Tight binding hamiltonian matrix

[gizzmo]

Can somebody explain to me why, when we work with fermions, the tight binding Hamiltonian matrix has a form
0 0 -t -t
0 0 +t +t
-t +t 0 0
-t +t 0 0
the basis are |\uparrow,\downarrow>, |\downarrow,\uparrow>, |\uparrow\downarrow,0>, |0,\uparrow\downarrow>,
Why there is +t and -t? (I think that this has something to do with the fact the the fermionic wave function is antisymmetric. But can somebody give me an example how to calculate this elements from the actual Hamiltonian. I always get -t.)

 

[kanato]

Tight binding is a very general method, and there are many systems where it doesn't have that form. You really should describe the system you are studying before asking a question like that. Also, when you make a statement of the form "I always get this answer and it's wrong" you should describe what it is that you're doing or you're just asking people to guess as what you have done wrong.

You need to write down your basis states in second quantized notation:
$$|\uparrow,\downarrow\rangle = c_{1\uparrow}^\dagger c_{2\downarrow}^\dagger |0\rangle$$
and you need to pick a normal ordering for your operators and make sure you are consistent when writing out your states. Then write out each inner product carefully and write out the contractions.