Modular transformations and topological orders in two dimensions

摘要: The string-net approach by Levin and Wen the local unitary transformation Chen, Gu provided ways to systematically label non-chiral topological orders in 2D. In those approaches, different topologically ordered many-body wave functions were characterized fixed-point tensors. Though extremely powerful, resulting tensors mathematical abstractions thus lacked a physical interpretation. As result it was hard judge if two described same quantum phase or not. We want improve that giving more description of orders. find non-Abelian Berry's phases, $T$- $S$-matrices, protected degenerate ground states on torus give rise Using Verlinde conjecture, we can even choose canonical basis for $S$-matrices. It is conjectured $T$ $S$-matrices form complete one-to-one characterization replace tensor us result, all properties be obtained from such as number quasiparticle types (from dimension $S$), statistics diagonal elements $T$), dimensions quasiparticles first row \etc.