# Berry phase in the toric code model and 2D chiral $p$-wave superconductors

When we derive the exchange statistics by moving quasiparticles around a circle in the toric code model we do not mention any Berry phase contribution. Is the Berry phase contribution trivial or it is nontrivial but does not alter the exchange statistics? This is also the case when we derive the nonabelian statistics for the vortices in 2D chiral $p$-wave superconductors. We seem to consider only the wave function monodromy but not the Berry phase contribution.

My question is:

(simpler one) what is the Berry phase contribution in both cases and why does it not alter the exchange statistics?

(harder one) is there any way to reach the conclusion (i.e. trivial vs.\ nontrivial, and alter vs.\ not alter the statistics) without calculation?

(challenge) could we find a general guideline as to whether and when we should account for the Berry phase contribution when deriving exchange statistics for any topological phase?

• Welcome to Physics Stack Exchange. This is a very interesting post! Please note, however, that it's important to ask one specific question per post. By asking several questions you reduce the probability of getting an answer because for someone to write an answer they have to read, understand, think about, solve, and write a solution for multiple things. – DanielSank Dec 7 '15 at 21:13
• Hi Daniel, thanks for the reminder. But these two questions are very related because the toric code model is the representative model for abelian anyon statistics and the 2D chiral $p$-wave superconductors is the representative model for nonabelian anyon statistics. – PhysicsMath Dec 7 '15 at 21:16

• On the other hand the "general result" that I was hoping for is indeed something that can help to tell whether Berry phase vanishes or merely contributes $2n\pi$. But from the papers that you mentioned I found out that it is highly nontrivial to make such a conclusion even for specific cases not to say in general. – PhysicsMath Dec 11 '15 at 0:54