  This article investigates the stability of the ground state subspace of a
canonical parent Hamiltonian of a Matrix product state against local
perturbations. We prove that the spectral gap of such a Hamiltonian remains
stable under weak local perturbations even in the thermodynamic limit, where
the entire perturbation might not be bounded. Our discussion is based on
preceding work by D.A. Yarotsky that develops a perturbation theory for
relatively bounded quantum perturbation of classical Hamiltonians. We exploit a
renormalization procedure, which on large scale transforms the parent
Hamiltonian of a Matrix product state into a classical Hamiltonian plus some
perturbation. We can thus extend D.A. Yarotsky's results to provide a
perturbation theory for parent Hamiltonians of Matrix product states and
recover some of the findings of independent contributions by S. Michalakis et
al. [arXiv:1109.1588] and J. I. Cirac et al. [arXiv:1306.4003].
