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Exactly Solvable Models for Fermionic Symmetry-Enriched Topological Phases and Fermionic ’t Hooft Anomaly

Jing-Ren Zhou and Zheng-Cheng Gu*

  • *Contact author: zcgu@phy.cuhk.edu.hk
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Phys. Rev. X 16, 011019 – Published 5 February, 2026

DOI: https://doi.org/10.1103/r6d4-b538

Abstract

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 nonanomalous nonchiral 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 wave functions of each fSET phase. Mathematically, we provide a partial definition to G-graded super fusion category (when the total symmetry Gf is a direct product of physical symmetry G and fermion-parity symmetry Z2f), which is the input data of a symmetry-enriched fermionic string-net model. Next, we construct exactly solvable models for nonchiral 2+1D fSET phases with ’t Hooft anomaly, especially the H3(G,Z2) fermionic ’t Hooft anomaly which is different from the well-known bosonic H4[G,U(1)T] anomaly. In our construction, this H3(G,Z2) fermionic ’t Hooft anomaly is characterized by a violation of fermion-parity conservation in some of the surface F moves (a type of renormalization moves for the ground state wave functions of surface fSET 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 Z4 gauge theory embedded into a fermion system and the total symmetry Gf=Z2f×Z2×Z4. We further conjecture that the H2(G,Z2) fermionic ’t Hooft anomaly is characterized by a possible violation of nonterminating q-type (called σ-type) string in the fusion rules of surface models.

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Popular Summary

The existence of systems exhibiting anyon excitations with fractional charge, such as fractional quantum Hall effect (FQHE) and proposed phases in high-Tc cuprates, highlights a fundamental challenge: understanding the interplay between topological phases and global symmetry. We study topological orders with symmetry, known as symmetry-enriched topological (SET) phases, particularly in systems composed of fermions. Our research contributes groundbreaking exactly solvable lattice models that describe a diverse array of 2+1D fermionic SET phases, including those exhibiting the important ’t Hooft anomaly. This work sheds light on the fundamental properties and mathematical descriptions of these phases, and paves the way for a deeper understanding of the relationships among symmetry, topology, and quantum effects.

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Phys. Rev. X 16, 011019– Published 5 February, 2026
  • Received 4 January 2025
  • Accepted 15 December 2025
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