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.
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.
We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with Z2 topological order, that is, on gapped Z2 spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group H2(G,Z2). This result leads us to a symmetry classification of gapped Z2 spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and SO(3) spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric Z2 spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.
Symmetry fractionalization on topological excitations is one of the most remarkable quantum phenomena in topological orders with symmetry, i.e., symmetry-enriched topological phases. While much progress has been theoretically and experimentally made in two dimensions (2D), the understanding on symmetry fractionalization in 3D is far from complete. A long-standing challenge is to understand symmetry fractionalization on looplike topological excitations, which are spatially extended objects. In this paper, we construct a powerful topological-field-theoretical framework approach for 3D topological orders, which leads to a systematic characterization and classification of symmetry fractionalization. For systems with Abelian gauge groups (Gg) and Abelian symmetry groups (Gs), we successfully establish equivalence classes that lead to a finite number of patterns of symmetry fractionalization, although there are notoriously infinite number of Lagrangian-descriptions of the system. Based on this, we compute topologically distinct types of fractional symmetry charges carried by particles. Then, for each type, we compute topologically distinct statistical phases of braiding processes among loop excitations and external symmetry fluxes. As a result, we are able to unambiguously list all physical observables for each pattern of symmetry fractionalization. We present detailed calculations on many concrete examples. As an example, we find that the symmetry fractionalization in an untwisted ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological order with Z2 symmetry is classified by ${({\mathbb{Z}}_{2})}^{6}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$. If the topological order is twisted, the classification reduces to (Z2)6 in which particle excitations always carry integer charge. Inspired by the field-theoretical analysis, we find that our classification of symmetry fractionalization with twist $\ensuremath{\omega}$ can be formally organized into an algebraic formalism: ${⨁}_{{\ensuremath{\nu}}_{i}}{\mathcal{H}}^{4}({G}_{g}{\ensuremath{\leftthreetimes}}_{{\ensuremath{\nu}}_{i}}{G}_{s},U(1))/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i}),$ where anomaly-free symmetry fractionalization on particles ${\ensuremath{\nu}}_{i}\ensuremath{\in}{\mathcal{H}}_{non}^{2}({G}_{s},{G}_{g})$. Despite the lack of detailed dependence of ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$ on $\ensuremath{\omega}$ and ${\ensuremath{\nu}}_{i}$, the present field-theoretical approach allows us to efficiently calculate and understand ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$. Several future directions are presented at the end of this paper.
The hallmark of a two-dimensional (2d) topologically ordered phase is the existence of deconfined ``anyon'' excitations that have exotic braiding and exchange statistics, different from those of ordinary bosons or fermions. As opposed to conventional Landau-Ginzburg-Wilson phases, which are classified on the basis of the spontaneous breaking of an underlying symmetry, topologically ordered phases, such as those occurring in the fractional quantum Hall effect, are absolutely stable, not requiring any such symmetry. Recently, though, it has been realized that symmetries, which may still be present in such systems, can give rise to a host of new, distinct, many-body phases, all of which share the same underlying topological order. These ``symmetry enriched'' topological (SET) phases are distinguished not on the basis of anyon braiding statistics alone, but also by the symmetry properties of the anyons, such as their fractional charges, or the way that different anyons are permuted by the symmetry. Thus a useful approach to classifying SETs is to determine all possible such symmetry actions on the anyons that are algebraically consistent with the anyon statistics. Remarkably, however, there exist symmetry actions that, despite being algebraically consistent, cannot be realized in any physical system, and hence do not lead to valid 2d SETs. One class of such ``anomalous'' SETs, characterized by certain disallowed symmetry fractionalization patterns, finds a physical interpretation as an allowed surface state of certain three-dimensional (3d) short-range entangled phases, but another, characterized by some seemingly valid but anomalous permutation actions of the symmetry on the anyons and encoded in an H3(G,A) group cohomology class, has so far eluded a physical interpretation. In this work, we find a way to physically realize these anomalously permuting SETs at the surfaces of certain 3d long-range entangled phases, expanding our understanding of general anomalous SETs in two dimensions.
In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways, leading to a variety of symmetry-enriched topological (SET) phases.While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems.Instead, certain "anomalous" SETs can only occur on the surface of a 3D symmetry-protected topological (SPT) phase.In this paper, we describe a procedure for determining whether a SET of a discrete, on-site, unitary symmetry group G is anomalous or not.The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes.Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions is captured by the fourth cohomology group H 4 ðG; Uð1ÞÞ, which also precisely labels the set of 3D SPT phases, with symmetry group G.An explicit procedure for calculating the cohomology data from a SET is given, with the corresponding physical intuition explained.We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3D bulk SPT whose surface termination realizes it.We illustrate this idea using the chiral spin liquid [Uð1Þ 2 ] topological order with a reduced symmetry Z 2 × Z 2 ⊂ SOð3Þ, which can act on the semion quasiparticle in an anomalous way.We construct exactly solved 3D SPT models realizing the anomalous surface terminations and demonstrate that they are nontrivial by computing three-loop braiding statistics.Possible extensions to antiunitary symmetries are also discussed.
We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group Gf. In general, Gf is a central extension of the bosonic symmetry group Gb by fermion parity, ${(\ensuremath{-}1)}^{F}$, characterized by a nontrivial cohomology class $[{\ensuremath{\omega}}_{2}]\ensuremath{\in}{\mathcal{H}}^{2}({G}_{b},{\mathbb{Z}}_{2})$. We show how the presence of local fermions places a number of constraints on the algebraic data that defines the action of the symmetry on the supermodular tensor category that characterizes the anyon content. We find two separate obstructions to defining symmetry fractionalization, which we refer to as the bosonic and fermionic symmetry localization obstructions. The former is valued in H3(Gb,K(C)), while the latter is valued in either ${\mathcal{H}}^{3}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$ or Z2(Gb,Z2) depending on additional details of the theory. K(C) is the Abelian group of functions from anyons to U(1) phases obeying the fusion rules, A is the Abelian group defined by fusion of Abelian anyons, and $\ensuremath{\psi}$ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over ${\mathcal{H}}^{2}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$. We study a number of examples in detail; in particular, we provide a characterization of fermionic Kramers degeneracy arising in symmetry class DIII within this general framework, and we discuss fractional quantum Hall and Z2 quantum spin liquid states of electrons.
We examine non-Abelian topological defects in an Abelian lattice model in two dimensions. We first construct an exact solvable lattice model that exhibits coexisting and intertwined topological and classical orders. The anyon types of quasiparticle excitations are permuted by lattice symmetry operations like translations, rotations, and reflections. The global anyon permutation symmetry has a group structure of S3, the permutation group of three elements. Topological crystalline defects---dislocations and disclinations---change the anyon type of an orbiting quasiparticle. They exhibit multichannel order-dependent fusion rules and projective braiding operations. Their braiding and exchange statistics breaks modular invariance and violates the conventional spin-statistics theorem. We develop a framework to characterize these unconventional properties that originate from the semiclassical nature of defects.