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arXiv:2109.10911

Characterization and Classification of Fermionic Symmetry Enriched Topological Phases

David Aasen, Parsa Bonderson, Christina Knapp

Published Sep 22, 2021• 27 citations
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Abstract

We examine the interplay of symmetry and topological order in $2+1$ 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 ${\cal G}^{\rm f}$ 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 $G= {\cal G}^{\rm f}/\mathbb{Z}_2^{\rm f}$ on the physical fermions is essentially the central extension of $G$ by the $\mathbb{Z}_2^{\rm f}$ fermion parity conservation that yields the fermionic symmetry ${\cal G}^{\rm f}$. We develop an algebraic theory of ${\cal G}^{\rm f}$ symmetry defects for fermionic topological phases using $G$-crossed braided tensor category theory. This formalism allows us to fully characterize and classify $2+1$ dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group ${\cal G}^{\rm f}$. 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 ${\cal G}^{\rm f}$ 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.

DOI
10.48550/arxiv.2109.10911

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Symmetry Fractionalization in Three-Dimensional Z2\mathbb{Z}_2Z2​ Topological Order and Fermionic Symmetry-Protected Phases
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This paper develops a comprehensive theory of symmetry fractionalization together with the properties of symmetry defects in topologically ordered phases of matter in two spatial dimensions. To do this, the authors also introduce the full set of data, consistency conditions, and equivalences for a mathematical theory, known as a G-crossed braided tensor category, that characterizes the algebraic braiding and fusion properties of symmetry defects. This theoretical framework can completely characterize and classify symmetry-enriched topological phases of matter in the presence of arbitrarily strong interactions in quantum many-body systems in two spatial dimensions.
Symmetry fractionalization on topological excitations is one of the most remarkable quantum phenomena in topological orders with symmetry, i.e., symmetry-enriched topological phases. While much progress has been theoretically and experimentally made in two dimensions (2D), the understanding on symmetry fractionalization in 3D is far from complete. A long-standing challenge is to understand symmetry fractionalization on looplike topological excitations, which are spatially extended objects. In this paper, we construct a powerful topological-field-theoretical framework approach for 3D topological orders, which leads to a systematic characterization and classification of symmetry fractionalization. For systems with Abelian gauge groups (Gg{G}_{g}Gg​) and Abelian symmetry groups (Gs{G}_{s}Gs​), we successfully establish equivalence classes that lead to a finite number of patterns of symmetry fractionalization, although there are notoriously infinite number of Lagrangian-descriptions of the system. Based on this, we compute topologically distinct types of fractional symmetry charges carried by particles. Then, for each type, we compute topologically distinct statistical phases of braiding processes among loop excitations and external symmetry fluxes. As a result, we are able to unambiguously list all physical observables for each pattern of symmetry fractionalization. We present detailed calculations on many concrete examples. As an example, we find that the symmetry fractionalization in an untwisted ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological order with Z2{\mathbb{Z}}_{2}Z2​ symmetry is classified by ${({\mathbb{Z}}_{2})}^{6}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$. If the topological order is twisted, the classification reduces to (Z2)6{({\mathbb{Z}}_{2})}^{6}(Z2​)6 in which particle excitations always carry integer charge. Inspired by the field-theoretical analysis, we find that our classification of symmetry fractionalization with twist $\ensuremath{\omega}$ can be formally organized into an algebraic formalism: ${⨁}_{{\ensuremath{\nu}}_{i}}{\mathcal{H}}^{4}({G}_{g}{\ensuremath{\leftthreetimes}}_{{\ensuremath{\nu}}_{i}}{G}_{s},U(1))/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i}),$ where anomaly-free symmetry fractionalization on particles ${\ensuremath{\nu}}_{i}\ensuremath{\in}{\mathcal{H}}_{non}^{2}({G}_{s},{G}_{g})$. Despite the lack of detailed dependence of ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$ on $\ensuremath{\omega}$ and ${\ensuremath{\nu}}_{i}$, the present field-theoretical approach allows us to efficiently calculate and understand ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$. Several future directions are presented at the end of this paper.