Diagonalizing the all-to-all quantum spin model (quantum Curie-Weiss) with uniform couplings

Question

I am interested in diagonalizing the all-to-all quantum spin model

\begin{align} \hat{H} = \frac{1}{2}\sum_{i,j \neq i} \hat{S}_i \cdot \hat{S}_j \end{align}

or, if possible, a more general form involving an arbitrary coupling $\tilde{H}$ between the spin states on sites $i$ and $j$:

\begin{align} \hat{H} = \frac{1}{2} \sum_{i, j\neq i} \tilde{H}_{ij} \end{align}

There are numerous treatments for how to approach this sort of problem using mean field techniques, but for the applications I'm interested in I would like to obtain the spectrum exactly. Is this a tractable problem? Is the answer known? Is it trivial?

I am imagining something along the lines of an adaptation of the Bethe Ansatz to this problem-- for instance, in the case of the first Hamiltonian when the couplings are instead restricted to be nearest-neighbor instead of all to all, the solution is well known.

Answer 1

Assuming that each spin has total spin $s$, then you can rewrite the Hamiltonian: $$ H = -\frac J2\hat S{}^2 +\frac {JN}2s(s+1)\\ \hat S = \sum_i \hat s_i $$ You therefore just need to diagonalize $\hat S{}^2$. For this, you'll need to decompose the product space of $N$ $s$ spin spaces into irreducible representations. The spectrum of $\hat S{}^2$ will therefore be of the form $S(S+1)$ with $S$ ranging the possible total spin number, namely $0, 1 ... Ns$. To diagonalise the Hamiltonian, you'll need to use Clebsch-Gordon coefficients.

Hope this helps.