Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions

TL;DR

The paper develops a defect-based, $G$-extension framework for classifying two-dimensional interacting fermionic SPT phases with symmetry $G\times\mathbb{Z}_2^f$, connecting to the bosonic $H^3(G,\mathrm{U}(1))$ classification and introducing an essential $H^1(G,\mathbb{Z}_2)$ datum. By enforcing consistency of $G$-extensions within a $G$-crossed braided fusion category, it identifies an obstruction $O(\mathbf{g,h,k,l})=(-1)^{n(\mathbf{g,h})n(\mathbf{k,l)}}$ whose triviality in cohomology is required for viable fSPT phases; when satisfied, bosonic stacking data $\nu\in H^3(G,\mathrm{U}(1))$ appear alongside $n\in H^2(G,\mathbb{Z}_2)$, yielding a full Abelian classification, with non-Abelian phases arising from nontrivial $\mathbb{Z}_2$-valued homomorphisms from $G$ to $\mathbb{Z}_2$ and Ising-like defect theories. Gauging fermion parity provides a cross-check via symmetry-enriched $\mathbb{Z}_2$ gauge theories, where $H^2(G,\mathbb{Z}_2)$ captures symmetry fractionalization on the fermion-parity flux; explicit results are worked out for $G=\mathbb{Z}_n$, $\mathbb{Z}_2\times\mathbb{Z}_2$, and $\mathbb{Z}_2^T$, yielding detailed group structures and physical interpretations. The framework unifies and extends bosonic SPT classifications to interacting fermionic systems, yields concrete classification tables, and offers practical insight into potential realizations in correlated 2D materials and models.

Abstract

Recently, it has been shown that two-dimensional bosonic symmetry-protected topological(SPT) phases with on-site unitary symmetry $G$ can be completely classified by the group cohomology class $H^3(G, \mathrm{U}(1))$. Later, group super-cohomology class was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the mathematical framework of $G$-extension of unitary braided tensor category(UBTC) theory. We first reproduce the partial classifications given by group super-cohomology, then we show that with an additional $H^1(G, \mathbb{Z}_2)$ structure, a complete classification of SPT phases for two-dimensional interacting fermion systems for a total symmetry group $G\times\mathbb{Z}_2^f$ can be achieved. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.

Figures (2)

• Figure 1: Heptagon equations which ensure consistency between $G$-crossed braiding and fusion of defects.
• Figure 2: The Pentagon equation enforces the consistency between different sequences of $F$ moves starting and ending with the same fusion trees.