Abstract
Tensor network states are presentations of wavefunctions of a wide range of many-body systems. In this thesis, we are interested in the interplay between some basic ingredients in many-body systems from the perspective of tensor network: fermions, symmetries and entanglement. First, we present a systematic construction and computations of fermionic tensor network states. The construction can be done for fermionic Gaussian states in any spatial dimensions and the obtained Gaussian tensor network can be transformed into manybody tensors for engaging the interactions. We present the construction of two dimensional fermionic Gaussian tensor network state and one-dimensional Gutzwiller projected Fermi sea states as proof of principle. Second, we prove a theorem on the lower bound for SO(3)- symmetric uniform matrix product states for integer spin chains with translation symmetry unbroken, and use the entanglement lower bound to show that the correlation length of such symmetric states cannot be exactly zero. Finally, we present a scheme of Jordan- Wigner transformations in higher dimensions, which serves as an alternative way of treating fermion systems in the tensor network framework.