  Based on a result by Yarotsky (J. Stat. Phys. 118, 2005), we prove that
localized but otherwise arbitrary perturbations of weakly interacting quantum
spin systems with uniformly gapped on-site terms change the ground state of
such a system only locally, even if they close the spectral gap. We call this a
strong version of the local perturbations perturb locally (LPPL) principle
which is known to hold for much more general gapped systems, but only for
perturbations that do not close the spectral gap of the Hamiltonian. We also
extend this strong LPPL-principle to Hamiltonians that have the appropriate
structure of gapped on-site terms and weak interactions only locally in some
region of space.
  While our results are technically corollaries to a theorem of Yarotsky, we
expect that the paradigm of systems with a locally gapped ground state that is
completely insensitive to the form of the Hamiltonian elsewhere extends to
other situations and has important physical consequences.
