  In closed quantum systems, strong randomness can localize many-body
excitations, preventing ergodicity. An interesting consequence is that high
energy excited states can exhibit quantum coherent properties, such as symmetry
protected topological (SPT) order, that otherwise only occur in equilibrium
ground states. Here, we ask: which types of SPT orders can be realized in
highly excited states of a many-body (MB) localized system? We argue that this
question is equivalent to whether an SPT order can be realized in an exactly
solvable lattice model of commuting projectors. This perspective enables a
sharp definition of MB localizability. Using this criterion, it is
straightforward to establish that whereas all bosonic SPTs in spatial
dimensions $d=1,\,2,\,3$ are MB localizable, chiral phases (e.g. quantum Hall
fluids) are not. We also show that free fermion SPTs in $d >1$ (including
topological insulators and superconductors) cannot be localized if interactions
are weak. A key question is whether strong interactions can render them MB
localizable, which we study in the context of a class of $d=2$ topological
superconductors. Using a decorated domain wall (DDW) approach we show that some
phases in this class are MB localizable, when they correspond to bosonic SPT
orders. However, a similar DDW approach faces a fatal obstruction to realizing
certain intrinsically fermionic SPT orders, an issue we argue may persist
beyond this specific construction.
