  Topological superconductors differ from topologically trivial ones for the
presence of topologically protected zero-energy modes. To date, experimental
evidence of topological superconductivity in nanostructures has been mainly
obtained by measuring the zero-bias conductance peak via tunneling
spectroscopy. Here, we propose an alternative and complementary experimental
recipe to detect topological phase transitions in these systems. We show in
fact that, for a finite-sized system with broken time-reversal symmetry,
discontinuities in the Josephson current-phase relation correspond to the
presence of zero-energy modes and to a change in the fermion parity of the
groundstate. Such discontinuities can be experimentally revealed by a
characteristic temperature dependence of the current, and can be related to a
finite anomalous current at zero phase in systems with broken phase-inversion
symmetry.
