  Topological phases of matter that depend for their existence on interactions
are fundamentally interesting and potentially useful as platforms for future
quantum computers. Despite the multitude of theoretical proposals the only
interaction-enabled topological phase experimentally observed is the fractional
quantum Hall liquid. To help identify other systems that can give rise to such
phases we present in this work a detailed study of the effect of interactions
on Majorana zero modes bound to vortices in a superconducting surface of a 3D
topological insulator. This system is of interest because, as was recently
pointed out, it can be tuned into the regime of strong interactions. We start
with a 0D system suggesting an experimental realization of the
interaction-induced $\mathbb{Z}_8$ ground state periodicity previously
discussed by Fidkowski and Kitaev. We argue that the periodicity is
experimentally observable using a tunnel probe. We then focus on
interaction-enabled crystalline topological phases that can be built with the
Majoranas in a vortex lattice in higher dimensions. In 1D we identify an
interesting exactly solvable model which is related to a previously discussed
one that exhibits an interaction-enabled topological phase. We study these
models using analytical techniques, exact numerical diagonalization (ED) and
density matrix renormalization group (DMRG). Our results confirm the existence
of the interaction-enabled topological phase and clarify the nature of the
quantum phase transition that leads to it. We finish with a discussion of
models in dimensions 2 and 3 that produce similar interaction-enabled
topological phases.
