# Most general form of a spin rotation invariant Hamiltonian?

I am told that the most general form of a spin rotation invariant Hamiltonian for two systems 1 and 2 both with spin $$S$$, i.e., the spin operators

\begin{align} (\hat{S}_1^x)^2 +(\hat{S}_1^y)^2 + (\hat{S}_1^z)^2 = (\hat{S}_2^x)^2 +(\hat{S}_2^y)^2 + (\hat{S}_2^z)^2 = S(S+1)\hbar^2 \end{align} is given by

$$$$\mathcal{H} = \sum_{j=0}^{2S} a_j \bigg(\frac{\mathbf{\hat{S}_1}\cdot\mathbf{\hat{S}_2}}{\hbar}\bigg)^j$$$$

I understand that it should be a function of $$\mathbf{\hat{S}_1}\cdot\mathbf{\hat{S}_2}$$ but I do not understand why should the sum terminate at $$j=2S$$. Can someone explain it to me. Thank you.

$$\newcommand{\bm}[1]{\mathbf{#1}}$$ You need to look at this in terms of spin quantum numbers (i.e., eigenvalues).
$$(\bm S_1+\bm S_2)$$ can take values $$S_\mathrm{tot} = 0,\dots,2S$$. Now if we restrict to the subspace with total spin $$S_\mathrm{tot}$$, we have that \begin{aligned} 2S_\mathrm{tot}(2S_\mathrm{tot}+1) = (\bm S_1+\bm S_2)^2 &= \bm S_1\cdot \bm S_1 + \bm S_2\cdot \bm S_2 + 2\, \bm S_1\cdot \bm S_2 \\ &= S(S+1) + S(S+1) + 2\,\bm S_1\cdot \bm S_2\ , \end{aligned} and thus $$\bm S_1\cdot \bm S_2 = S_\mathrm{tot}(2S_\mathrm{tot}+1)-S(S+1) \tag{1}$$ can take $$2S+1$$ possible values. (Note that this means that $$\bm S_1\cdot \bm S_2$$ and $$\bm S_1+\bm S_2$$ are diagonal in the same basis, that is, we can reason about them as if they were just numbers which can take the corresponding set of values.)
A $$\mathrm{SU}(2)$$ invariant Hamiltonian of the two spins will take a different value of each subspace of total spin $$S_\mathrm{tot}$$, i.e., it is of the form (with $$\Pi_{S_\mathrm{tot}}$$ the projector onto the subspace with total spin $$S_\mathrm{tot}$$) $$\mathcal H = \sum_{S_\mathrm{tot}=0}^{2S} E_{S_\mathrm{tot}} \Pi_{S_\mathrm{tot}}\ . \tag{2}$$ Since there is a one-to-one relation between the total spin and the value of $$\bm S_1\cdot \bm S_2$$ -- Eq (1) --, each projector $$\Pi_{S_\mathrm{tot}}$$ can be expressed as a function of $$\bm S_1\cdot \bm S_2$$: $$\Pi_{S_\mathrm{tot}}=f_{S_\mathrm{tot}}(\bm S_1\cdot \bm S_2)\ . \tag{3}$$ This function must be $$f_{S_\mathrm{tot}}(\bm S_1\cdot \bm S_2)=1$$ for the desired $$S_\mathrm{tot}$$ (using (1)), and $$f(\bm S_1\cdot \bm S_2)=0$$ for all other values the total spin can take (again using (1)). This means that we only need to fix the value of $$f$$ at $$2S+1$$ points, and thus, it can be choosen to be a polynomial of degree $$2S$$: $$f_{S_\mathrm{tot}}(\bm S_1\cdot \bm S_2) = \sum_{j=0}^{2S} a_{j,S_\mathrm{tot}} (\bm S_1\cdot \bm S_2)^j\ .$$ Substituting this into (3) and then into (2) gives that $$\mathcal H = \sum_{j=0}^{2S} a_{j} (\bm S_1\cdot \bm S_2)^j\ .$$