# 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

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.

• Great answer, Dr Cheng! It looks like people do pay attention to Berry contribution. But could you please elaborate a bit more on your comment that the separation into the monodromy and Berry phase is artificial? Do you mean that we do not really care too much about how much each contribute because eventually the physical property that they would impact on is the exchange statistics and therefore only their sum is of physical interest? – PhysicsMath Dec 11 '15 at 0:50
• 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
• We do not care too much about each contributions because only the sum of them, the holonomy of the wavefunction, is physically meaningful. For each of these examples, you can always redefine the wavefunction in such a way that the monodromy is trivial, and everything comes from the Berry phase (although not very natural). And yes, proving the Berry phase vanishes for the Moore-Read wavefunction is a true tour de force, and after all the nontrivial analytics is done one still requires input from numerical simulations to make the case (this is true even in the much simpler Laughlin case). – Meng Cheng Dec 11 '15 at 1:10
• I see. Thanks for the clarification Dr Cheng! Look forward to seeing some example of trivializing the monodromy by redefining the wave function (of course there must be a good reason for doing so). – PhysicsMath Dec 13 '15 at 23:59