Abstract
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 $G_f=\mathbb{Z}_2^f \times_{\omega_2} G_b$ which is a central extension of bosonic symmetry group $G_b$ (may contain time reversal symmetry) by the fermion parity symmetry group $\mathbb{Z}_2^f = \{1,P_f\}$. 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.
Highlights
The concept of an equivalence class of finite-depth symmetric local unitary (LU) (SLU) transformations suggests that, in the presence of global symmetry, even short-range entangled (SRE) states still can belong to many different phases if they do not break any symmetry of the system. (It is well known that the traditional Landau symmetry-breaking states are characterized by different broken symmetries.)
We show that the equivalence class of finite-depth fermionic SLU (FSLU) transformation and decorated symmetry domain wall picture still applies for generic cases, subjected to much more complicated obstruction conditions
We show that, if Gb is time-reversal symmetry, an additional layer of p þ ip topological superconducting state decoration on the symmetry domain wall leads to new fermionic SPT (FSPT) states, which is the analogy of decorating the E8 state onto the symmetry domain wall for bosonic SPT (BSPT) phases with time-reversal symmetry [8]
Summary
Topological phases of quantum matter have become a fascinating subject in the past three decades. Very recently, based on the concept of an equivalence class of finite-depth fermionic SLU (FSLU) transformation and decorated symmetry domain wall picture, a breakthrough has been made on the full construction and classification of FSPT states with a total symmetry Gf 1⁄4. The fixed-point wave functions generated by FSLU transformations can be realized by exactly solvable lattice models, and the resulting classification results all agree with previous studies in 1D and 2D using other methods [43,52,54,55,56,57] Most surprisingly, such a completely different physical approach precisely matches the potential global anomaly for interacting fermion systems classified by spin cobordism theory [10,58,59,60,61,62,63,64]. We show that, if Gb is time-reversal symmetry, an additional layer of p þ ip topological superconducting state decoration on the symmetry domain wall leads to new FSPT states, which is the analogy of decorating the E8 state onto the symmetry domain wall for BSPT phases with time-reversal symmetry [8]
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