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arXiv:2604.09503

Classification of 2D Fermionic Systems with a $\mathbb Z_2$ Flavor Symmetry

Chi‐Ming Chang, Jin Chen, Fengjun Xu

Published Apr 10, 2026• 0 citations
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Abstract

We classify superfusion categories describing two-dimensional fermionic systems equipped with the universal fermion-parity symmetry, implemented by a topological defect line (TDL) $Z$, and an additional $\mathbb{Z}_2$ flavor symmetry generated by a $W$ TDL. Depending on whether $W$ is m-type or q-type, its fusion rules lead to three distinct classes, and solving the super-pentagon equations yields 16 consistent superfusion categories. These are labeled by invariants $(ν_W,ν_Z,ν_{WZ})$, which determine the $\mathbb{Z}_8$ anomaly classes of the symmetries generated by $W$, $Z$, and $WZ$. We also provide explicit realizations using multiple Majorana fermions and comment on implications for fermionic CFTs and gapped phases.

DOI
10.48550/arxiv.2604.09503

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Classification of 2D Fermionic Systems with a Z2\mathbb Z_2Z2​ Flavor Symmetry
We classify superfusion categories describing two-dimensional fermionic systems equipped with the universal fermion-parity symmetry, implemented by a topological defect line (TDL) ZZZ, and an additional Z2\mathbb{Z}_2Z2​ flavor symmetry generated by a WWW TDL. Depending on whether WWW is m-type or q-type, its fusion rules lead to three distinct classes, and solving the super-pentagon equations yields 16 consistent superfusion categories. These are labeled by invariants (νW,νZ,νWZ)(ν_W,ν_Z,ν_{WZ})(νW​,νZ​,νWZ​), which determine the Z8\mathbb{Z}_8Z8​ anomaly classes of the symmetries generated by WWW, ZZZ, and WZWZWZ. We also provide explicit realizations using multiple Majorana fermions and comment on implications for fermionic CFTs and gapped phases.
We consider topological defect lines (TDLs) in two-dimensional fermionic conformal field theories (CFTs). Besides inheriting all the properties of TDLs in bosonic CFTs, TDLs in fermionic CFTs could host fermionic defect operators at their endpoints and junctions. Furthermore, there is a new type of TDLs, called q-type TDLs, that have no analog in bosonic CFTs. Their distinguishing feature is an extra one-dimensional Majorana fermion living on the worldline of the TDLs. The properties of TDLs in fermionic CFTs are captured in the mathematical language of the super fusion category. We propose a classification of the rank-2 super fusion categories generalizing the \mathbb{Z}_8 ℤ 8 classification for the anomalies of \mathbb{Z}_2 ℤ 2 symmetries. We explicitly solve the F-moves for all the nontrivial categories, and derive the corresponding spin selection rules that constrain the spectrum of the defect operators. We find the full set of TDLs in the standard fermionic minimal models and a partial set of TDLs in the two exceptional models, which give CFT realizations to the rank-2 super fusion categories. Finally, we discuss a constraint on the renormalization group flow that preserves a q-type TDL.
A bstract We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category Rep( H ) of a semisimple weak Hopf algebra H , the fermionized TQFT has a superfusion category symmetry SRep( H \mathcal{H} H H u ), which is the supercategory of super representations of a weak Hopf superalgebra H \mathcal{H} H H u . The weak Hopf superalgebra H \mathcal{H} H H u depends not only on H but also on a choice of a non-anomalous ℤ 2 subgroup of Rep( H ) that is used for the fermionization. We derive a general formula for H \mathcal{H} H H u by explicitly constructing fermionic TQFTs with SRep( H \mathcal{H} H H u ) symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on F -symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs.
We introduce the concept of fermionic matrix product operators, and show that they provide a natural representation of fermionic fusion tensor categories. This allows for the classification of two dimensional fermionic topological phases in terms of matrix product operator algebras. Using this approach we give a classification of fermionic symmetry protected topological phases with respect to a group GGG in terms of three cohomology groups: H1(G,Z2)H^1(G,\mathbb{Z}_2)H1(G,Z2​), describing which matrix product operators are of Majorana type, H2(G,Z2)H^2(G,\mathbb{Z}_2)H2(G,Z2​), describing the fermionic nature of the fusion tensors that arise when two matrix product operators are multiplied, and the supercohomolgy group Hˉ3(G,U(1))\bar{H}^3(G,U(1))Hˉ3(G,U(1)) which corresponds to the associator that changes the order of fusion. We also generalize the tensor network description of the string-net ground states to the fermionic setting, yielding simple representations of a class that includes the fermionic toric code.
This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories E⊂C⊂M\mathcal E\subset\mathcal C\subset\mathcal ME⊂C⊂M plus the chiral central charge ccc. Here E\mathcal EE is a symmetric fusion category, E=Rep(G)\mathcal E=\mathrm{Rep}(G)E=Rep(G) for boson systems or E=sRep(Gf)\mathcal E=\mathrm{sRep}(G^f)E=sRep(Gf) for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry; C\mathcal CC is the category of all the quasiparticle excitations in the bulk, containing E\mathcal EE as its Müger center; M\mathcal MM is a minimal modular extension of C\mathcal CC, that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.
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 + 1 D 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 G f is a direct product of physical symmetry G and fermion-parity symmetry Z 2 f ), which is the input data of a symmetry-enriched fermionic string-net model. Next, we construct exactly solvable models for nonchiral 2 + 1 D fSET phases with ’t Hooft anomaly, especially the H 3 ( G , 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 , Z 2 ) 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 Z 4 gauge theory embedded into a fermion system and the total symmetry G f = Z 2 f × Z 2 × Z 4 . We further conjecture that the H 2 ( G , Z 2 ) 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.
We discuss an isomorphism between the possible anomalies of (d+1d+1d+1)-dimensional quantum field theories with Z2{\mathbb{Z}}_{2}Z2​ unitary global symmetry, and those of ddd-dimensional quantum field theories with time-reversal symmetry T\mathsf{T}T. This correspondence is an instance of symmetry defect decoration. The worldvolume of a Z2{\mathbb{Z}}_{2}Z2​ symmetry defect is naturally invariant under T\mathsf{T}T, and bulk Z2{\mathbb{Z}}_{2}Z2​ anomalies descend to T\mathsf{T}T anomalies on these defects. We illustrate this correspondence in detail for (1+1)d(1+1)d(1+1)d bosonic systems where the bulk Z2{\mathbb{Z}}_{2}Z2​ anomaly leads to a Kramers degeneracy in the symmetry defect Hilbert space and exhibits examples. We also discuss (1+1)d(1+1)d(1+1)d fermion systems protected by Z2{\mathbb{Z}}_{2}Z2​ global symmetry where interactions lead to a Z8{\mathbb{Z}}_{8}Z8​ classification of anomalies. Under the correspondence, this is directly related to the Z8{\mathbb{Z}}_{8}Z8​ classification of (0+1)d(0+1)d(0+1)d fermions protected by T\mathsf{T}T. Finally, we consider (3+1)d(3+1)d(3+1)d bosonic systems with Z2{\mathbb{Z}}_{2}Z2​ symmetry where the possible anomalies are classified by ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We construct topological field theories realizing these anomalies and show that their associated symmetry defects support anyons that can be either fermions or Kramers doublets.