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arXiv:1210.5601 · Physical Review Letters

Topological Order in the Projected Entangled-Pair States Formalism: Transfer Operator and Boundary Hamiltonians

Norbert Schuch, Didier Poilblanc, J. I. Cirac, David Pérez-Garcı́a

Published Aug 27, 2013• 127 citations
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Abstract

We study the structure of topological phases and their boundaries in the projected entangled-pair states (PEPS) formalism. We show how topological order in a system can be identified from the structure of the PEPS transfer operator and subsequently use these findings to analyze the structure of the boundary Hamiltonian, acting on the bond variables, which reflects the entanglement properties of the system. We find that in a topological phase, the boundary Hamiltonian consists of two parts: A universal nonlocal part which encodes the nature of the topological phase and a nonuniversal part which is local and inherits the symmetries of the topological model, which helps to infer the structure of the boundary Hamiltonian and thus possibly of the physical edge modes.

DOI
10.1103/physrevlett.111.090501

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We study the structure of topological phases and their boundaries in the projected entangled-pair states (PEPS) formalism. We show how topological order in a system can be identified from the structure of the PEPS transfer operator and subsequently use these findings to analyze the structure of the boundary Hamiltonian, acting on the bond variables, which reflects the entanglement properties of the system. We find that in a topological phase, the boundary Hamiltonian consists of two parts: A universal nonlocal part which encodes the nature of the topological phase and a nonuniversal part which is local and inherits the symmetries of the topological model, which helps to infer the structure of the boundary Hamiltonian and thus possibly of the physical edge modes.
Projected entangled pair states (PEPS) provide a natural ansatz for the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We develop a framework to describe onsite symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, in terms of virtual symmetries of the local tensors expressed as a set of matrix product operators (MPOs) labeled by distinct group elements. These MPOs describe the possibly anomalous symmetry of the edge theory, whose local degrees of freedom are concretely identified in a PEPS. A classification of SPT phases is obtained by studying the obstructions to continuously deforming one set of MPOs into another, recovering the results derived for fixed-point models [Chen et al., Phys. Rev. B 87, 155114 (2013)]. Our formalism accommodates perturbations away from fixed-point models, opening the possibility of studying phase transitions between different SPT phases. We also demonstrate that applying the recently developed quantum state gauging procedure to a SPT PEPS yields a PEPS with topological order determined by the initial symmetry MPOs. The MPO framework thus unifies the different approaches to classifying SPT phases, via fixed-point models, boundary anomalies, or gauging the symmetry, into the single problem of classifying inequivalent sets of matrix product operator symmetries that are defined purely in terms of a PEPS.
We study projected entangled pair states (PEPS) with continuous virtual symmetries, i.e., symmetries in the virtual degrees of freedom, through an elementary class of models with SU(2)\mathrm{SU}(2)SU(2) symmetry. Discrete symmetries of that kind have previously allowed for a comprehensive explanation of topological order in the PEPS formalism. We construct local parent Hamiltonians whose ground space with open boundaries is exactly parametrized by the PEPS wave function, and show how the ground state can be made unique by a suitable choice of boundary conditions. We also find that these models exhibit a logarithmic correction to the entanglement entropy and an extensive ground space degeneracy on systems with periodic boundaries, which suggests that they do not describe conventional gapped topological phases, but either critical models or some other exotic phase.
We investigate the topological character of lattice chiral Gaussian fermionic states in two dimensions possessing the simplest descriptions in terms of projected entangled pair states (PEPS). They are ground states of two different kinds of Hamiltonians. The first one, Hff{\mathcal{H}}_{\mathrm{ff}}Hff​, is local, frustration free, and gapless. It can be interpreted as describing a quantum phase transition between different topological phases. The second one, Hfb{\mathcal{H}}_{\mathrm{fb}}Hfb​, is gapped, and has hopping terms scaling as 1/r31/{r}^{3}1/r3 with the distance rrr. The gap is robust against local perturbations, which allows us to define a Chern number for the PEPS. As for (nonchiral) topological PEPS, the nontrivial topological properties can be traced down to the existence of a symmetry in the virtual modes that are used to build the state. Based on that symmetry, we construct stringlike operators acting on the virtual modes that can be continuously deformed without changing the state. On the torus, the symmetry implies that the ground state space of the local parent Hamiltonian is twofold degenerate. By adding a string wrapping around the torus, one can change one of the ground states into the other. We use the special properties of PEPS to build the boundary theory and show how the symmetry results in the appearance of chiral modes, and a universal correction to the area law for the zero R\'enyi entropy.
Abstract We consider projected entangled pair state (PEPS) models with a global Z N symmetry, which are constructed from Z N -symmetric tensors and are thus Z N -invariant wavefunctions, and study the occurence of long-range order and symmetry breaking in these systems. First, we show that long-range order in those models is accompanied by a degeneracy in the so-called transfer operator of the system. We subsequently use this degeneracy to determine the nature of the symmetry broken states, i.e. those stable under arbitrary perturbations, and provide a succinct characterization in terms of the fixed points of the transfer operator (i.e. the different boundary conditions) in the individual symmetry sectors. We verify our findings numerically through the study of a Z 3 -symmetric model, and show that the entanglement Hamiltonian derived from the symmetry broken states is quasi-local (unlike the one derived from the symmetric state), reinforcing the locality of the entanglement Hamiltonian for gapped phases.
We investigate the relations between the bulk and boundaries properties of short-range resonating valence bond wave functions using Projected Entangled Pair States. We find a gapless (gapped) entanglement spectrum for kagome (square) lattices, which reflects the properties of the edge modes rather than those of the bulk correlations. We also derive the topological entropy and show that the topological nature of the wave function manifests itself in a non-local many-body boundary Hamiltonian which we identify completely. We propose to use this feature as a probe to identify topological phases in microscopic Hamiltonians.