We study the parton construction of chiral spin liquids (CSLs) using projected Gaussian fermionic entangled pair states (GfPEPSs). First, we show that GfPEPSs can represent generic spinless Chern insulators faithfully with finite bond dimensions. Then by applying the Gutzwiller projection to a bilayer GfPEPSs, spin-$\frac{1}{2}$ Abelian and non-Abelian CSLs are obtained for Chern number $C=1$ and 2, respectively. As a consequence of the topological obstruction for GfPEPSs, very weak Gossamer tails are observed in the correlation functions of the fermionic projected entangled pair state (PEPS) Ansätze, suggesting that the no-go theorem for chiral PEPS is universal but does not bring any practical limitation. Remarkably, without finetuning, all topological sectors can be constructed showing the expected number of chiral branches in the respective entanglement spectra, providing a sharp improvement with respect to the known bosonic PEPS approach.

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• Edge states
• Quantum spin liquid
• Topological phases of matter
• Projected entangled pair states
• Tensor network methods