  The surface of a higher order topological insulator (HOTI) comprises a
two-dimensional topological insulator (TI) with broken inversion symmetry,
whose mass is determined by the microscopic details of the surface such as
surface potentials and termination. It hosts a helical mode pinned to selected
hinges where the surface gap changes its sign. We study the effect of
perturbations that break time-reversal and particle-conservation on this
helical mode, such as a Zeeman field and a proximate superconductor. We find
that in contrast to the helical modes of inversion symmetric TIs, which are
gapped by these couplings, the helical modes at the hinges can remain gapless
and spatially split. When this happens, the Zeeman field splits the helical
mode into a chiral mode surrounding the magnetized region; and a superconductor
results in a helical Majorana mode surrounding the superconducting region. The
combination of the two might lead to the gapping of one of the chiral Majorana
modes, and leave a single one-dimensional chiral Majorana around the
superconducting island. We propose that the different topological states can be
measured in electrical transport.
