Tight binding simulation issues
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Hello,
This may be too far on the physics side of things, but I figured I may try and ask anyways. My main issue comes from the eigenvalues I am recieving from the code when I diagonalize a matrix.
I am trying to diagonalize a 2D NxN square lattice Hamiltonian which contains a uniform d-wave super conducting order parameter and a nearest neighbor hoping term. We can solve this model analtically with a fourier transform due to translational invariance, so it gives me an opertunity to check my answer. This next part is not too important, but here is the explicit model:
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/251811/image.png)
Here the delta is the d-wave superconductivity term which connects nearest neighbors on the lattice: it can be treated very similar to the first term. Basically we just need to throw in a negative sign for terms which are above a given lattice point (can go into more detail if needed). Doing the math we can find the following dispersion relation:
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/251812/image.png)
Now for the simulation. We can rewrite the above summations in the following form:
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/251813/image.png)
Where
; H is a matrix choosen to make the two definitions consitent. Finding the eigen energies here amounts to diagonalizing H.
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/251814/image.png)
My question is I get all of the energies I see with my analytic solution, but also quite a few extra I do not. Also, it seems like my degeneracy is way too high!
I am unsure why this could be the case. I am certain my Hamiltonian looks right. I can provide anyone who wants it with my code. Hopefully this is not too vague, and I can expand if needed.
Thanks!
4 Comments
Vladimir Sovkov
on 3 Dec 2019
From elementary Physics courses, I have always thought that the superconductivity was produced by the Cooper pairs, i.e., couples of electrons exchanging phonons, but I have never been an expert in the subject. The idea that the same effect can be achieved by an electron-hole pair (can we call it "exciton"?) is new for me. It is interesting.
Is it possible that your extra states all possess a wrong symmetry, so that imposing the symmetry conditions would cancell them totally?
I think that there is nothing else I can advise.
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