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The projective construction (the slave-particle approach) has played an veryimportant role in understanding strongly correlated systems, such as theemergence of fermions, anyons, and gauge theory in quantum spin liquids andquantum Hall states. Recently, fermionic Projected Entangled Pair States(fPEPS) have been introduced to effciently represent many-body fermionicstates. In this paper, we show that the strongly correlated bosonic/fermionicstates obtained both from the projective construction and the fPEPS approachcan be represented systematically as Grassmann tensor product states. Thisconstruction can also be applied to all other tensor network states approaches.The Grassmann tensor product states allow us to encode many-bodybosonic/fermionic states effciently with a polynomial number of parameters. Wealso generalize the tensor-entanglement renormalization group (TERG) method forcomplex tensor networks to Grassmann tensor networks. This allows us toapproximate the norm and average local operators of Grassmann tensor productstates in polynomial time, and hence leads to a variational approach fordescribing strongly correlated bosonic/fermionic systems in higher dimensions.

Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states