- #1
SalomeH
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I have various 1/2-spin systems and I should find energetic spectra (eigenvalues of the Hamiltonian matrix).
Hamiltonian I use is in the form:
H=JSiSj+t(c+ic-j+c+jc-i)
The first part is interaction between two spins, so sum over every spin-pair, the second part is interaction between spin and a "hole", that is for electron moving from one place to another, also sum over spin-hole pairs.
My systems are closed chains of 2, 3 or 4 electrons, or there is an additional hole to these.
I have already solved systems with only spins, because that's where the second part disappears, so I can use the fact, that H has the same eigenfunctions as total spin squared, single spins squared or I can use symmetry, so this part is OK.
But I have troubles with these holes. So let's take triangle with two spins and one hole. The basis has 12 vectors, so counting determinant wouldn't be nice - and I need a method I could use for bigger systems too.
My first idea was, that I could basicly use the result for bare 2 spins and just add 2t constant for two spin-hole pairs. But I don't think that's right, because if I try to write the Hamiltonian matrix, t appears not on the diagonal...
I am also not sure about this c+ operator - does it creates the same spin, that c- deletes on the other position, or can it creates both spins? I would say both from the Hamiltonian form, but if that should ilustrate moving of electron?
Is there any trick as with squared operators in spin systems? When there is a hole, that destroys the symmetry, so I can't use this trick, can I?
I would appreciate any suggestions that would move me further or any related articles - all I found were basic problems with only spin systems.
Hamiltonian I use is in the form:
H=JSiSj+t(c+ic-j+c+jc-i)
The first part is interaction between two spins, so sum over every spin-pair, the second part is interaction between spin and a "hole", that is for electron moving from one place to another, also sum over spin-hole pairs.
My systems are closed chains of 2, 3 or 4 electrons, or there is an additional hole to these.
I have already solved systems with only spins, because that's where the second part disappears, so I can use the fact, that H has the same eigenfunctions as total spin squared, single spins squared or I can use symmetry, so this part is OK.
But I have troubles with these holes. So let's take triangle with two spins and one hole. The basis has 12 vectors, so counting determinant wouldn't be nice - and I need a method I could use for bigger systems too.
My first idea was, that I could basicly use the result for bare 2 spins and just add 2t constant for two spin-hole pairs. But I don't think that's right, because if I try to write the Hamiltonian matrix, t appears not on the diagonal...
I am also not sure about this c+ operator - does it creates the same spin, that c- deletes on the other position, or can it creates both spins? I would say both from the Hamiltonian form, but if that should ilustrate moving of electron?
Is there any trick as with squared operators in spin systems? When there is a hole, that destroys the symmetry, so I can't use this trick, can I?
I would appreciate any suggestions that would move me further or any related articles - all I found were basic problems with only spin systems.