(3+1)D topological phases of matter can host a broad class of nontrivial topological defects of codimension1, 2, and 3, of which the wellknown 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 faulttolerant logical operations in topological quantum errorcorrecting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension2 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 ℤ 2 gauge theory with fermionic charges, in \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 gauge theory with bosonic charges, and also in nonAbelian discrete gauge theories based on dihedral ( D_n D n ) and alternating ( A_6 A 6 ) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H^4 H 4 cohomology class that characterizes part of an underlying 3group symmetry of the topological order. The equations involving background gauge fields for the 3group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with nonAbelian flux loops (defining part of a noninvertible higher symmetry), examples of noninvertible codimension2 defects, and examples of the interplay of codimension2 defects with codimension1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D A_6 A 6 gauge theory.
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Dihedral twist liquid models from emergent Majorana fermions
We present a family of electronbased coupledwire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by manybody interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are nonlocal (fractional) and constitute the S O ( N ) KacMoody WessZuminoWitten algebra at level 1. We focus on the dihedral D k symmetry of S O ( 2 n ) 1 , and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the C / G twist liquids, where G = Z 2 for C = U ( 1 ) l , S U ( n ) 1 , and G = Z 2 , Z k , D k for C = S O ( 2 n ) 1 . We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the nonAbelian metaplectic anyons and a new class of quasiparticles referred to as Isingfluxons. We show an eightfold periodic gauging pattern in S O ( 2 n ) 1 / G by identifying the nonchiral components of the twist liquids with discrete gauge theories.
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 Award ID(s):
 1653535
 NSFPAR ID:
 10436733
 Date Published:
 Journal Name:
 Quantum
 Volume:
 7
 ISSN:
 2521327X
 Page Range / eLocation ID:
 967
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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