# How do you fourier transform a tight binding hamiltonian numerically?

The task is to do a fourier transformation of a tight binding hamiltonian of a 1D-chain with unit cell size 2, but even after many tries and googling I still don't have a idea how to do it correctly.

EDIT:I know that for a chain with unit size 1, no potential and only nearest-neighbor hopping the matrix looks like

0  -t ... -t
-t 0  -t
...
-t


the elements in the corners are there if you have periodic boundary conditions. For the unit cell I tried

• alternating t's
• Hl x Hu where Hu is the hopping hamiltonian for 2 sites, Hl for the chain and x the kronecker product.

I also can do analytical fourier transformation (transforming c and c dagger) and tried many different matrix formulations for them but none worked out when fourier transformed. And for the fourier transformation (matlab transforms every column when given a matrix) I tried to write matrices for c dagger for different sites as single row and put then under each other, then do the transformation and rearrange the rows to matrices again, but did not work.

For c2 for example I tried

0 1 0 ...
0 1 0 ...
...


I know that they don't obey the anticommutator relations, but using

H = - sum tij ci dagger cj


with the sum over the nearest neighbors you get the right hamiltonian.

• Hi! This will work much better if you edit your question to show us one of your "tries"; this is not a homework help site but a place to answer the deep conceptual questions that are getting in your way. So we'd need to know, for example, whether you can write down the Hamiltonian, and whether you correctly wrote down the DFT, and what your expressions look like -- then we can advise you maybe on how to write your computer program. – CR Drost Nov 8 '15 at 0:52
• @CRDrost DFT? This kind of Fourier transform can be done analytically. The problem of OP lies somewhere with expressing the Hamiltonian with $c_k \propto \sum_i e^{ikx} c_i$. Also, Daniel, please use TeX to write your question. – Praan Nov 12 '15 at 13:33