Symmetry Enrichment in Three-Dimensional Topological Phases

Shang-Qiang Ning Tsinghua University Zheng-Xin Liu Renmin University of China Peng Ye University of Illinois at Urbana-Champaign

Publications of CMSA of Harvard mathscidoc:1702.38071

While two-dimensional symmetry-enriched topological phases (š–²š–¤š–³s) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry Gs on gauge theories (denoted by š–¦š–³) with gauge group Gg. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" (š–²š–¤š–¦), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on š–²š–¤š–¦s with gauge group Gg=ā„¤N1Ɨā„¤N2Ɨā‹Æ and on-site unitary symmetry group Gs=ā„¤K1Ɨā„¤K2Ɨā‹Æ or Gs=U(1)Ɨā„¤K1Ɨā‹Æ. Each š–²š–¤š–¦(Gg,Gs) is described in the path integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground state properties (i.e., š–²š–¤š–³ orders) of š–²š–¤š–¦s in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the \emph{mixed} multi-loop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from š–²š–¤š–¦s to š–²š–¤š–³s. By giving full dynamics to background gauge fields, š–²š–¤š–¦s may be eventually promoted to a set of new gauge theories (denoted by š–¦š–³āˆ—). Based on their gauge groups, š–¦š–³āˆ—s may be further regrouped into different classes each of which is labeled by a gauge group Gāˆ—g. Finally, a web of gauge theories involving š–¦š–³, š–²š–¤š–¦, š–²š–¤š–³ and š–¦š–³āˆ— is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.
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@inproceedings{shang-qiangsymmetry,
  title={Symmetry Enrichment in Three-Dimensional Topological Phases},
  author={Shang-Qiang Ning, Zheng-Xin Liu, and Peng Ye},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208004108198655298},
}
Shang-Qiang Ning, Zheng-Xin Liu, and Peng Ye. Symmetry Enrichment in Three-Dimensional Topological Phases. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208004108198655298.
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