We introduce a framework for characterizing Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) in terms of symme-tries. This allows us to understand how PEPS appear as ground states of local Hamiltonians with finitely degenerate ground states and to charac-terize the ground state subspace. Subsequently, we apply our framework to show how the topological properties of these ground states can be ex-plained solely from the symmetry: We prove that ground states are locally indistinguishable and can be transformed into each other by acting on a restricted region, we explain the origin of the topological entropy, and we discuss how to renormalize these states based on their symmetries. Fi-nally, we show how the anyonic character of excitations can be understood as a consequence of the underlying symmetries. 1
We introduce a framework for characterizing Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) in terms of symme-tries. This allows us to understand how PEPS appear as ground states of local Hamiltonians with finitely degenerate ground states and to charac-terize the ground state subspace. Subsequently, we apply our framework to show how the topological properties of these ground states can be ex-plained solely from the symmetry: We prove that ground states are locally indistinguishable and can be transformed into each other by acting on a restricted region, we explain the origin of the topological entropy, and we discuss how to renormalize these states based on their symmetries. Fi-nally, we show how the anyonic character of excitations can be understood as a consequence of the underlying symmetries. 1
We give a classification of gapped quantum phases of one-dimensional systems in the framework of matrix product states (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states and in both the absence and the presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground-state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, that is, the equivalence classes of its projective representations, a result first derived by Chen, Gu, and Wen [Phys. Rev. B 83, 035107 (2011)]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labeled by both the permutation action of the former and the cohomology class of the latter. Using projected entangled pair states (PEPS), we subsequently extend our framework to the classification of two-dimensional phases in the neighborhood of a number of important cases, in particular, systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS, which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus we can focus on to the classification of isometric forms.
Tensor Networks are a tool from quantum information theory to describe many-body
quantum systems. It has been shown that states arising from Tensor Networks in 1D (Matrix
Product States - MPS) accurately describe ground states of local, gapped Hamiltonians.
The same project for higher dimensional systems is currently the subject of great efforts.
However, quantum many-body systems in dimension 2 or greater can exhibit much richer
behaviour than their one-dimensional counterparts. One of the hallmarks of this is the
scarcely understood notion of topological order. It has been shown recently, that Projected
Entangled-Pair States (PEPS - Tensor Networks in 2D) can describe chiral topological
states, that lie in the ground state manifold of a certain class of gapless Hamiltonians. In
this work, we try to perturb the Hamiltonian into a gapped chiral phase. We will then
study, whether this gapped, chiral phase can be described in terms of PEPS or how the
PEPS-construction has to be modified to accommodate for this kind of behaviour.
Projected entangled pair states (PEPS) provide a natural description of the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We show that on-site symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, can be captured by a virtual symmetry of the tensors expressed as a set of matrix product operators labelled by the different group elements. A classification of SPT phases can hence be obtained by studying the topological obstructions to continuously deforming one set of matrix product operators into another. This leads to the classification of bosonic SPT states in terms of group cohomology, as originally derived by Chen et al. in [1106.4772]. Our formalism accommodates perturbations away from fixed point models, and hence opens up the possibility of studying phase transitions between different SPT phases. We furthermore show how the global symmetries of SPT PEPS can be promoted into a set of local gauge constraints by introducing bosonic degrees of freedom on the links of the PEPS lattice, thereby providing a natural and general mapping between PEPS in SPT phases and topologically ordered phases.
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 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, is local, frustration free, and gapless. It can be interpreted as describing a quantum phase transition between different topological phases. The second one, Hfb, is gapped, and has hopping terms scaling as 1/r3 with the distance r. 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.
This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states. The Projected Entangled Pair State (PEPS) formalism allows us to locally encode the main properties of the models (like topological order, symmetries and their interplay) in the tensors. We have used that encoding to classify, construct and detect some classes of symmetry-enriched topological phases in 2D PEPS. For that purpose, we have studied what the allowed freedom in the tensors generating the same tensor network state is. These results are the so-called 'fundamental theorems' and they give the necessary knowledge to properly study symmetries (actions that leave the states invariant). We focus our study on the family of PEPS describing quantum double models of G, the so-called G-injective PEPS, and on global on-site symmetries coming from a finite group Q: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.