  We study the excited states of interacting fermions in one dimension with
particle-hole symmetric disorder (equivalently, random-bond XXZ chains) using a
combination of renormalization group methods and exact diagonalization. Absent
interactions, the entire many-body spectrum exhibits infinite-randomness
quantum critical behavior with highly degenerate excited states. We show that
though interactions are an irrelevant perturbation in the ground state, they
drastically affect the structure of excited states: even arbitrarily weak
interactions split the degeneracies in favor of thermalization (weak disorder)
or spontaneously broken particle-hole symmetry, driving the system into a
many-body localized spin glass phase (strong disorder). In both cases, the
quantum critical properties of the non-interacting model are destroyed, either
by thermal decoherence or spontaneous symmetry breaking. This system then has
the interesting and counterintuitive property that edges of the many-body
spectrum are less localized than the center of the spectrum. We argue that our
results rule out the existence of certain excited state symmetry-protected
topological orders.
