# Correct Visualization of a Spin Wave in a Ferromagnetic material

One way of describe a ferromagnetic material is the Heisenberg hamiltonian

$H = -\frac{J}{2} \sum_{<i,j>}$S$_i$S$_j$

where $J$ measures the interaction between spins (positive for ferromagnetism and negative for antiferromagnetism) and S$_i$ is the Spin operator in the site $i$.

The Ground state has all spins aligned in the same direction but lowering spins to $S-1$ is not an eigenstate of the hamiltonian. The correct eigenstates are Spin Waves in which the perturbation travels across the lattice.

One usually visualizes this waves as a precesion around the $z$ axis where each site has a phase difference with its neighbours. Here is when I start to get confused.

I understand that the existence of a Spin Wave lowers each site's spin by a unit so that, if the system was in the ground state where $s_z=S$ in every site, now each site has spin $S-1$.

If this is correct... can there exist spin waves in lattices with $S=\frac{1}{2}$? I don't understand how this would happen if the spins are supposed to precess the $z$ axis.

I would be grateful if someone could clarify this precession picture and explain if it needs S to be large.

Spin waves usually emerge when you assume that $S$ is large. In this regime one can see that the quantum mechanical equations of motion (in Heisenberg picture and in the continuum limit) become the wave equation for $\vec{S}$.
For the case $S=\frac{1}{2}$ the large $S$ approximation doesn't seem to be accurate, in this special case you can exactly solve the problem using Jordan-Wigner transformation and find the energy spectrum. (However there is this Mermin Wagner theorem which states that the completely ordered state is unstable due to the effect of fluctuations)