Computing the Classification of Fermionic Symmetry-Protected Topological States

The bosonic symmetry protected topological (SPT) states are classified by the cohomology groups of the symmetry group. However, for interacting fermionic systems, the classification is much more complicated. It was untill very recent that a breakthrough was maded on the classification and construction of fermionic SPT states based on the concept of equivalence class under finite-depth symmetric fermionic local transformations. However, though layers of data for classification and construction can be obtained theoretically, the formalism based on the bar resolution of symmetry groups (also known as the inhomogeneous cochains) makes it impossible for realistic computation due to high computational costs. To solve these problems, we first design reduced resolutions for general groups by group extension, then we apply the chain maps between the bar resolution and reduced resolution to solve the obstruction function to get the unobstructed layers of data. Meanwhile, to simplify the computation, we apply the technique of spectral sequence. Finally, we give a classification of fermionic SPTs protected by 2D wallpaper-group symmetries.