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arXiv:2112.06124 · Physical review. B./Physical review. B

Towards a complete classification of nonchiral topological phases in two-dimensional fermion systems

Jing-Ren Zhou, Qing-Rui Wang, Zheng‐Cheng Gu

Published Dec 13, 2022• 7 citations
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Abstract

In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang, and Wen, the concept of equivalence classes of fermionic local unitary (FLU) transformations was proposed to systematically understand nonchiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang, and Wen construction to include those fermionic topological phases with q-type anyon excitations. We argue that all nonchiral fermionic topological phases in $2+1\mathrm{D}$ are characterized by a set of tensors $({N}_{k}^{ij},{F}_{k}^{ij},{F}_{kln,\ensuremath{\chi}\ensuremath{\delta}}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}},{n}_{i},{d}_{i})$, which satisfy a set of nonlinear algebraic equations parameterized by phase factors ${\mathrm{\ensuremath{\Xi}}}_{kl}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}}$ and ${\mathrm{\ensuremath{\Xi}}}_{kln,\ensuremath{\chi}\ensuremath{\delta}}^{ij}$. Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors, which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the fermionic topological phase from Tambara-Yamagami category for ${\mathbb{Z}}_{2N}$, which can be regarded as the ${\mathbb{Z}}_{2N}$ parafermion generalization of Ising fermionic topological phase.

DOI
10.1103/physrevb.106.245120

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In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang, and Wen, the concept of equivalence classes of fermionic local unitary (FLU) transformations was proposed to systematically understand nonchiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang, and Wen construction to include those fermionic topological phases with q-type anyon excitations. We argue that all nonchiral fermionic topological phases in 2+1D2+1\mathrm{D}2+1D are characterized by a set of tensors (Nkij,Fkij,Fkln,\ensuremathχ\ensuremathδijm,\ensuremathα\ensuremathβ,ni,di)({N}_{k}^{ij},{F}_{k}^{ij},{F}_{kln,\ensuremath{\chi}\ensuremath{\delta}}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}},{n}_{i},{d}_{i})(Nkij​,Fkij​,Fkln,\ensuremathχ\ensuremathδijm,\ensuremathα\ensuremathβ​,ni​,di​), which satisfy a set of nonlinear algebraic equations parameterized by phase factors \ensuremathΞklijm,\ensuremathα\ensuremathβ{\mathrm{\ensuremath{\Xi}}}_{kl}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}}\ensuremathΞklijm,\ensuremathα\ensuremathβ​ and \ensuremathΞkln,\ensuremathχ\ensuremathδij{\mathrm{\ensuremath{\Xi}}}_{kln,\ensuremath{\chi}\ensuremath{\delta}}^{ij}\ensuremathΞkln,\ensuremathχ\ensuremathδij​. Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors, which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the fermionic topological phase from Tambara-Yamagami category for Z2N{\mathbb{Z}}_{2N}Z2N​, which can be regarded as the Z2N{\mathbb{Z}}_{2N}Z2N​ parafermion generalization of Ising fermionic topological phase.
The string-net approach by Levin and Wen, and the local unitary transformation approach by Chen, Gu, and Wen, provide ways to classify topological orders with gappable edge in two-dimensional (2D) bosonic systems. The two approaches reveal that the mathematical framework for (2+1)2+1)2+1)-dimensional (2+1)D(2+1)\mathrm{D}(2+1)D bosonic topological order with gappable edge is closely related to unitary fusion category theory. In this paper, we generalize these systematic descriptions of topological orders to 2D fermion systems. We find a classification of (2+1)D(2+1)\mathrm{D}(2+1)D fermionic topological orders with gappable edge in terms of the following set of data $({N}_{k}^{ij},{F}_{k}^{ij},{F}_{jkn,\ensuremath{\chi}\ensuremath{\delta}}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}},{d}_{i})$, which satisfy a set of nonlinear algebraic equations. The exactly soluble Hamiltonians can be constructed from the above data on any lattices to realize the corresponding topological orders. When Fkij=0{F}_{k}^{ij}=0Fkij​=0, our result recovers the previous classification of 2+1D2+1\mathrm{D}2+1D bosonic topological orders with gappable edge.
We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases that contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions and condensing pairs of physical and emergent fermions. There are two distinct types of objects in the resulting fermionic fusion categories, which we call “m-type” and “q-type” objects. The endomorphism algebras of q-type objects are complex Clifford algebras, and they have no analogs in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations arising from the condensed theories. We prove a series of results relating data in fermionic theories to data in their parent bosonic theories; for example, if C is a modular tensor category containing a fermion, then the tube category constructed from the condensed theory satisfies Tube(C/ψ)≅C×(C/ψ). We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the SO(3)6 theory, and the 12E6 theory and compute the quasiparticle excitation spectrum in each of the condensed theories.
We examine the interplay of symmetry and topological order in 2+12+12+1 dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the Gf{\cal G}^{\rm f}Gf fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry G=Gf/Z2fG= {\cal G}^{\rm f}/\mathbb{Z}_2^{\rm f}G=Gf/Z2f​ on the physical fermions is essentially the central extension of GGG by the Z2f\mathbb{Z}_2^{\rm f}Z2f​ fermion parity conservation that yields the fermionic symmetry Gf{\cal G}^{\rm f}Gf. We develop an algebraic theory of Gf{\cal G}^{\rm f}Gf symmetry defects for fermionic topological phases using GGG-crossed braided tensor category theory. This formalism allows us to fully characterize and classify 2+12+12+1 dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group Gf{\cal G}^{\rm f}Gf. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed Gf{\cal G}^{\rm f}Gf form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.
We describe a systematic way of producing fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain an emergent fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call “m-type” and “q-type” particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if C is a modular tensor category containing a fermion, then the tube category of the condensed theory satisfies Tube(C/ψ) ≅ C × C/ψ. We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum.
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.