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arXiv:1610.05384

A Short Guide to Anyons and Modular Functors

Simon Burton

Published Oct 17, 2016• 4 citations
arXivHTMLPDF

Abstract

To the working physicist, anyon theory is meant to describe certain quasi-particle excitations occurring in two dimensional topologically ordered systems. A typical calculation using this theory will involve operations such as $\otimes$ to combine anyons, $F^{abc}_{d}$ to re-associate such combinations, and $R^{ab}_c$ to commute or braid these anyons. Although there is a powerful string-diagram notation that greatly assists these manipulations, we still appear to be operating on particles arranged on a one-dimensional line, algebraically ordered from left to right. The obvious question is, where is the other dimension? The topological framework for considering these anyons as truly living in a two dimensional space is known as a modular functor, or topological quantum field theory. In this work we show how the apparently one-dimensional algebraic anyon theory is secretly the theory of anyons living in a fully two-dimensional system. The mathematical literature covering this secret is vast, and we try to distill this down into something more manageable.

DOI
10.48550/arxiv.1610.05384

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To the working physicist, anyon theory is meant to describe certain quasi-particle excitations occurring in two dimensional topologically ordered systems. A typical calculation using this theory will involve operations such as ⊗\otimes⊗ to combine anyons, FdabcF^{abc}_{d}Fdabc​ to re-associate such combinations, and RcabR^{ab}_cRcab​ to commute or braid these anyons. Although there is a powerful string-diagram notation that greatly assists these manipulations, we still appear to be operating on particles arranged on a one-dimensional line, algebraically ordered from left to right. The obvious question is, where is the other dimension? The topological framework for considering these anyons as truly living in a two dimensional space is known as a modular functor, or topological quantum field theory. In this work we show how the apparently one-dimensional algebraic anyon theory is secretly the theory of anyons living in a fully two-dimensional system. The mathematical literature covering this secret is vast, and we try to distill this down into something more manageable.
Abstract Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.
The theory of anyon systems, as modular functors topologically and unitary modular tensor categories algebraically, is mature. To go beyond anyons, our first step is the interplay of anyons with conventional group symmetry due to the paramount importance of group symmetry in physics. This led to the theory of symmetry-enriched topological order. Another direction is the boundary physics of topological phases, both gapless as in the fractional quantum Hall physics and gapped as in the toric code. A more speculative and interesting direction is the study of Banados–Teitelboim–Zanelli (BTZ) black holes and quantum gravity in 3d. The clearly defined physical and mathematical issues require a far-reaching generalization of anyons and seem to be within reach. In this short survey, I will first cover the extensions of anyon theory to symmetry defects and gapped boundaries. Then, I will discuss a desired generalization of anyons to anyon-like objects — the BTZ black holes — in 3d quantum gravity.
Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.
Anyons, comprising a special type of particles that exhibit braid statistics,\nhave received considerable attention in the past decade, from physicists,\nmathematicians, and computer scientists. It turns out that so-called modular\ntensor categories naturally appear in the description of the key properties of\nanyons, which partly explains the interest of mathematicians in the subject.\nIn this thesis, we study the way such modular tensor categories emerge in two\nclasses of models, viz. quantum field theory in low-dimensional Minkowski\nspace-time, and quantum spin systems on an infinite spatial lattice. Although\nthe nature of these two classes of models is quite different, we show in this\nthesis that they may both be discussed in a similar (mathematically rigorous)\nframework. This approach is inspired by the work of Dpplicher, Haag and\nRoberts (DHR) in algebraic quantum field theory.\nThe DHR approach leads to braided tensor categories, which describe the charges\nin the theory (in our case, these are anyons). These categories need not be\nmodular. In this thesis we present a method to remove an obstruction for\nmodularity, in the case of a space-time of dimension three. We then study (a\nvariant of) Kitaev's toric code, regarded as a quantum spin model on an\ninfinite lattice. Using techniques inspired by the DHR programme, we show that\none can obtain a modular tensor category describing all relevant properties of\nthe anyons in this system. Finally, some generalisations to non-abelian models\nare discussed.
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like $e^{-\ał}$, where łłł is a length scale, and ααα is some positive constant. In contrast, the $\q$presumptive" qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about 10−410^{-4}10−4) before computation can be stabilized.