We construct in the
K matrix formalism concrete examples of symmetry-enriched topological phases, namely intrinsically topological phases with global symmetries. We focus on the Abelian and nonchiral topological phases and demonstrate by our examples how the interplay between the global symmetry and the fusion algebra of the anyons of a topologically ordered system determines the existence of gapless edge modes protected by the symmetry and that a (quasi)group structure can be defined among these phases. Our examples include phases that display charge fractionalization and more exotic nonlocal anyon exchange under global symmetry that correspond to general group extensions of the global symmetry group.
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.
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.
A bstract Symmetry-enriched topological (SET) phases combine intrinsic topological order with global symmetries, giving rise to novel symmetry phenomena. While SET phases with Abelian anyons are relatively well understood, those involving nonabelian anyons remain elusive. This obscurity stems from the multi-dimensional internal gauge spaces intrinsic to nonabelian anyons — a feature first made explicit in [1] and further explored in our recent works [2–7]. These internal spaces can transform in highly nontrivial ways under global symmetries. In this work, we employ an exactly solvable model — the multifusion Hu-Geer-Wu string-net model introduced in a companion paper [8] — to reveal how the internal gauge spaces of nonabelian anyons transform under symmetries. We uncover a universal phenomenon, global symmetry fragmentation (GSF), whereby symmetry-invariant anyons exhibit internal Hilbert space decompositions into eigensubspaces labeled by generally fractional symmetry charges. Meanwhile, symmetry-permuted anyons hybridize and fragment their internal spaces in accordance with their symmetry behavior. These fragmented structures realize genuinely nonlinear symmetry representations that transcend conventional linear and projective classifications. Our results identify nonlinear global symmetry fragmentation as a hallmark of SETs and may shed new light on symmetry-enabled control in topological quantum computation.
In topological phases in
2+1 dimensions, anyons fall into representations of quantum group symmetries. As proposed in our work [Hung and Wan, Int. J. Mod. Phys. B 28, 1450172 (2014)], the physics of a symmetry enriched phase can be extracted by the mathematics of (hidden) quantum group symmetry breaking of a ``parent phase.'' This offers a unified framework and classification of the symmetry enriched (topological) phases, including symmetry protected trivial phases as well. In this paper, we extend our investigation to the case where the ``parent'' phases are non-Abelian topological phases. We show explicitly how one can obtain the topological data and symmetry transformations of the symmetry enriched phases from that of the ``parent'' non-Abelian phase. Two examples are computed: (1) the $\text{Ising}\ifmmode\times\else\texttimes\fi{}\overline{\text{Ising}}$ phase breaks into the
Z2 toric code with
Z2 global symmetry; (2) the
SU(2)8 phase breaks into the chiral Fibonacci $\ifmmode\times\else\texttimes\fi{}$ Fibonacci phase with a
Z2 symmetry, a first non-Abelian example of symmetry enriched topological phase beyond the gauge-theory construction.
We develop a folding approach to study two-dimensional symmetry-enriched topological (SET) phases with the mirror reflection symmetry. Our folding approach significantly transforms the mirror SETs, such that their properties can be conveniently studied through previously known tools: (i) it maps the nonlocal mirror symmetry to an onsite Z[subscript 2] layer-exchange symmetry after folding the SET along the mirror axis, so that we can gauge the symmetry; (ii) it maps all mirror SET information into the boundary properties of the folded system, so that they can be studied by the anyon condensation theory—a general theory for studying gapped boundaries of topological orders; and (iii) it makes the mirror anomalies explicitly exposed in the boundary properties, i.e., strictly 2D SETs and those that can only live on the surface of a 3D system can be easily distinguished through the folding approach. With the folding approach, we derive a set of physical constraints on data that describes mirror SET, namely, mirror permutation and mirror symmetry fractionalization on the anyon excitations in the topological order. We conjecture that these constraints may be complete, in the sense that all solutions are realizable in physical systems. Several examples are discussed to justify this. Previously known general results on the classification and anomalies are also reproduced through our approach.