# 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.

The state in which all spins are parallel is not an eigenstate of the Hamiltonian since you never can assign an exact direction to any spin degree of freedom, because different components of spin don't commute with each other.

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)