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arXiv:2410.19126 · Physical Review X

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

Jing-Ren Zhou, Zheng‐Cheng Gu

Published Dec 15, 2025• 0 citations
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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 + 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.

DOI
10.1103/r6d4-b538

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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 construct exactly solvable models for a wide class of symmetry-enriched topological (SET) phases. Our construction applies to two-dimensional (2D) bosonic SET phases with finite unitary on-site symmetry group GGG and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: If we couple them to a dynamical lattice gauge field with gauge group GGG, the resulting gauge theories are equivalent to string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2{\mathbb{Z}}_{2}Z2​ symmetry which exchanges the eee and mmm type anyons. We further illustrate our construction with a number of additional examples.
A bstract We present a systematic framework for constructing exactly-solvable lattice models of symmetry-enriched topological (SET) phases based on an enlarged version of the string-net model. We also gauge the global symmetries of our SET models to obtain string-net models of pure topological phases. Without invoking externally imposed onsite symmetry actions, our approach promotes the string-net model of a pure topological order, specified by an input unitary fusion category F \mathcal{F} F F , to an SET model, specified by a multifusion category together with a set of isomorphisms. Two complementary construction strategies are developed in the main text: (i) promotion via outer automorphisms of F \mathcal{F} F F and (ii) promotion via the Frobenius algebras of F \mathcal{F} F F . The global symmetries derived via these two strategies are intrinsic to topological phases and are thus termed blood symmetries, as opposed to adopted symmetries, which can be arbitrarily imposed on topological phases. We propose the concept of symmetry-gauging family of topological phases, which are related by gauging their blood symmetries. With our approach, we construct the first explicit lattice realization of a nonabelian-symmetry-enriched topological phase — the S 3 symmetry-enriched ℤ 2 × ℤ 2 quantum-double phase. The approach further reveals the role of local excitations in SET phases and establishes their symmetry constraints.
Symmetry-enriched topological phases (SET) exhibit both long-range entanglement and intricate interplay with global symmetries, such as anyon permutation symmetry or fractionalization of quantum numbers, going beyond the Landau paradigm of symmetry breaking. Here, the authors study two-dimensional SET phases based on the novel concept of symmetric local unitary transformations. Using the fixed-point structure of wave functions, they construct systematically exactly solvable models that can describe generic nonchiral SET phases. The technique applies to on-site as well as spatial symmetries and, furthermore, to anomalous symmetries that can only be realized on the surface of three-dimensional symmetry-protected phases.
The classification and lattice model construction of symmetry protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group Gf=Z2f×ω2GbG_f=\mathbb{Z}_2^f \times_{\omega_2} G_bGf​=Z2f​×ω2​​Gb​ which is a central extension of bosonic symmetry group GbG_bGb​ (may contain time reversal symmetry) by the fermion parity symmetry group Z2f={1,Pf}\mathbb{Z}_2^f = \{1,P_f\}Z2f​={1,Pf​}. Our construction is based on the concept of equivalence class of finite depth fermionic symmetric local unitary (FSLU) transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We will also discuss the systematical construction and classification of boundary anomalous SPT (ASPT) states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction-free and trivialization-free constructions naturally lead to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point/space group symmetry as well as continuum Lie group symmetry.
In this thesis we investigate topological phases of matter that have a global, unbroken symmetry group -- also known as symmetry enriched topological (SET) phases. We address three questions about these phases: (1) how can we build exactly solvable models that realize them? (2) how can we determine if their edge theories can be gapped without breaking the symmetry? and (3) how do we understand the phenomenon of decoupled charge and neutral modes which occurs in certain fractional quantum Hall states? ,More specifically, we address the first question by constructing exactly solvable models for a wide class of symmetry enriched topological (SET) phases, which we call symmetry-enriched string nets. The construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group G, and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2 symmetry which exchanges the 'e' and 'm' type anyons. We further illustrate our construction with a number of additional examples. ,For the second question, we focus on the edge theories of 2D SET phases with Z2 symmetry. The central problem we seek to solve is to determine which edge theories can be gapped without breaking the symmetry. Previous attempts to answer this question in special cases relied on constructing perturbations of a particular type to gap the edge. This method proves the edge can be gapped when the appropriate perturbations can be found, but is inconclusive if they cannot be found. We build on this previous work by deriving a necessary and sufficient algebraic condition for when the edge can be gapped. Our results apply to Z2 symmetry protected topological phases as well as Abelian Z2 SET phases,Finally, in the fourth chapter, we describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode --- i.e. an edge mode that does not carry electric charge. These models consist of two counter-propagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the nu = 2/3 state is the most prominent example.
Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach [L. Bhardwaj et al., SciPost Phys. 19, 056 (2025); L. Bhardwaj et al., arXiv: 2502.20440]. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider “all-boson type” fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups G G (possibly with an ’t Hooft anomaly) via generalized gauging—that is, by stacking with an H H -symmetric TFT and gauging a subgroup H H . The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.