Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly

The interplay between symmetry and topological properties plays a very important role in modern physics. In the past decade, the concept of symmetry-enriched topological (SET) phases was proposed and their classifications have been systematically studied for bosonic systems. Very recently, the concept of SET phases has been generalized into fermionic systems and their corresponding classification schemes are also proposed. Nevertheless, how to realize all these fermionic SET (fSET) phases in lattice models remains to be a difficult open problem. In this paper, we first construct exactly solvable models for non-anomalous non-chiral 2+1D fSET phases, namely, the symmetry-enriched fermionic string-net models, which are described by commuting-projector Hamiltonians whose ground states are the fixed-point wavefunctions of each fSET phase. Mathematically, we provide a partial definition to $G$-graded super fusion category, which is the input data of a symmetry-enriched fermionic string-net model. Next, we construct exactly solvable models for non-chiral 2+1D fSET phases with ’t Hooft anomaly, especially the $H^3(G,\mathbb{Z}_2)$ fermionic ’t Hooft anomaly which is different from the well known bosonic $H^4(G,U(1)_T)$ anomaly. In our construction, this $H^3(G,\mathbb{Z}_2)$ fermionic ’t Hooft anomaly is characterized by a violation of fermion-parity conservation in some of the surface ${F}$-moves (a kind of renormalization moves for the ground state wavefunctions of surface SET phases), and also by a new fermionic obstruction $Θ$ in the surface pentagon equation. We demonstrate this construction in a concrete example that the surface topological order is a $\mathbb{Z}_4$ gauge theory embedded into a fermion system and the total symmetry $G^f=\mathbb{Z}_2^f\times\mathbb{Z}_2\times\mathbb{Z}_4$.

💡 Research Summary

This paper presents a significant advancement in the theoretical understanding and concrete modeling of fermionic symmetry-enriched topological (fSET) phases. The central challenge addressed is the realization of such phases in explicit, exactly solvable lattice models, moving beyond abstract classification schemes.

The authors first construct exactly solvable models for non-anomalous, non-chiral (2+1)-dimensional fSET phases. These are dubbed “symmetry-enriched fermionic string-net models.” They are commuting-projector Hamiltonians whose ground states represent the fixed-point wavefunctions for each fSET phase. The key mathematical input for these models is a partially defined “G-graded super fusion category.” This structure encodes the topological data (string types, fusion rules) enriched by a global symmetry group G, where the total fermionic symmetry is G^f = Z^f_2 × G (with Z^f_2 being the always-present fermion parity symmetry). The construction generalizes the bosonic symmetry-enriched string-net model framework to fermionic systems by carefully incorporating the fermionic degrees of freedom and their parity.

The second major contribution is the construction of exactly solvable models for fSET phases that possess a ’t Hooft anomaly, specifically focusing on the H^3(G, Z_2) fermionic anomaly. This anomaly is distinct from the well-known bosonic H^4(G, U(1)_T) anomaly. The authors show that this fermionic anomaly manifests concretely in the surface theory of a (3+1)D fermionic symmetry-protected topological (fSPT) phase. In their model, the anomaly is characterized by a violation of fermion parity conservation in some of the surface “F-moves” (a type of local unitary transformation that renormalizes the ground state wavefunction). Mathematically, this violation corresponds to the appearance of a new fermionic obstruction factor, denoted Θ, in the pentagon equation governing the consistency of these F-moves. This provides a direct lattice-model realization of how an anomaly constrains the possible surface topological orders.

The paper demonstrates this anomalous construction through a detailed example where the surface topological order is a Z_4 gauge theory embedded in a fermionic system with total symmetry G^f = Z^f_2 × Z_2 × Z_4. Furthermore, the authors conjecture on the nature of the H^2(G, Z_2) fermionic anomaly, suggesting it would be characterized by a violation in the fusion rules involving non-terminating “σ-type” strings on the surface.

The work provides extensive technical background, including a discussion of super fusion categories, their Drinfeld centers (which describe the anyon excitations), and the relationship to bosonic models via fermion condensation. Appendices review relevant concepts like super modular categories, the classification of anyons and symmetry defects in bosonic SET phases, and offer additional details on renormalization moves and partition function approaches.

In summary, this paper bridges the gap between the algebraic classification of fermionic SET phases and their physical realization in lattice models. It provides a generic framework for building exactly solvable models for both anomaly-free and anomalous fSET phases, offering concrete tools for numerical studies and deepening the understanding of symmetry-anomaly interplay in fermionic topological matter.