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We study higher dimensional models with symmetric correlated hoppings, whichgeneralize a one-dimensional model introduced in the context ofdipole-conserving dynamics. We prove rigorously that whenever the localconfiguration space takes its smallest non-trivial value, these models exhibitlocalized behavior due to fragmentation, in any dimension. For the same classof models, we then construct a hierarchy of conserved quantities that arepower-law localized at the boundary of the system with increasing powers.Combining these with Mazur's bound, we prove that boundary correlations areinfinitely long lived, even when the bulk is not localized. We use our resultsto construct quantum Hamiltonians that exhibit the analogues of strong zeromodes in two and higher dimensions.

Fragmentation-induced localization and boundary charges in dimensions two and above