# Calculating the Hamiltonain kernel of a Haldane model in zigzag graphene

Currently I'm learning how to calculate the edge effect of a material. I try to calculate edge state of a Haldane model in a zigzag graphene. Since Im writing in a second quantisation basis, each element in my hamiltonian kernel representing total hopping contribution on different site. Considering the simplest cases: for example : after doing the Fourier transform in x direction, the element in the first row and second column representing the hopping from site 1 to site 0(referring to the figure). There are totally 2 contributions from it's neighbour, so it would be $$$$t_{1}e^{ika/2} + t_{1}e^{-ika/2}$$$$ ( a is a translational vector of the unit cell) My question is about setting this element in Haldane model. Because the Haldane model is about setting the Next nearest neighbour with complex hopping strength, the new element will appear in the diagonal site j and j+2 and j-2 element with strength $$$$t_{2}e^{I\Phi}$$$$. I suppose that the nearest neighbouring term will not be affect, but I found that it was changed from $$$$t_{1}e^{ik} + t_{1}e^{-ik}$$ \quad$$ to $$\quad -t(1+e^{ik})$$ ( assuming a/2 = 1) . The explicit form of a 4 site model was shown in below I suppose that these NN elements in the matrix will not be modified

May be Ive completely mess up the concept. Sorry for my naive question.

• It's not clear to me exactly what you are asking here. Could you please edit your post to clarify?
– Chris
Commented Jul 13, 2020 at 20:05
• Is it possible include an actual *question mark *in your question somewhere? It's hard to formulate an answer when there's nothing obvious to answer.
– uhoh
Commented Sep 21, 2020 at 2:49