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arXiv:2410.23380

An operator algebraic approach to symmetry defects and fractionalization

Kyle Kawagoe, Siddharth Vadnerkar, Daniel Wallick

Published Oct 30, 2024• 0 citations
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Abstract

We provide a superselection theory of symmetry defects in 2+1D symmetry enriched topological (SET) order in the infinite volume setting. For a finite symmetry group $G$ with a unitary on-site action, our formalism produces a $G$-crossed braided tensor category $G\mathsf{Sec}$. This superselection theory is a direct generalization of the usual superselection theory of anyons, and thus is consistent with this standard analysis in the trivially graded component $G\mathsf{Sec}_1$. This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute $G\mathsf{Sec}$ explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.

DOI
10.48550/arxiv.2410.23380

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We provide a superselection theory of symmetry defects in 2+1D symmetry enriched topological (SET) order in the infinite volume setting. For a finite symmetry group GGG with a unitary on-site action, our formalism produces a GGG-crossed braided tensor category GSecG\mathsf{Sec}GSec. This superselection theory is a direct generalization of the usual superselection theory of anyons, and thus is consistent with this standard analysis in the trivially graded component GSec1G\mathsf{Sec}_1GSec1​. This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute GSecG\mathsf{Sec}GSec explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.
In this dissertation, we detail an operator algebraic approach to studying topological order in the infinite volume setting. We give a thorough and self-contained review of the DHR-style approach on quantum spin systems, which builds a category DHR\mathrm{\textbf{DHR}}DHR of anyon sectors starting from microscopic lattice spin systems. In general, this category has the structure of a braided C∗\mathrm{C}^*C∗-tensor category. We will verify in full detail that DHR\mathrm{\textbf{DHR}}DHR is the expected category in Kitaev's Quantum Double model, a paradigmatic model for studying topological order on the lattice. We will then extend the DHR-style analysis to systems in the presence of a global on-site symmetry, and introduce a category of symmetry defects, GSecG\mathsf{Sec}GSec, and show that it has the structure of a GGG-crossed braided C∗\mathrm{C}^*C∗-tensor category.
This paper develops a comprehensive theory of symmetry fractionalization together with the properties of symmetry defects in topologically ordered phases of matter in two spatial dimensions. To do this, the authors also introduce the full set of data, consistency conditions, and equivalences for a mathematical theory, known as a G-crossed braided tensor category, that characterizes the algebraic braiding and fusion properties of symmetry defects. This theoretical framework can completely characterize and classify symmetry-enriched topological phases of matter in the presence of arbitrarily strong interactions in quantum many-body systems in two spatial dimensions.
Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G , which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group H 2 ( G , A a b e l i a n ) being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G , one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic ‘twist’ defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418 ), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.
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