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I got stuck on Homework problem, where I need to construct Hamiltonian of 2D Graphene layer and obtain Dispersion graph from it.

I already went trough a lot of materials but all I find is $2\times 2$ Hamiltonian of one unit cell of graphene and in my problem I need to construct a arbitary $N$, $2N\times 2N$Hamiltonian.

So essence of my question is how should Hamiltonian of an arbitrary $N$ graphene atoms should look like. $t = -2.7$eV only for nearest neighbors.

I'm not sure what are appropriate phase factors in d) part.

Here is the problem:

https://www.docdroid.net/pmzEiVX/project3.pdf.html

Here is my solution for N = 3: E-kx Dispersion

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Can't you just sum over all nearest neighbors?

Something like

$$ H = -t\sum_{<i,j>,\sigma}(a^\dagger_{i,\sigma}b^\dagger_{j,\sigma} + H.c.), $$

where $<i,h>$ means all nearest neighbours (c.f. http://web.physics.ucsb.edu/~phys123B/w2015/leggett-lecture.pdf).

Your matrix would basically just represent the sum, e.g. entry $h_{12}$ would be the hopping between lattice site 1 and site 2

For the phase-factors, as far as I understand it, they come from the difference in distance for the three different sub-Hamiltonians.

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