# How does on-site energy M influence Berry curvature and topological transitions in Haldane's model?

SOLUTION: The following papers almost fully-answer my question:

https://arxiv.org/pdf/0904.2117.pdf

https://arxiv.org/pdf/1111.5020.pdf

Essentially, the Dirac points move and merge as M changes.

I am using this review on topological insulators to study Haldane's model. I used their Hamiltonian and MATLAB to numerically calculate Berry curvature. The Hamiltonian goes as:

The review says that M acts as a symmetry-breaking term, so I wanted to visualize how exactly it behaves by animating changes in Berry curvature as M is varied.

For a single Brillouin zone, I see:

and for several unit cells:

It appears that, within the region prescribed by Haldane - $-3\sqrt{3}\leq M \leq 3\sqrt{3}$ - as M increases, a big peak in the figure above ejects two smaller peaks which travel and combine with another peak just before changing sign. I do not understand this mechanism well.

According to this course, gap closings at Dirac points are related to M via the ratio $\frac{t_{2}}{M}$, but the hopping strength $t_{2}$ is kept fixed here.

I am having trouble understanding the mechanism behind these smaller peaks that split off from the big peak only to join the small peaks from another. So, could someone give me some insight on this?

I left out details of how I calculate Berry curvature because I assumed that this phenomenon is well-understood by experts. However, I wasn't able to understand this through any of the resources I consulted. I am guessing that this has something to do with charge pumping (which the Chern number helps quantify overall), but how exactly do these mini-peaks form and how do they propagate? Are there equations that describe this behavior of Berry curvature? I am guessing that NN and NNN hoppings are both observed here (I first thought that it’s a bilayer effect, with the smaller peaks coming from a different layer, but that seems too far-fetched).

Additionally, I would appreciate insight on what the splitting means physically, because my understanding of Haldane's model is still very abstract. For instance, does this somehow demonstrate the hopping-path of an electron? If so, what determines the value of the Berry curvature? I understand holonomy and Berry phase sufficiently, but how do I explain what I see by going from the physical interpretation to the theoretical explanation (and not vice versa).

Despite all this, the 'splitting' in my animations could simply be artifacts of my code.

• Cool. It does look like net Berry curvature is being transported over the Brillouin zone. Is there a way to close this interval of $M$ into a circle so we can discuss Berry flux pumping? – Ryan Thorngren Sep 8 '18 at 22:46
• @RyanThorngren hmmm, did you mean whether I could make an animation that started with $M=-3\sqrt{3}$ and terminated at $M=3\sqrt{3}$? It doesn't look too continuous because the starting peak positions (and signs) and ending ones are very different, unless I reverse the direction of the iteration after one cycle. Also, if it is Berry flux pumping, do you know what consequences such pumping has in Haldane's model? Preliminary searches only gave me the obvious explanation involving the Chern number. – TribalChief Sep 8 '18 at 23:27