Tensor Product States and 2D Gapped Phases

Abstract

Tensor product state is a natural generalization of matrix product state to two and higher dimensions. It is similar to matrix product states in many ways, like satisfying the entanglement area law and having a projected entangled pair interpretation. However, it is also different from matrix product states in that it represents not only states with short-range entanglement but topologically ordered states with long-range entanglement as well. In this chapter, we first introduce the basic properties of tensor product states and then proceed to discuss how it represents symmetry-breaking phases and topological phases with explicit examples. In particular, we are going to focus on the general structural properties of the local tensors which are responsible for the corresponding symmetry-breaking or topological order. Other forms of tensor network state, including the tree tensor network state and MERA (multiscale entanglement renormalization ansatz), are also discussed.