Published April 11, 2023 | Published
Journal Article Open

Codimension-2 defects and higher symmetries in (3+1)D topological phases

  • 1. ROR icon Joint Quantum Institute
  • 2. ROR icon University of Maryland, College Park
  • 3. ROR icon Harvard University
  • 4. ROR icon California Institute of Technology
  • 5. IBM Quantum

Abstract

(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error-correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of ℤ₂ gauge theory with fermionic charges, in ℤ₂ ×ℤ₂ gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral (D_n) and alternating (A₆) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H⁴ cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of the interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D A₆ gauge theory.

Copyright and License

Copyright M. Barkeshli et al. This work is licensed under the Creative Commons Attribution 4.0 International License.
Published by the SciPost Foundation.

Acknowledgement

We thank David Aasen, Yichul Choi, Sahand Seifnashri, Shu-Heng Shao, and Dominic Williamson for discussions. MB thanks David Penneys and Corey Jones for ongoing discussions and explanations of higher category theory. YC and NT thank Tyler Ellison for discussions of the e ↔ m twist string in the (2+1)D toric code. YC also thanks Po-Shen Hsin and Anton Kapustin for discussions of defects and 3-groups.

Funding

MB acknowledges financial support from NSF CAREER DMR-1753240. YC, SH, and RK are supported by the JQI postdoctoral fellowship at the University of Maryland. NT is supported by the Walter Burke Institute for Theoretical Physics at Caltech. GZ is supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number
DE-SC0012704.

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Created:
December 11, 2024
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December 11, 2024