Perturbation Theory for Parent Hamiltonians of Matrix Product States

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].