Anyons and matrix product operator algebras

By Nick Bultinck, Michael Marien, Dominic J. Williamson, M. B. cSahinouglu, J. Haegeman, Frank Verstraete. 2015

146 citations recorded by the source corpus.

Quantum tensor network states and more particularly projected entangled-pair 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.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/35a54216205b8f37134386db7bb27a1a43aefd66

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Citations in the corpus

Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases (2019, background): “These symmetries may also be realized by matrix-product operators (MPOs) [8,11,54], and were even found in some related 2d statistical mechanical models [2].”
Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward (2022, background): “In this case, the Rep(G) symmetry has an intuitive interpretation as string operators whose end points are gauge charges for the quantum double of G [86].”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, methodology): “Third, MPOs play a fundamental role in describing symmetries of PEPS (Bultinck et al., 2017; Chen et al., 2011c; Şahinoğlu et al., 2014).”