  We consider a family of quantum many-body Hamiltonians that show exact
Hilbert space fragmentation in certain limits. The question arises whether
fragmentation has implications for Hamiltonians in the vicinity of the subset
defined by these exactly fragmented models, in particular in the thermodynamic
limit. We attempt to illuminate this issue by considering distinguishable
classes of transitional behavior between fragmented and nonfragmented regimes
and employing a set of numerical observables that indicate this transition. As
one of these observables we present a modified inverse participation ratio
(IPR) that is designed to capture the emergence of fragmented block structures.
We compare this block IPR to other definitions of inverse participation ratios,
as well as to the more traditional measures of level-spacing statistics and
entanglement entropy. In order to resolve subtleties that arise in the
numerics, we use perturbation theory around the fragmented limit as a basis for
defining an effective block structure. We find that our block IPR predicts a
boundary between fragmented and nonfragmented regimes that is compatible with
results based on level statistics and bipartite entanglement. A scaling
analysis indicates that a finite region around the exactly fragmented limit is
dominated by effects of approximate fragmentation, even in the thermodynamic
limit, and suggests that fragmentation constitutes a phase. We provide evidence
for the universality of our approach by applying it to a different family of
Hamiltonians, that features a fragmented limit due to emergent dipole
conservation.
