Discrete spin structures and commuting projector models for two-dimensional fermionic symmetry-protected topological phases

We construct exactly solved commuting projector Hamiltonian lattice models for all known (2+1)-dimensional (2+1D) fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group G sub([functionof])= (ProQuest: Formulae and/or non-USASCII text omitted), where G is finite an...

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Bibliographic Details
Published in:Physical review. B 2016-09, Vol.94 (11), Article 115115
Main Authors: Tarantino, Nicolas, Fidkowski, Lukasz
Format: Article
Language:English
Subjects:
Construction
Mathematical models
Phases
Projectors
Spin structure
Symmetry
Texts
Two dimensional models
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Summary:We construct exactly solved commuting projector Hamiltonian lattice models for all known (2+1)-dimensional (2+1D) fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group G sub([functionof])= (ProQuest: Formulae and/or non-USASCII text omitted), where G is finite and (ProQuest: Formulae and/or non-USASCII text omitted) is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1D. A natural ingredient in our construction is a discrete form of the spin structure of the 2D spatial surface M on which our model is defined, namely a "Kasteleyn" orientation of a certain graph associated with the lattice. As a special case, our construction yields commuting projector models for all eight members of the [dbl-struck Z] sub(8) classification of 2D fermionic SPTs with G= [dbl-struck Z] sub(2).
ISSN:2469-9950
2469-9969
DOI:10.1103/PhysRevB.94.115115