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Abstract

We apply the framework of Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) and their associated parent Hamiltonians to the classification of quantum phases in one and higher dimensions, where we define two systems to be in the same phase if they can be connected via a path of gapped Hamiltonians. In one dimension, we prove that any two Hamiltonians with MPS ground states are in the same phase if and only if they have the same ground state degeneracy. Subsequently, we extend our framework to the classification of two-dimensional quantum phases in the neighborhood of a number of important cases, such as systems with unique ground states, local symmetry breaking, and topological order. As a central tool in our derivation, we introduce the isometric form of MPS and PEPS. Isometric forms are renormalization fixed points both for MPS and for relevant classes of PEPS, and are connected to the original MPS via a gapped path in Hamiltonian space. Our construction thus yields a way to implement renormalization flows locally, this is, without actual renormalization.