Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory

TL;DR

This work develops a comprehensive framework to classify interacting fermionic SPT phases in 3D by formulating equivalence classes of fermionic symmetric local unitaries (FSLU) and constructing general fixed-point wavefunctions. It shows that 2D FSPT states are captured by $H^1(G_b,\mathbb Z_2)$, $BH^2(G_b,\mathbb Z_2)$, and $H^3(G_b,U_T(1))$, while 3D states require obstruction-free subgroups $\tilde{B}H^2(G_b,\mathbb Z_2)$, $BH^3(G_b,\mathbb Z_2)$, and $H^4_{rigid}(G_b,U_T(1))$, organized via short exact sequences that define a general group super-cohomology theory. The construction relies on discrete spin structures implemented through Kasteleyn orientations and Majorana chain decorations on symmetry-domain walls, with fermionic pentagon/hexagon equations enforcing consistency. The results unify known 2D classifications and predict a complete 3D classification for unitary bosonic symmetries, while revealing the role of obstructions and $\mathbb Z_8$-valued stacking structure. This framework provides commuting-projector Hamiltonians on arbitrary triangulations, linking algebraic data to explicit lattice realizations and guiding future explorations of nontrivial 3D FSPT phases and their braiding properties.

Abstract

Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this work, we revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. We construct very general fixed point SPT wavefunctions for interacting fermion systems. We naturally reproduce the partial classifications given by special group super-cohomology theory, and we show that with an additional $\tilde{B}H^2(G_b, \mathbb Z_2)$ (the so-called obstruction free subgroup of $H^2(G_b, \mathbb Z_2)$) structure, a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group $G_f=G_b\times \mathbb Z_2^f$ can be obtained for unitary symmetry group $G_b$. We also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions.

Figures (35)

• Figure 1: Bosonic and fermionic degrees of freedoms for 1D fixed-point FSPT states on a link. The black dots are bosonic degrees of freedom labelled by $g_i\in G$ on sites. The blue ball represents the complex fermion $c_{(ij)}$ at the center of the link $\langle ij\rangle$. The arrow represents the local order of two sites.
• Figure 2: Fermionic degrees of freedom in a triangle. The red dots represent Majorana fermions at the two sides of each link. The blue ball represents the complex fermion of the special group super-cohomology model at the center of the triangle. The green strip is the decorated Kitaev's Majorana chain onto the dual lattice $\mathcal{P}$.
• Figure 3: Fermionic degrees of freedom in a tetrahedron. The red dots represent Majorana fermions on the two sides of each triangle. The blue ball represents the complex fermion of the (special) group super-cohomology model at the center of the tetrahedron. The green line is the decorated Kitaev's Majorana chain on the dual lattice $\mathcal{P}$.
• Figure 4: Example of triangulation $\mathcal{T}$ of torus and Kitaev chain decoration. All vertices $\langle1\rangle$, $\langle2\rangle$, $\langle3\rangle$, and $\langle4\rangle$ (blue dots) are singular vertices, i.e., $w_0=\langle1\rangle+\langle2\rangle+\langle3\rangle+\langle4\rangle$. We choose link $\langle 13\rangle$ and $\langle 24\rangle$ (blue lines) to be singular lines, i.e., $w_0=\partial(\langle 13\rangle+\langle 24\rangle)$. The direction of the red links dual to $\langle 13\rangle$ and $\langle 24\rangle$ are changed. Vertices $i=1,2,3$, and $4$ of $\mathcal{T}$ are labelled by group elements $g_i\in G$. Majorana fermions (red dots) reside on the vertices of the resolved dual lattice $\tilde{\mathcal{P}}$ (solid and dashed red links). The solid red links and gray ellipses indicate that the two Majorana fermions at their two ends are paired with respect to the link direction. The green strip is the $\mathbb Z_2$ domain wall of the "spin" configuration $\{g_i\}$ and is decorated by a Kitaev's Majorana chain (Majorana fermions along the domain wall are paired differently from the "vacuum").
• Figure 5: Regular pair $v\subseteq\sigma^i$ ($i=0,1,2$) for vertex $v$.