  We study higher dimensional models with symmetric correlated hoppings, which
generalize a one-dimensional model introduced in the context of
dipole-conserving dynamics. We prove rigorously that whenever the local
configuration space takes its smallest non-trivial value, these models exhibit
localized behavior due to fragmentation, in any dimension. For the same class
of models, we then construct a hierarchy of conserved quantities that are
power-law localized at the boundary of the system with increasing powers.
Combining these with Mazur's bound, we prove that boundary correlations are
infinitely long lived, even when the bulk is not localized. We use our results
to construct quantum Hamiltonians that exhibit the analogues of strong zero
modes in two and higher dimensions.
