  We introduce higher-order topological Dirac superconductor (HOTDSC) as a new
gapless topological phase of matter in three dimensions, which extends the
notion of Dirac phase to a higher-order topological version. Topologically
distinct from the traditional topological superconductors and known Dirac
superconductors, a HOTDSC features helical Majorana hinge modes between
adjacent surfaces, which are direct consequences of the symmetry-protected
higher-order band topology manifesting in the system. Specifically, we show
that rotational, spatial inversion, and time-reversal symmetries together
protect the coexistence of bulk Dirac nodes and hinge Majorana modes in a
seamless way. We define a set of topological indices that fully characterizes
the HOTDSC. We further show that a practical way to realize the HOTDSC phase is
to introduce unconventional odd-parity pairing to a three-dimensional Dirac
semimetal while preserving the necessary symmetries. As a concrete
demonstration of our idea, we construct a corresponding minimal lattice model
for HOTDSC obeying the symmetry constraints. Our model exhibits the expected
topological invariants in the bulk and the defining spectroscopic features on
an open geometry, as we explicitly verify both analytically and numerically.
Remarkably, the HOTDSC phase offers an example of a higher-order topological
quantum critical point, which enables realizations of various higher-order
topological phases under different symmetry-breaking patterns. In particular,
by breaking the inversion symmetry of a HOTDSC, we arrive at a higher-order
Weyl superconductor, which is yet another new gapless topological state that
exhibits hybrid higher-order topology.
