  We examine the stability of marginally Anderson localized phase transitions
between localized phases to the addition of many-body interactions, focusing in
particular on the spin-glass to paramagnet transition in a disordered
transverse field Ising model in one dimension. We find evidence for a
perturbative instability of localization at finite energy densities once
interactions are added, i.e. evidence for the relevance of interactions - in a
renormalization group sense - to the non-interacting critical point governed by
infinite randomness scaling. We introduce a novel diagnostic, the
"susceptibility of entanglement", which allows us to perturbatively probe the
effect of adding interactions on the entanglement properties of eigenstates,
and helps us elucidate the resonant processes that can cause thermalization.
The susceptibility serves as a much more sensitive probe, and its divergence
can detect the perturbative beginnings of an incipient instability even in
regimes and system sizes for which conventional diagnostics point towards
localization. We expect this new measure to be of independent interest for
analyzing the stability of localization in a variety of different settings.
