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

(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

\mathbb{Z}_2

gauge theory with fermionic charges, in

\mathbb{Z}_2 \times \mathbb{Z}_2

gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral (

D_n

) and alternating (

A_6

) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an

H^4

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

A_6

gauge theory.

Note:

70 pages, 42 figures

string: twist
phase: topological
defect: topological
string: interaction
fermion: charge
defect: condensation
gauge field theory: discrete
flux: nonabelian
symmetry: topological
abelian

References(101)

Figures(64)

[1]

Non-Abelian anyons and topological quantum computation

Chetan Nayak
(
- UC, Santa Barbara
)
,
Steven H. Simon
(
- Bell Labs
)
,
Ady Stern
(
- Weizmann Inst.
)
,
Michael Freedman
(
- UC, Santa Barbara
)
,
Sankar Das Sarma
(
- Maryland U.
)

- Rev.Mod.Phys. 80 (2008) 1083-1159
•
e-Print:
- 0707.1889
•
DOI:
- 10.1103/RevModPhys.80.1083

[18]

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

et al.

- SciPost Phys. 14 (2023) 4, 065,
- SciPost Phys. 14 (2023) 065
•
e-Print:
- 2208.07367
•
DOI:
- 10.21468/SciPostPhys.14.4.065

[2]

Topological quantum computation, American Mathematical Society, Providence, Rhode Island, USA

Z. Wang

[3]

Classification of (3+1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons

Tian Lan
(
- Perimeter Inst. Theor. Phys. and
- Waterloo U.
)
,
Liang Kong
(
- Tsinghua U., Beijing
)
,
Xiao-Gang Wen
(
- MIT, Cambridge, Dept. Phys.
)

- Phys.Rev.X 8 (2018) 2, 021074
•
DOI:
- 10.1103/PhysRevX.8.021074

[4]

Classification of 3+1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions

- Phys.Rev.X 9 (2019) 2, 021005
•
e-Print:
- 1801.08530
•
DOI:
- 10.1103/PhysRevX.9.021005

[5]

Topological Order with a Twist: Ising Anyons from an Abelian Model

H. Bombin
(
- Perimeter Inst. Theor. Phys.
)

- Phys.Rev.Lett. 105 (2010) 030403
•
e-Print:
- 1004.1838
•
DOI:
- 10.1103/PhysRevLett.105.030403

[6]

Topological field theory, higher categories, and their applications, in Proceedings of the international congress of mathematicians(ICM), Hindustan Book Agency, New Delhi, India