Abstract
Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this work, we revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. We construct very general fixed point SPT wavefunctions for interacting fermion systems. We naturally reproduce the partial classifications given by special group super-cohomology theory, and we show that with an additional $\tilde{B}H^2(G_b, \mathbb Z_2)$ (the so-called obstruction free subgroup of $H^2(G_b, \mathbb Z_2)$) structure, a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group $G_f=G_b\times \mathbb Z_2^f$ can be obtained for unitary symmetry group $G_b$. We also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions.
Highlights
In recent years, a new type of topological order— symmetry-protected topological (SPT) order [1,2,3]—has been proposed and intensively studied in interacting boson and fermion systems
We construct fixed-point wave functions for fermionic SPT (FSPT) phases in two and three dimensions based on the novel concept of fermionic symmetric local unitary (FSLU) transformations
We believe that our construction will give rise to a complete classification for FSPT states with total symmetry Gf 1⁄4 Gb × Zf2 when Gb is a unitary symmetry group
Summary
A new type of topological order— symmetry-protected topological (SPT) order [1,2,3]—has been proposed and intensively studied in interacting boson and fermion systems. Achieved using generalized group cohomology theory [2,3,8] or cobordism theory [9] This systematic classification essentially classifies the quantum anomalies associated with the corresponding global symmetries in interacting bosonic systems. [63], it was shown that fermionic local unitary (FLU) transformations can be used to define and classify intrinsic topological phases for interacting fermion systems. In the presence of global symmetry, we can further introduce the notion of fermionic symmetric local unitary (FSLU) transformations to define and classify fermionic SET (FSET) phases in interacting fermion systems. To represent a fermionic symmetric unitary operator acting on a region labeled by i.) Again, FSPT phases are a special class of FSET phases that have trivial bulk excitation and can be adiabatically connected to a product state in the absence of global symmetry. We only need to enforce the FSLU transformations to be one dimensional (when acting on the support space ρA for any region A) to classify all FSPT states
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