View the article online for updates and enhancements. AbstractThe curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes.These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states.The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.

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We use the multiscale entanglement renormalisation ansatz (MERA) to numerically investigate three critical quantum spin chains with Z2 × Z2 on-site symmetry: a staggered XXZ model, a transverse field cluster model, and the quantum Ashkin-Teller model. All three models possess a continuous one-parameter family of critical points. Along this critical line, the thermodynamic limit of these models is expected to be described by classes of c = 1 conformal field theories (CFTs) of two possible types: the S 1 free boson and its Z2-orbifold. Our numerics using MERA with explicitly enforced Z2 × Z2 symmetry allow us to extract conformal data for each model, with strong evidence supporting the identification of the staggered XXZ model and critical transverse field cluster model with the S 1 boson CFT, and the Ashkin-Teller model with the Z2-orbifold boson CFT. Our first two models describe the phase transitions between symmetry protected topologically ordered phases and trivial phases, which lie outside the usual Landau-Ginsburg-Wilson paradigm of symmetry breaking. Our results show that a range of critical theories can arise at the boundary of a single symmetry protected phase. arXiv:1501.02817v2 [cond-mat.str-el]

A binary interface defect is any interface between two (not necessarily invertible) domain walls. We compute all possible binary interface defects in Kitaev's Z/pZ model and all possible fusions between them. Our methods can be applied to any Levin-Wen model. We also give physical interpretations for each of the defects in the Z/pZ model. These physical interpretations provide a new graphical calculus which can be used to compute defect fusion. *

We study 't Hooft anomalies of discrete groups in the framework of (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It is numerically demonstrated that this line is a protected gapless phase. Finally, we implement a duality transformation between a pair of critical lines using our subclass of MERA.Comment: 12+18 pages, 6+5 figures, 0+2 tables, v2 published versio

We demonstrate how to do many computations for doubled topological phases with defects. These defects may be 1-dimensional domain walls or 0-dimensional point defects.Using Vec⁡(S3) as a guiding example, we demonstrate how domain wall fusion and associators can be computed using generalized tube algebra techniques. These domain walls can be both between distinct or identical phases. Additionally, we show how to compute all possible point defects, and the fusion and associator data of these. Worked examples, tabulated data and Mathematica code are provided.

We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with abelian topological order by finding ribbon operators for Z d quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting nonabelian topological order.Despite the apparent simplicity of quantum spin models, they can exhibit a wide variety of interesting and potentially useful phenomena. These range from conventional magnetic order to the more novel topological [1] and symmetry-protected[2] and symmetry-enriched [3] topological orders which are of interest in both condensed matter physics [4] and quantum information theory. [5] These states are disordered in the sense of LandauGinzburg-Wilson, however they do exhibit properties distinct from the usual disordered phases. For example, topological phases possess quasiparticle excitations, known as anyons, whose braid relations can be far more exotic than those of fermions or bosons.[6] These phases also have ground state degeneracy which depends on the topology of the lattice. [7] This protected degeneracy has prompted the investigation of topologically ordered models as quantum memories. [5,8] Quantum information stored in the degenerate subspace can be protected from arbitrary local noise when error correction techniques are employed. [9][10][11] Distinguishing topological phases can be an especially challenging task precisely because of their topological nature: there is no broken symmetry and no local order parameter signalling the phase transition.[1] There has been a large amount of previous work which attempts to identify topological order (TO). The existing key techniques, such as the topological entanglement entropy, [12][13][14][15][16][17] [26,27] have been highly successful in various domains of applicability. We review these methods below. A common feature of these methods is that they utilize the ground state of the model, which is unfortunately a challenging computational task in general.Here we propose a numerical method that we call the ribbon operators method for identifying TO in the ground state of a given 2D Hamiltonian using only the Hamiltonian, without reference to the ground state. We reduce the search for TO to a 1D problem through the bulk of the material, and we p...

We study the realization of anyon-permuting symmetries of topological phases on the lattice using tensor networks. Working on the virtual level of a projected entangled pair state, we find matrix product operators (MPOs) that realize all unitary topological symmetries for the toric and color codes. These operators act as domain walls that enact the symmetry transformation on anyons as they cross. By considering open boundary conditions for these domain wall MPOs, we show how to introduce symmetry twists and defect lines into the state. I. REVIEW: TOPOLOGICAL ORDER AND PEPSIn this section, we review some key concepts, notation and conventions required for the remainder of the paper.We begin with a discussion of topologically ordered phases, the kind of symmetries they support and the connections to fault-tolerant quantum computation. This motivates the discussion of locality preserving APS actions, domain walls, and defects. These topics form the primary objects of study in this paper.We introduce PEPS, the main tool used in this work, and a streamlined notation we use throughout the pa-arXiv:1708.08930v2 [quant-ph]