Many-body entanglement : topological orders, tensor networks and superconductivity

Abstract

In this thesis, we discuss the characterization and application of quantum many-body entanglements. We try to establish a non-local "order parameter" description of different patterns of many-body entanglement, which are also named topological orders. In 2+ 1D, we show that this could be achieved by calculating the non-Abelian geometric phase (S, T)-matrices from the fixed-point wave functions, obtained in the string-net approach by Levin and Wen and the local unitary transformation approach by Chen, Gu and Wen. In doing so, (S, T)-matrices act as our non-local "order parameter" and give a full characterization of 2+ 1D exact topological orders (topological orders that have a gappable edge). For a generic non-fixed-point wave function, however, obtaining the (S, T)-matrices is numerically formidable. To go around this problem, we introduce a new tensor-network method that works on any generic wave function, and obtain the "environment matrix" as a less powerful "order parameter" description. The "environment matrix" can characterize topological orders described by any gauge theory as well as ID symmetry protected topological (SPT) orders. As an application of both the concept of many-body entanglement and the new tensor-network method developed earlier, in the last chapter of the thesis, we propose a non-BCS mechanism for superconductivity, in which the driving force is not traditional pair-attraction, but statistical confusion of charge carrier induced by strong many-body entanglement. This may open new doors for identifying and constructing new superconducting states.