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

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

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.


1 Answer 1


$\newcommand{\bm}[1]{\mathbf{#1}}$ 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{equation} \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} \end{equation} 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 $$ \mathcal H = \sum_{j=0}^{2S} a_{j} (\bm S_1\cdot \bm S_2)^j\ . $$


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