Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory

By Z. Gu, X. Wen. 2012

265 citations recorded by the source corpus.

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of super cocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (ie, non-on-site) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric} boundary must be gapless for 1+1D boundary, and must be gapless or topologically ordered beyond 1+1D. As an application of this general result, we construct a new SPT phase in 3D, for interacting fermionic superconductors with coplanar spin order (which have $T^2=1$ time-reversal $Z_2^T$ and fermion-number parity $Z_2^f$ symmetries described by a full symmetry group $Z_2^T\times Z_2^f$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion-pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in 2D with a full symmetry group $Z_2\times Z_2^f$. Those 2D fermionic SPT phases all have central-charge $c=1$ gapless edge excitations, if the symmetry is not broken.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/6371a3e1fefa54651dec35c56568609b67914ca5

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Symmetry protected topological phases and generalized cohomology (2017, background): “In a seminal paper [GW14], Gu and Wen gave a detailed construction of a class of fermionic SPT phases with unitary symmetry group.”
Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases (2021, methodology): “It would be a supercohomology class but it does not satisfy the Gu-Wen equation [108] (also see [115]).”
Spin TQFTs and fermionic phases of matter (2015, unknown): “(7) Although z[X, η, βd−1] does not appear to admit a d-dimensional bosonic description, we also find that it is a quadratic refinement of a bosonic pairing: z[X, η, βd−1 + β ′ d−1] = z[X, η, βd−1]z[X, η, β ′ d−1](−1) ∫ X βd−1∪d−2β d−1 .”
Gapless Symmetry-Protected Topological Order (2017, unknown): “[38] Zheng-Cheng Gu and Xiao-Gang Wen, “Symmetry-”
State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter (2016, methodology): “The calculation involves some crucial signs involving the GuWen Grassmann integral [11] and a choice of spin structure on M .”
Gauging spatial symmetries and the classification of topological crystalline phases (2016, methodology): “Applying the principle to fermionic systems, one obtains a partial classification of fermionic SPTs protected by space-group symmetries from “group supercohomology” [22] and a complete classification of fermionic crystalline SPTs from cobordism theory [27], with some caveats.”
Generalized global symmetries (2014, background): “This approach can be generalized to SPT phases with time-reversing symmetries and fermionic degrees of freedom [28, 29] (see [30] and [31] for alternative proposals).”
Fermionic symmetry protected topological phases and cobordisms (2014, background): “This improves on the previous proposal that group cohomology classifies interacting bosonic SPT phases [6], while “group supercohomology” [7] classifies interacting fermionic SPT phases.”
Pre-thermal phases of matter protected by time-translation symmetry (2016, background): “[15] Zheng-Cheng Gu and Xiao-Gang Wen, “Symmetryprotected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory,” Phys.”