Expressibility of Tensor Network States for 2D Free Fermion Systems

Tensor network states describe a class of variational wavefunction ansatzes which have been shown to be useful for modelling strongly interacting systems. The quantum many body states which they are able to represent describe the ground states of local quantum Hamiltonians. While the ability of tensor networks to represent 1D states is well understood in terms of just states with area law entanglement, it is not yet clear what classes of 2D states can be represented by tensor networks with an efficient number of parameters. We discuss the representation of 2D free fermion systems which are the first step towards studying systems of interacting fermions. In the free fermion case, we demonstrate a measure of the expressibility of tensor network states which is derived from properties of trigonometric polynomials.