Fermionic SPT phases in higher dimensions and bosonization

TL;DR

This work develops a bosonization-based framework to classify fermionic SRE (FSRE) phases across 1–3 spatial dimensions, extending known 1d/2d results to 3d by introducing Kitaev strings and a 3-group symmetry structure. The authors construct bosonic shadows for 3d FSREs, formulated through a 3-group $E$ with a nontrivial anomaly and a G-equivariant 2-Ising model (the 4d shadow), and propose a G-dependent classification in terms of data $( u, ho,\sigma)$ constrained by generalized Gu–Wen equations. They show how gauging the 2-form and 1-form ${ m Z}_2$ symmetries, via a sequence of shadow-to-FSRE constructions, yields a comprehensive scheme that recovers Gu–Wen supercohomology as a special case and extends it to non-supercohomology phases, including Kitaev-string–carrying configurations. The framework highlights a deep link between 3-group symmetry, spin structure, and higher-form anomalies, with potential extensions to higher dimensions and explicit lattice models, and it clarifies how fermionic statistics emerge from the interplay of particle and string sectors. Overall, the paper provides a concrete, physically meaningful path to realize and classify 3d FSREs in the presence of arbitrary finite symmetry $G$ via bosonization and higher-form symmetries.

Abstract

We discuss bosonization and Fermionic Short-Range-Entangled (FSRE) phases of matter in one, two, and three spatial dimensions, emphasizing the physical meaning of the cohomological parameters which label such phases and the connection with higher-form symmetries. We propose a classification scheme for fermionic SPT phases in three spatial dimensions with an arbitrary finite point symmetry G. It generalizes the supercohomology of Gu and Wen. We argue that the most general such phase can be obtained from a bosonic "shadow" by condensing both fermionic particles and strings.

Figures (4)

• Figure 1: A picture of the F-junction or $A_3$ singularity, where four zippers meet. With the x axis along the blue-grey junction and the y axis along the green-grey junction, the planes $x+y = c$ cut through this picture to give a movie of the F move as we vary $c$ through zero.
• Figure 3: A configuration of two $\sigma$ anyons (black circles) in a Hopf link formation. The $\sigma$'s are the boundary of the $O$ surface (orange discs). Where the $\sigma$'s intersect the orange disc (red stars), we have a $\psi$ anyon (wavy black curve) being born.
• Figure 4: A configuration of two $\sigma$ anyons (black circles) in a Hopf link formation given by the boundary of a twice-twisted ribbon of $O$ surface (orange skeleton). With this configuration of the $O$ surface, the self-linking of each component is even, so there is no need for a $\psi$ line connecting them. This contrasts with the into-the-page framed Hopf link we drew above, where the framing induced by the $O$ surface has odd self-linking in each component, so the two components are fermionic and there must be a $\psi$ line connecting them.
• Figure 5: We revisit the 15j symbol in the presence of $C$ with non-zero $\rho$ but $\sigma = 0$. The tetrahedra where $\rho \neq 0$ have a non-conservation of $\psi$ lines, indicated by red curves coming out of the resolved 4-way junctions ($A_3$ singularities) dual to the tetrahedra. The $\psi$ lines go and join "the condensate", represented by a red ball which may absorb any number of $\psi$ lines. In evaluating the diagram according to the rules of the Ising category, we get contributions from crossings. The red with black give a sign contribution of $-1$ to power $\epsilon_2(034)\rho(0123)+\epsilon_2(014)\rho(1234) = (\epsilon_2 \cup_1 \rho)(01234)$. The black with black crossing gives a contribution of $-1$ to power $\epsilon_2(012)\epsilon_2(234) = (\epsilon_2 \cup \epsilon_2)(01234)$.