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.
Two-dimensional mirror symmetry enriched topological (SET) orders can be studied using the folding approach: it can be folded along the mirror axis and turned into a bilayer system on which the mirror symmetry acts as a
Z2 layer-exchange symmetry. How mirror symmetry enriches the topological order is then encoded at the mirror axis, which is a gapped boundary of the folded bilayer system. Based on anyon-condensation theory, we classify possible
Z2-symmetric gapped boundaries of the folded system. In particular, we derive an
H2 obstruction function, which corresponds to an
H3 obstruction for topological orders enriched by the time-reversal symmetry instead of mirror symmetry. We demonstrate that states with a nontrivial
H2 obstruction function can be constructed on the surface of a three-dimensional mirror SET order.
We present a systematic framework to classify symmetry-enriched topological quantum spin liquids in two spatial dimensions. This framework can deal with all topological quantum spin liquids, which may be either Abelian or non-Abelian and chiral or nonchiral. It can systematically treat a general symmetry, which may include both lattice symmetry and internal symmetry, may contain antiunitary symmetry, and may permute anyons. The framework applies to all types of lattices and can systematically distinguish different lattice systems with the same symmetry group using their quantum anomalies, which are sometimes known as Lieb-Schultz-Mattis anomalies. We apply this framework to classify U ( 1 ) 2 N chiral states and non-Abelian Ising ( ν ) states enriched by a p 6 × SO ( 3 ) or p 4 × SO ( 3 ) symmetry and Z N topological orders and U ( 1 ) 2 N × U ( 1 ) − 2 N topological orders enriched by a p 6 m × SO ( 3 ) × Z 2 T , p 4 m × SO ( 3 ) × Z 2 T , p 6 m × Z 2 T , or p 4 m × Z 2 T symmetry, where p 6 , p 4 , p 6 m , and p 4 m are lattice symmetries while SO(3) and Z 2 T are spin rotation and time-reversal symmetries, respectively. In particular, we identify symmetry-enriched topological quantum spin liquids that are not easily captured by the usual parton-mean-field approach, including examples with the familiar Z 2 topological order. Published by the American Physical Society 2024
We study the classification of two-dimensional (2D) topological orders enriched by point-group symmetries, by generalizing the folding approach, which was previously developed for mirror-symmetry-enriched topological orders. We fold the 2D plane hosting the topological order into the fundamental domain of the point group, which is a sector with an angle $2\ensuremath{\pi}/n$ for the cyclic point group
Cn and a sector with an angle $\ensuremath{\pi}/n$ for the dihedral point group
D2n, and the point-group symmetries become onsite unitary symmetries on the sector. The enrichment of the point-group symmetries is then fully encoded at the boundary of the sector and the apex of the section, which forms a junction between the two boundaries. The mirror-symmetry enrichment encoded on the boundaries is analyzed by the classification theory of symmetric gapped boundaries, and the point-group-symmetry enrichment encoded on the junction is analyzed by a framework for classifying symmetric gapped junctions between boundaries which we develop in this work. We show that at the junction, there are two potential obstructions, which we refer to as an
H1 obstruction and an
H2 obstruction, respectively. When the obstruction vanishes, the junction, and therefore the point-group-symmetry-enriched topological orders, are classified by an
H0 cohomology class and an
H1 cohomology class, which can be understood as an additional Abelian anyon and a symmetry charge attached to the rotation center, respectively. These results are consistent with the classification of onsite-symmetry-enriched topological orders, where the
H1 and
H2 obstructions and the junction correspond to the
H3 and
H4 obstructions for onsite symmetries, respectively.
We propose a definition of a non-invertible symmetry enriched topological order (NI-SETO), and we implement our definition for string net models. We do so in two ways, using full inclusions of unitary fusion categories (UFCs), as well as anyon condensation. In both cases, the NI-SETO is a relative center of UFCs. All NI-SETOs can be realized in either model, where we can use enriched UFCs to get chiral examples on the boundary of a 3D Walker-Wang model representing the anomaly. We describe several examples of NI-SETOs and compute the qualitative symmetry action on anyons and symmetry defects using tube algebra techniques.
We propose a framework to describe topological phases with global internal symmetries. This involves embedding the phase of interest into an intrinsic topological-order. Such an embedding is related to the physics of anyon condensation, although no physical condensation actually occurs here. Crucial physical properties of the symmetric phase, such as edge states and global charges of anyons are easily recovered from the embedding. The framework naturally reproduces all phases that are currently known in 2+1 d, and potentially provides a complete classification of symmetric topological phases in 2+1 d. We illustrate our idea with simple examples.