Solving Heisenberg Hamiltonian for 1/2-Spin Systems

[SalomeH]

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=JS_iS_j+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.

 

[mark726]

Thank you for sharing your research with us. It sounds like you are working on a very interesting problem involving spin and hole interactions. I have a few suggestions that may help you with your calculations.

Firstly, it is important to understand the meaning of the Hamiltonian you are using. The first part represents the interaction between two spins, while the second part represents the interaction between a spin and a hole. This means that the first part will contribute to the energy of the system if there are two spins in close proximity, while the second part will contribute if there is a spin and a hole in close proximity.

To simplify your calculations, you can consider the two parts separately. For example, for the first part, you can use the fact that the total spin squared is a conserved quantity and use its eigenfunctions as a basis for your calculations. This will greatly reduce the number of basis states you need to consider.

For the second part, you can use the same approach, but now you will need to consider the spin and hole together as a composite object. This means that the basis states will be a combination of the spin and hole states. For example, for a triangle with two spins and one hole, you will have a basis of 12 states, as you mentioned. To simplify the calculation, you can use symmetries and other properties of the Hamiltonian to reduce the number of states you need to consider even further.

In regards to the c+ operator, it creates a spin at the position of the hole, while c- destroys a spin at the position of the hole. So, if you have a spin at one position and a hole at another, applying c+ will create a spin at the position of the hole and leave the spin at the original position unchanged.

As for using tricks with squared operators, it depends on the specific problem you are working on. It may be possible to use similar tricks, but it is important to carefully consider the symmetries and properties of your system before applying them.

I hope these suggestions will help you with your calculations. Good luck with your research!