Norbert Schuch, California Institute of Technology
Abstract
We apply the framework of Matrix Product States (MPS) and their parent Hamiltonians to the classification of gapped quantum phases in one-dimensional systems and beyond. We discuss both the case of systems with unique ground states and with degenerate ground states, i.e., symmetry breaking, both in the absence and in the presence of symmetries. We find that without symmetries, all systems (up to ground state degeneracy) are in the same phase. If symmetries are imposed, the classification is based on the different types of projective representations of the symmetry group, together with its permutation action on the symmetry broken ground states. We also discuss how to extend our results to two dimensional systems using Projected Entangled Pair States (PEPS), and show that one cannot expect the same type of symmetry protection as in one-dimension. As a central tool, we introduce the isometric form of an MPS which gives a rigorous framework for studying phases using renormalization fixed points, without a need to actually renormalize the system.
Joint work with D. Perez-Garcia and I. Cirac.