Anyons and matrix product operator algebras

Abstract

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]

Diagonalizing Transfer Matrices and Matrix Product Operators: A Medley of Exact and Computational Methods

“…This construction allows us to define string nets [46] and quantum doubles [70] on arbitrary lattices. Furthermore, this construction decomposes the physical Hilbert space into a direct sum of anyonic sectors: the MPO formalism gives an explicit expression for the elementary excitations (anyons) and their braiding properties in terms of central idempotents of the algebra generated by the ensuing MPOs [50]. It is truly remarkable that such a rich algebraic structure emerges out of a very innocuous looking set of MPOs.…”

“…PEPS turn out to be also very useful for characterizing systems with topological quantum order. In that case, PEPS exhibit nontrivial symmetries on the virtual level [68, 69,48,50]. Such systems with topological order are currently under intense study as they present a natural framework for constructing quantum error correcting codes which allow to build fault tolerant quantum memories and quantum computers.…”

“…This hidden symmetry is characterized by a set of MPOs (see figure 7), and the full richness of the topological theory (including e.g. anyon statistics and braiding properties) can be deduced from studying the algebra generated by those MPOs [48,50]. The so-called quantum double models and string nets [46] provide systems for which those MPOs can be constructed analytically.…”

“…Similarly, the central ingredient in solving integrable spin systems makes essential use of matrix product operators: the core of the algebraic Bethe ansatz [44,45] is the construction of an algebra of matrix product operators. Matrix product operators can also be used to construct explicit representations of tensor fusion categories [46,47,48,49,50], which makes them central to the study of topologically ordered systems. MPOs have also been studied extensively in the context of non-equilibrium systems [51,52], as one-dimensional cellular automata are MPOs of a special kind [53].…”

Matrix product operators for symmetry-protected topological phases: Gauging and edge theories

“…[39], and they should describe the emergent symmetry defects and their G-graded fusion and G-crossed braiding properties. Note this data subsumes the underlying anyon theory and the possibly fractional symmetry transformation of the defects.…”