Josephson junctions of 2D 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 non-topological superconductors (TRITOPS-S). We focus on long junctions for which several channels intervene in the tunneling coupling through the junction. We present a description of the topological edge modes for different TRITOPS models including $p$-wave pairing and the combination of $s$-wave pairing with spin-orbit coupling. We derive effective low-energy Hamiltonians to describe the Josephson junction, which can be solved analytically to explain the contribution of the edge states to the Josephson current as a function of the phase bias. We find that edge-modes yield singular corrections to the CPR for both junction types. The primary effects occur for the response of the Majorana zero-modes at half-flux quantum phase $\phi\approx \pi$ in TRITOPS-TRITOPS junctions and for integer flux quantum phase $\phi \approx 0$ for TRITOPS-S junctions, respectively. The former effect is particularly strong two-component nematic superconductors. The latter effect leads to a spontaneously broken time-reversal symmetry in the TRITOPS-S junction and to a breakdown of the bulk-boundary correspondence