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.