  We investigate the tensor network representations of fermionic crystalline
symmetry-protected topological (SPT) phases on two-dimensional lattices. As a
mapping from virtual indices to physical indices, projected entangled-pair
state (PEPS) serves as a concrete way to construct the wavefunctions of 2D
crystalline fermionic SPT (fSPT) phases protected by 17 wallpaper group
symmetries, for both spinless and spin-1/2 fermions. Based on PEPS, the full
classification of 2D crystalline fSPT phases with wallpaper groups can be
obtained. Tensor network states provide a natural framework for studying 2D
crystalline fSPT phases.
