Holstein-Primakoff bosons

There are several useful formalisms for spin systems. We start with Holstein-Primakoff bosons, which will be useful for when we want to consider a small perturbation around an ordered state (e.g., spin-wave excitations).
$$\begin{eqnarray} S^+ &=& (\sqrt{2 S - n_b}) b,\cr S^- &=& b^\dagger (\sqrt{2 S - n_b}),\cr S^z &=& S - n_b. \end{eqnarray}$$
Using $[b,b^\dagger]=1$ it is simple to check the commutation relations
$$\begin{equation} [S^\alpha,S^\beta]=i \epsilon^{\alpha \beta \gamma} S^\gamma. \end{equation}$$
We also need to project out all the unphysical states with $n_b > 2 S$, which is consistent since the raising and lowering operators will not take one out of the physical subspace if one starts there. These are useful as a way to expand systematically around the nearly classical large-$S$ limit.

The second formalism introduced was that of coherent states. Here the formulas are quite lengthy, and I will just refer the reader to Auerbach chapter 7. The coherent-state representation will be used to understand general spin systems (rather than just ordered states and low-energy excitations) when we set up a spin path integral in a few lectures.
The basic idea is to represent spins in a basis of states $|{\hat \Omega}\rangle$ that in the classical limit behave like classical vectors: the states are obtained by rotating the north pole $|S,S\rangle$ with a rotation operator $R(\theta,\phi,\chi)$ that is a function of three Euler angles. Actually $\chi$ is essentially an arbitrary phase, which will not be important for most of what we do.

Note that both HP bosons and coherent states become much simpler in the large-$S$ limit. You should be aware of the difference between large-$S$ representations of $SU(2)$, which we will study in this course, and representations of the enlarged rotation groups $SU(N)$. Large-$N$ is also a very useful technique but not one that we will discuss.