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