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Based on a result by Yarotsky (J. Stat. Phys. 118, 2005), we prove thatlocalized but otherwise arbitrary perturbations of weakly interacting quantumspin systems with uniformly gapped on-site terms change the ground state ofsuch a system only locally, even if they close the spectral gap. We call this astrong version of the local perturbations perturb locally (LPPL) principlewhich is known to hold for much more general gapped systems, but only forperturbations that do not close the spectral gap of the Hamiltonian. We alsoextend this strong LPPL-principle to Hamiltonians that have the appropriatestructure of gapped on-site terms and weak interactions only locally in someregion of space.  While our results are technically corollaries to a theorem of Yarotsky, weexpect that the paradigm of systems with a locally gapped ground state that iscompletely insensitive to the form of the Hamiltonian elsewhere extends toother situations and has important physical consequences.

Local stability of ground states in locally gapped and weakly interacting quantum spin systems