# The most general $SU(2)$ invariant spin $\frac{1}{2}$ hamiltonian on 5 sites

I have periodic chain of spins $$s=\frac{1}{2}$$. I want to know what is the most general $$SU(2)$$ invariant and translation invariant hamiltonian. My guess is: $$\sum_i (j_1 S_i \cdot S_{i+1}+j_2 S_i \cdot S_{i+2} + j_3 (S_i \cdot S_{i+1})( S_{i+2} \cdot S_{i+3}) + j_4 (S_i \cdot S_{i+1})( S_{i+2} \cdot S_{i+4}) +j_5 (S_i \cdot S_{i+2})( S_{i+1} \cdot S_{i+3})).$$ Can somebody confirm this or point me to some literature where this is discussed? Also I would be interested in a derivation of this result.

• You are specifically interested in 5 sites? Where is the relevance that it is a "periodic chain" then? – Norbert Schuch May 17 '20 at 12:30
• I am interested in small systems. So first, I am trying to construct this hamiltonian on 5 sites. The relevance of periodicity is that it imposes translation symmetry. – Frank May 18 '20 at 11:46
• Ah - so you are looking for an SU(2) invariant and translational invariant Hamiltonian. You should state that. --- Note that if you e.g. want to parametrize this Hamiltonian numerically, there should be ways to automatize this without deriving the explicit form. – Norbert Schuch May 18 '20 at 16:14
• I don't think you Hamiltonian includes terms like $\vec S\cdot(\vec S\times \vec S)$. – Norbert Schuch May 18 '20 at 22:22
• Indeed, you are right about this term. Time reversal symmerty removes it, right? – Frank May 19 '20 at 18:20

You must impose some constraints otherwise the space of invariant Hamiltonians would be much larger (then the five parameters that you consider). It's dimension would be the dimension of the $$J=0$$ sector which is exponentially large.

Typical constraints are

1) geometric locality (interaction takes place only between nearby particles) and

2) few body interaction. This expresses the fact that interaction between many particles at the same time are rare and the most usual choice is to have only two particles per interaction. But of course in principle $$k$$-body interactions are possible.

• Thanks for the response. I understand that nonlocal interactions don't describe real physics very well, but I am really interested in a most general hamiltonian that only respects SU2 symmetry and translational symmetry. Therefore I do not wish to impose locality. I also understand that such hamiltonian is very complex, that's why I restricted myself to a small system, i.e 5 sites. – Frank May 18 '20 at 11:54
• Then the answer is: any Hermitian matrix in the $J=0$ sector. – lcv May 18 '20 at 19:16
• By the way, I should add that my argument works for an even number of sites – lcv May 18 '20 at 19:48
• Ah and then you wanna impose translational symmetry too. I didn't see that. If you want a practical way to build a Hamiltonian with such symmetries (or find the value of a certain Hamiltonian in this sector) I could do that – lcv May 18 '20 at 19:53
• Well, I have a su2 and translation invariant hamiltonian is some weird basis and I want to interpret it in terms of spin-$\frac{1}{2}$ interactions. In short, I am looking for a basis of operators. – Frank May 18 '20 at 22:43