I know long-range entanglement is the essence of nontrivial topological ordered
ID: 1380111 • Letter: I
Question
I know long-range entanglement is the essence of nontrivial topological ordered states. (Trivial refers to short range entangled and nontrivial refers to long range.) So, entanglement measure at large scale compared with correlation length is a tool to distinguish topological trivial and nontrivial states. One of the ways to measure entanglement is through topological entanglement entropy, Stop=logD, where D is the quantum dimension. The state is topological trivial if and only if D=1. But it seems that this method can not be applied to distinguish nontrivial states. For example, for FQH state with filling factor v=1/q, D=q?, indicating that it is nontrivial. But we know that FQH ground state has q fold degeneracy, all of them are nontrivial. My question is how to distinguish between long range entangled states in a quantitative way.
Explanation / Answer
"How to distinguish between long range entangled states in a quantitative way?"
I know four quantitative ways to describe long-range entangled states.
1) Modular transformation (ground state topological degeneracy and non-Abelian geometric phases) for 2+1D topological orders. (This may be most general.)
2) K-matrix for all Abelian topological orders.
3) String-net (spherical fusion category) for 2+1D non-chiral topological orders
4) Patterns of zeros an Zn chiral algebra for chiral topological orders.