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

Question

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?

This post imported from StackExchange Physics at 2015-12-08 22:41 (UTC), posted by SE-user PhysicsMath

Comments

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.

This post imported from StackExchange Physics at 2015-12-08 22:41 (UTC), posted by SE-user DanielSank

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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.

This post imported from StackExchange Physics at 2015-12-08 22:41 (UTC), posted by SE-user PhysicsMath

Answer 1

For 2D chiral p-wave, it has been shown that the Berry phase contribution vanishes and therefore the exchange statistics is entirely given by the monodromy. This was done in http://arxiv.org/abs/cond-mat/0505515. For the closely related case of Moore-Read wavefunctions, the problem is much harder and was finally resolved in http://arxiv.org/abs/1008.5194 (plus some numerical verification of two-component plasma screening).

In general, the separation of exchange/braiding statistics into the monodromy and Berry phase is by itself artificial, so it is not clear what kind of "general result" one should expect. In the context of FQH where wavefunctions are obtained from conformal blocks of rational conformal field theory, we expect when the wavefunction represents a gapped phase (i.e. certain plasma is screening), the monodromy coming from the conformal blocks is the whole story and Berry phase contribution should vanish, but I don't think there is a proof.

This post imported from StackExchange Physics at 2015-12-08 22:41 (UTC), posted by SE-user Meng Cheng