On gapped boundaries for SPT phases beyond group cohomology

Abstract

We discuss a strategy to construct gapped boundaries for a large class of symmetry-protected topological phases (SPT phases) beyond group cohomology. This is done by a generalization of the symmetry extension method previously used for cohomological SPT phases. We find that this method allows us to construct gapped boundaries for time-reversal-invariant bosonic SPT phases and for fermionic Gu-Wen SPT phases for arbitrary finite internal symmetry groups.

Pin TQFT and Grassmann integral

“…But to make sense of this expression we have to extend the definition of the Gu-Wen Grassmann integral σ(α d−1 ) to the case when α d−1 ∈ C d−1 (M, Z 2 ) is not necessarily closed. This generalization was performed for the spin case in [32]. By slightly generalizing the analysis in [32] to the pin case, we will see that the extended Gu-Wen integral nicely couples to the bulk in a gauge invariant fashion.…”

“…This generalization was performed for the spin case in [32]. By slightly generalizing the analysis in [32] to the pin case, we will see that the extended Gu-Wen integral nicely couples to the bulk in a gauge invariant fashion.…”

“…When we naively use the above definition (2.1) when α is not closed: δα = β, the resulting expression is problematic since u(t) can become Grassmann-odd. Following [32], we avoid this conundrum by coupling with the Gu-Wen integral σ(N, β) in (d + 1) dimensional bulk N such that ∂N = M , making all components in the path integral Grassmann-even. Now let us write down the boundary Gu-Wen integral coupled with bulk; we denote the entire integral by σ(α; β).…”

Symmetry-enriched quantum spin liquids in (3 + 1)d

“…This comes from the last term in (2.64), where H 3 (H, U(1)) represents the 0-form symmetry domain wall defects that support H gauge theories in (2 + 1)d [34]. Some of these 0-form symmetry defects permute the types of the surface operators that correspond to non-trivial conjugacy classes of H. 41 Consider such a surface operator piercing the wall at a line intersection. There is a non-trivial holonomy of the H connection around the line intersection on the wall that equals the holonomy around the surface operator, and thus the intersection represents a magnetic line in the H gauge theory on the wall.…”

“…This can be shown explicitly for Abelian H. As an example, consider H = Z 3 2 gauge theory as discussed in section 2.2.4. The wall corresponding to the nontrivial element in H 3 (H, U(1)) A = Z 2 ⊂ H 3 (H, U(1)) permutes the surface operators that 40 A non-trivial η2 ∈ H 2 (G, Z(H)) corresponds to a non-trivial group extension of G by H. 41 In general, the symmetry G can permute non-local operators even when the symmetry group is a product G global × Hgauge. 42 The set of line operators in finite group H gauge theory in (2 + 1)d depends on the topological action classified by H 3 (H, U(1)) [34] (see e.g.…”