctions between topological and nontopological superconductors (TRITOPS-S). We focus on wide junctions for which several channels intervene in the tunneling coupling. We derive effective Hamiltonians to describe the topological edge modes for different TRITOPS models, including Hamiltonians with $p$-wave pairing and Hamiltonians combining $s$-wave pairing with spin-orbit coupling. We also derive effective low-energy Hamiltonians to describe the Josephson junction. These can be solved analytically and explain the contribution of the edge states to the Josephson current as a function of the phase bias. We find that edge modes yield peculiar features to the CPR for both junction types. The primary effects occur for the response of the Majorana zero modes at half-flux quantum phase $\ensuremath{\phi}\ensuremath{\approx}\ensuremath{\pi}$ in TRITOPS-TRITOPS junctions and for integer flux quantum phase $\ensuremath{\phi}\ensuremath{\approx}0$ for TRITOPS-S junctions, respectively. The former effect is particularly strong for two-component nematic superconductors. The second effect leads to a spontaneously broken time-reversal symmetry in the TRITOPS-S junction and to a breakdown of the bulk-boundary correspondence. We analyze in this case the role of the phase fluctuations. For weakly coupled junctions, we show that time-reversal symmetry is restored for large enough stiffness in these fluctuations.