Modular transformation and bosonic/fermionic topological orders in Abelian fractional quantum Hall states

Understand

The non-Abelian geometric phases of the robust degenerate ground states were proposed as physically measurable defining properties of topological order in 1990.

• In this paper we discuss in detail such a quantitative characterization of topological order, using generic Abelian fractional quantum Hall states as examples.
• We find a way to determine even the pure U(1) phase of the non-Abelian geometric phases.
• We show that the non-Abelian geometric phases not only contain information about the quasi-particle statistics, they also contain information about the chiral central charge Delta c mod 24 of the edge states, as well as the Hall viscosity.