We study (2+1)-dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry-enriched topological (SET) phases. Here, we present a
K-matrix Chern-Simons approach to identify distinct phases with Abelian topological order, in the presence of unitary or antiunitary global symmetries. A key step is the identification of a smooth edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method by classifying
Z2 topological order (
Z2 spin liquids) in the presence of an internal
Z2 global symmetry for which we find six distinct phases. These include two phases with an unconventional action of symmetry that permutes anyons leading to symmetry-protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry-enriched Laughlin states and double-semion theories are also discussed. Somewhat surprisingly, we observe that (i) gauging the global symmetry of distinct SET phases leads to topological orders with the same total quantum dimension, and (ii) a pair of distinct SET phases can yield the same topological order on gauging the symmetry.
Symmetry-enriched topological phases (SET) exhibit both long-range entanglement and intricate interplay with global symmetries, such as anyon permutation symmetry or fractionalization of quantum numbers, going beyond the Landau paradigm of symmetry breaking. Here, the authors study two-dimensional SET phases based on the novel concept of symmetric local unitary transformations. Using the fixed-point structure of wave functions, they construct systematically exactly solvable models that can describe generic nonchiral SET phases. The technique applies to on-site as well as spatial symmetries and, furthermore, to anomalous symmetries that can only be realized on the surface of three-dimensional symmetry-protected phases.
Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a
G-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension
E of
G by
N: an
E-graded fusion category induces a
G-SET input by passing to the quotient symmetry
G, and the resulting input naturally carries a compatible
N-grading that implements the further condensation inside the SET phase while preserving
G. We illustrate the construction using three quantum-double examples: the trivial extension
Z2×Z2, the non-Abelian semidirect product
S3, and the nontrivial central extension
Z4. The
Z4 example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.
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.
Symmetry plays a crucial role in enriching topological phases of matter. Phases with intrinsic topological order that are symmetric are called symmetry-enriched topological phases (SET). In this paper, we focus on SETs in three spatial dimensions, where the intrinsic topological orders are described by Abelian gauge theory and the symmetry groups are also Abelian. As a series work of our previous research [Phys. Rev. B 94, 245120 (2016); (arXiv:1609.00985)], we study these topological phases described by twisted gauge theories with global symmetry and consider all possible topologically inequivalent charge matrices. Within each equivalence class, there is a unique pattern of symmetry fractionalization on both point-like and string-like topological excitations. In this way, we classify Abelian topological order enriched by Abelian symmetry within our field-theoretic approach. To illustrate, we concretely calculate many representative examples of SETs and discuss future directions.