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arXiv:1801.01210

A Classification of (2+1)D Topological Phases with Symmetries

Tian Lan

Published Jan 4, 2018• 3 citations
arXivHTMLPDF

Abstract

This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories $\mathcal E\subset\mathcal C\subset\mathcal M$ plus the chiral central charge $c$. Here $\mathcal E$ is a symmetric fusion category, $\mathcal E=\mathrm{Rep}(G)$ for boson systems or $\mathcal E=\mathrm{sRep}(G^f)$ for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry; $\mathcal C$ is the category of all the quasiparticle excitations in the bulk, containing $\mathcal E$ as its Müger center; $\mathcal M$ is a minimal modular extension of $\mathcal C$, that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.

DOI
10.48550/arxiv.1801.01210

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This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories E⊂C⊂M\mathcal E\subset\mathcal C\subset\mathcal ME⊂C⊂M plus the chiral central charge ccc. Here E\mathcal EE is a symmetric fusion category, E=Rep(G)\mathcal E=\mathrm{Rep}(G)E=Rep(G) for boson systems or E=sRep(Gf)\mathcal E=\mathrm{sRep}(G^f)E=sRep(Gf) for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry; C\mathcal CC is the category of all the quasiparticle excitations in the bulk, containing E\mathcal EE as its Müger center; M\mathcal MM is a minimal modular extension of C\mathcal CC, that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.
We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry GGG. The key is to use the so-called symmetric fusion category E\mathcal{E}E to describe the symmetry. Here, E=sRep(Z2f)\mathcal{E}=\mathrm{sRep}({\mathbb{Z}}_{2}^{f})E=sRep(Z2f​) describing particles in a fermionic product state without symmetry, or E=sRep(Gf)[E=Rep(G)]\mathcal{E}=\mathrm{sRep}({G}^{f}) [\mathcal{E}=\mathrm{Rep}(G)]E=sRep(Gf)[E=Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry GGG. Then, topological orders with symmetry E\mathcal{E}E are classified by nondegenerate unitary braided fusion categories over E\mathcal{E}E, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to p+ipp+\phantom{\rule{1.0pt}{0ex}}\mathrm{i}\phantom{\rule{1.0pt}{0ex}}pp+ip fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge c=14c=\frac{1}{4}c=41​, whose only topological excitation has non-Abelian statistics with spin s=14s=\frac{1}{4}s=41​ and quantum dimension d=1+2d=1+\sqrt{2}d=1+2​.
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.
Large classes of matter have been systematized before by using a pair of groups to classify all spontaneous symmetry breaking phases or by using a pair of groups plus projective representations to classify all gapped quantum phases of one-dimensional boson and fermion systems with various symmetries. In this paper, the authors go beyond this and use a pair of groups plus three braided fusion categories and a chiral central charge in order to classify gapped quantum phases of two-dimensional boson and fermion systems with various finite symmetries. This allows, among other things, to generate many group tables of two-dimensional gapped phases.
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group GGG and its group 4-cocycle ω4∈H4[G;U(1)]ω_4 \in \mathcal H^4[G;U(1)]ω4​∈H4[G;U(1)] up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
A bstract We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.
Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach [L. Bhardwaj et al., SciPost Phys. 19, 056 (2025); L. Bhardwaj et al., arXiv: 2502.20440]. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider “all-boson type” fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups G G (possibly with an ’t Hooft anomaly) via generalized gauging—that is, by stacking with an H H -symmetric TFT and gauging a subgroup H H . The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.