A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions

TL;DR

The paper advances a comprehensive 3+1D classification of bosonic topological orders with emergent fermions (EF orders) by showing every EF order has a gappable boundary described by a unitary fusion 2-category and that pointlike excitations realize a Z2^f-extended group G_f. Majorana zero modes at triple-string intersections reveal how the boundary data encodes a 2-cocycle ρ2 that distinguishes EF1 from EF2 orders, with EF1 captured by a simplified pointed 2-category and EF2 requiring nontrivial Majorana-linked structure. The EF classification is tied to gauged 3+1D fermionic SPT phases, and the authors advocate a general gauging framework to obtain a full classification of 3+1D topological orders with finite unitary symmetry for both bosonic and fermionic systems, including SETs. The results unify bulk-boundary correspondence in higher categories and connect intrinsic orders to symmetry-protected phases, offering a path toward a complete higher-categorical understanding of 3+1D topological matter.

Abstract

In this paper, we classify EF topological orders for 3+1D bosonic systems where some emergent pointlike excitations are fermions. (1) We argue that all 3+1D bosonic topological orders have gappable boundary. (2) All the pointlike excitations in EF topological orders are described by the representations of $G_f=Z_2^f\leftthreetimes_{e_2} G_b$ -- a $Z_2^f$ central extension of a finite group $G_b$ characterized by $e_2\in H^2(G_b,Z_2)$. (3) We find that the EF topological orders are classified by 2+1D anomalous topological orders $\mathcal{A}_b^3$ on their unique canonical boundary. Here $\mathcal{A}_b^3$ is a unitary fusion 2-category with simple objects labeled by $\hat G_b=Z_2^m\leftthreetimes G_b$. $\mathcal{A}_b^3$ also has one invertible fermionic 1-morphism for each object as well as quantum-dimension-$\sqrt 2$ 1-morphisms that connect two objects $g$ and $gm$, where $g\in \hat G_b$ and $m$ is the generator of $Z_2^m$. (4) When $\hat G_b$ is the trivial $Z_2^m$ extension, the EF topological orders are called EF1 topological orders, which is classified by simple data $(G_b,e_2,n_3,ν_4)$. (5) When $\hat G_b$ is a non-trivial $Z_2^m$ extension, the EF topological orders are called EF2 topological orders, where some intersections of three stringlike excitations must carry Majorana zero modes. (6) Every EF2 topological order with $G_f=Z_2^f\leftthreetimes G_b$ can be associated with a EF1 topological order with $G_f=Z_2^f\leftthreetimes \hat G_b$. (7) We find that all EF topological orders correspond to gauged 3+1D fermionic symmetry protected topological (SPT) orders with a finite unitary symmetry group. (8) We further propose that the general classification of 3+1D topological orders with finite unitary symmetries for bosonic and fermionic systems can be obtained by gauging or partially gauging the finite symmetry group of 3+1D SPT phases of bosonic and fermionic systems.

Figures (17)

• Figure 1: A string configuration in the bulk described by a triple $(\chi_{g^f_1},\chi_{g^f_2},[g_3^f])$, where $\chi_{g^f}$ is a conjugacy class in $G_f$ containing $g^f\in G_f$ and the triple satisfy $g_1^f g_2^f=g_3^f$.
• Figure 2: Condensing all bosonic pointlike excitations in a 3+1D EF topological order $\EuScript{C}^4_{EF}$ gives rise to 3+1D $Z_2^f$ topological order $\EuScript{C}^4_{Z_2^f}$. $\EuScript{C}^4_{EF}$ contain a fermionic pointlike excitation $f$, and a stringlike excitation, $Z_2^f$-flux, which behave like the $\pi$-flux line for the fermion $f$. The domain wall $\EuScript{A}_w^3$ between $\EuScript{C}^4_{EF}$ and $\EuScript{C}^4_{Z_2^f}$ contain strings labeled by elements $g\in G_f$ and only one fermionic particle $f$. The strings and the fermion have quantum dimension 1.
• Figure 3: (Color online) The dimension reduction of 3D space $M^2\times S^1$ to 2D space $M^2$. The top and the bottom surfaces are identified and the vertical direction is the compactified $S^1$ direction. A 3D pointlike excitation (the blue dot) becomes an anyon particle in 2D. A 3D stringlike excitation wrapping around $S^1$ (the red line) also becomes an anyon particle in 2D.
• Figure 4: (Color online) The untwisted sector in the dimension reduction can be realized directly on a 2D sub-manifold in 3D space without compactification.
• Figure 5: (Color online) If a (composite) boundary excitations can be lifted in to the bulk, its half braiding with other boundary excitations must satisfy some self consistent conditions. The above illustrates the hexagon equation $b_{A,Y} b_{A,X} = b_{A,X\otimes Y}$.