Quantum tensor network states and more particularly projected entangledpair states provide a natural framework for representing ground states of gapped, topologically ordered systems. The defining feature of these representations is that topological order is a consequence of the symmetry of the underlying tensors in terms of matrix product operators. In this paper, we present a systematic study of those matrix product operators, and show how this relates entanglement properties of projected entangled-pair states to the formalism of fusion tensor categories. From the matrix product operators we construct a C * -algebra and find that topological sectors can be identified with the central idempotents of this algebra. This allows us to construct projected entangled-pair states containing an arbitrary number of anyons. Properties such as topological spin, the S matrix, fusion and braiding relations can readily be extracted from the idempotents. As the matrix product operator symmetries are acting purely on the virtual level of the tensor network, the ensuing Wilson loops are not fattened when perturbing the system, and this opens up the possibility of simulating topological theories away from renormalization group fixed points. We illustrate the general formalism for the special cases of discrete gauge theories and string-net models.1 arXiv:1511.08090v2 [cond-mat.str-el]

We prove an upper bound on the maximal rate at which a Hamiltonian interaction can generate entanglement in a bipartite system. The scaling of this bound as a function of the subsystem dimension on which the Hamiltonian acts nontrivially is optimal and is exponentially improved over previously known bounds. As an application, we show that a gapped quantum many-body spin system on an arbitrary lattice satisfies an area law for the entanglement entropy if and only if any other state with which it is adiabatically connected (i.e. any state in the same phase) also satisfies an area law.PACS numbers: 03.67. Mn, 03.67.Bg, 03.65.Vf Introduction Entanglement is one of the defining trademarks of quantum mechanics, appearing ubiquitously, both at the theoretical and experimental level. For many of the applications, notably in quantum optics, nuclear magnetic resonance and condensed matter physics, it is the optimal creation of entanglement that is important, and much experimental effort has been devoted to this process. A fundamental question then is, given some Hamiltonian interaction between two subsystems A and B, what is the maximal rate at which the Hamiltonian evolution can generate entanglement [1][2][3]? It is this dynamical question that we study in this paper, and we will provide a tight upper bound for the maximal entanglement rate in the most general setting. But as we will show in the second part of the paper, this issue also has important consequences for the static entanglement properties of quantum many-body systems.Over the past decade it has been realized that looking at entanglement properties provides a new window on these systems, allowing an improved classification of different quantum phases of matter [4][5][6]. In particular it was found that the entanglement distribution for gapped systems always seems to have the same characteristic behavior [7]: for two arbitrary connected subsystems, the entanglement entropy scales like the boundary area (with small logarithmic corrections for critical systems), rather than the volume as one would expect for random states. This so called area law has important consequences for the efficient representability of quantum many-body states. And as such it has served as a guiding principle for the formulation of the different successful numerical tensor network methods that have emerged in the last years [8]. However, so far the area law has only been proven for gapped one-dimensional systems [9,10]. Our results provide a step in the direction of a general proof for higher dimensions. Using the formalism of quasi-adiabatic continuation [11], our bound on the entanglement rate allows us to show that within a gapped quantum phase, a subsystem's

We introduce tangent space methods for projected entangled-pair states (PEPS) that provide direct access to the low-energy sector of strongly-correlated two-dimensional quantum systems. More specifically, we construct a variational ansatz for elementary excitations on top of PEPS ground states that allows for computing gaps, dispersion relations and spectral weights directly in the thermodynamic limit. Solving the corresponding variational problem requires the evaluation of momentum transformed two-point and three-point correlation functions on a PEPS background, which we can compute efficiently by introducing a new contraction scheme. As an application we study the spectral properties of the magnons of the Affleck-Kennedy-Lieb-Tasaki model on the square lattice and the anyonic excitatons in a perturbed version of Kitaev's toric code.In the last decades, low-dimensional quantum systems have been at the forefront of both theoretical and experimental physics. With the reduced dimensionality allowing for stronger quantum correlations, these systems show an extreme variety of exotic phenomena but are notoriously difficult to simulate 1 . In particular, the lowenergy physics of such systems typically cannot be apprehended by perturbing around some free or otherwise exactly solvable point.For one-dimensional quantum spin chains, the advent of the density matrix renormalization group (DMRG) 2 has proven revolutionary. DMRG has developed into the de facto standard method for reliably and efficiently probing the low-energy behaviour of strongly correlated systems in one dimension. A better understanding of this success was realized through the identification of the variational class over which DMRG optimizes as matrix product states (MPS) 3,4 . More specifically, it was understood how the entanglement structure of MPS allows for a natural parametrization of the low-energy states of gapped, local one-dimensional quantum systems 5 . The characterizing feature of the entanglement structure in gapped local quantum systems is the observed area law for entanglement entropy 6 . With this insight the natural extension of MPS to higher dimensions is given by the class of projected entangled-pair states (PEPS) 7,8 . This set of states has shown to capture the low-energy physics of several interesting systems in two dimensions and is competing with more established methods in determining their ground-state properties 9,10 . Yet, in contrast to the one-dimensional case, the PEPS simulation of two-dimensional systems is computationally challenging, thus making the development of new algorithms highly desirable. Furthermore, existing PEPS algorithms focus exclusively on capturing ground state wave functions, whereas experimentally relevant low-energy properties such as excitation spectra remain inaccessible.In this paper we initiate a new set of methods based on the tangent space of the PEPS manifold of states, a concept which has proven extremely versatile in the context of matrix product states 11 . We apply this methodology for con...

We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e. the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a Kähler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.

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 on-site 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 [X. Chen et al., Phys. Rev. B 87, 155114 (2013)] [1]. 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-points 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.