Matrix Product State and 1D Gapped Phases

Abstract

Based on the general notions introduced in the previous chapters, including local unitary transformations and short-/long-range entanglement, we study gapped phases in one spatial dimension in this chapter. Our goal is to understand what short-/long-range entangled phases exist in 1D and for this purpose, a useful tool is the matrix product state representation. The matrix product state representation provides an efficient description of the ground-state wave function of 1D gapped systems. We introduce this formalism in this chapter and discuss its various properties. By mapping matrix product states to their fixed-point form through renormalization group transformations, we show that there is actually no long-range entangled phase, hence no intrinsic topological order, in one-dimensional spin systems.