  The classification of topological states of matter in terms of unitary
symmetries and dimensionality predicts the existence of nontrivial topological
states even in zero-dimensional systems, i.e., a system with a discrete energy
spectrum. Here, we show that a quantum dot coupled with two superconducting
leads can realize a nontrivial zero-dimensional topological superconductor with
broken time-reversal symmetry, which corresponds to the finite size limit of
the one-dimensional topological superconductor. Topological phase transitions
corresponds to a change of the fermion parity, and to the presence of
zero-energy modes and discontinuities in the current-phase relation at zero
temperature. These fermion parity transitions therefore can be revealed by the
current discontinuities or by a measure of the critical current at low
temperatures.
