# How do we determine the statistics and spin of quasi-particles?

I am considering the Heisenberg XXZ model at the moment. In the literature it says that (in the $J\Delta\rightarrow\infty$ limit, i.e. the ferromagnetic Ising regime) one can either view low-energy excitations as magnons (single overturned spins), or as domains of consecutive flipped spins, named spinons. The latter is supposed to have spin $S=1$. My question is two-fold:

-How do we easily note the statistics of the introduced particle and what spin it has?

-When do we choose what quasi-particle?

• If the quasi-particle is a "compound" object, you just add the individual spins according to the appropriate rules for adding angular momenta. An example would be the polaron, which is an electron dressed with a bunch of phonons. The electron has spin $1/2$, the phonons have spin $0$, so the total object has spin $1/2$.
• If the quasi-particle is more like a collective excitation (magnon, spinon), just compare the total spin of a system with and without the thing. The difference must then be the spin of your particle. Consider a ferromagnet of spin $1/2$ magnetic moments in its ground-state with all spins pointing in the same direction. The total spin is then $N/2$ where $N$ is the number of moments. Flipping a single spin means the total spin is now $(N-1)/2 - 1/2 = N/2 - 1$, so the newly created excitation has spin 1.