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arXiv:2502.11106 · Physical review. B./Physical review. B

Point-group symmetry enriched topological orders

Zhaoyang Ding, Yang Qi

Published Jul 17, 2025• 0 citations
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Abstract

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 ${C}_{n}$ and a sector with an angle $\ensuremath{\pi}/n$ for the dihedral point group ${D}_{2n}$, 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 ${H}^{1}$ obstruction and an ${H}^{2}$ obstruction, respectively. When the obstruction vanishes, the junction, and therefore the point-group-symmetry-enriched topological orders, are classified by an ${H}^{0}$ cohomology class and an ${H}^{1}$ 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 ${H}^{1}$ and ${H}^{2}$ obstructions and the junction correspond to the ${H}^{3}$ and ${H}^{4}$ obstructions for onsite symmetries, respectively.

DOI
10.1103/ghds-2st8

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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{\mathbb{Z}}_{2}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{\mathbb{Z}}_{2}Z2​-symmetric gapped boundaries of the folded system. In particular, we derive an H2{H}^{2}H2 obstruction function, which corresponds to an H3{H}^{3}H3 obstruction for topological orders enriched by the time-reversal symmetry instead of mirror symmetry. We demonstrate that states with a nontrivial H2{H}^{2}H2 obstruction function can be constructed on the surface of a three-dimensional mirror SET order.
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.
Recently, a new class of quantum phases of matter---symmetry protected topological states, such as topological insulators---has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. These phases have short-ranged entanglement and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in the presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement in the presence or absence of on-site global symmetries. In 2+12+12+1 dimensions, the quantum phases in the presence of a global symmetry group SGSGSG, and with topological order described by a finite gauge group GGGGGG, are classified by the cohomology group ${H}^{3}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Generally, in d+1d+1d+1 dimensions, such quantum phases are classified by ${H}^{d+1}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [Chen et al., Phys. Rev. B 87, 155114 (2013)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond, the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases and generalizations of our classification are discussed.
We examine the interplay of symmetry and topological order in 2+12+12+1 dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the Gf{\cal G}^{\rm f}Gf fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry G=Gf/Z2fG= {\cal G}^{\rm f}/\mathbb{Z}_2^{\rm f}G=Gf/Z2f​ on the physical fermions is essentially the central extension of GGG by the Z2f\mathbb{Z}_2^{\rm f}Z2f​ fermion parity conservation that yields the fermionic symmetry Gf{\cal G}^{\rm f}Gf. We develop an algebraic theory of Gf{\cal G}^{\rm f}Gf symmetry defects for fermionic topological phases using GGG-crossed braided tensor category theory. This formalism allows us to fully characterize and classify 2+12+12+1 dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group Gf{\cal G}^{\rm f}Gf. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed Gf{\cal G}^{\rm f}Gf form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.
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.
This works uses the two dimensional rotational invariant plane in bosonic symmetry protected topological phases with C2{C}_{2}C2​ rotational symmetry in four dimensions to classify the phases and obtain a boundary field theory as well as an anomalous topological order for the phase that is not captured by the group cohomology classification.