# Differences between spin waves and spin density waves

Roughly speaking, in condensed matter systems, spin waves and spin density waves are both low-energy states with spin that varies spatially. What precisely are their differences?

-

In spin waves, the quantity that depends on the spatial coordinates in the wave-like way, e.g. $\sin kz$, is the (average) spin $\vec J$ itself (a vector meaning the angular momentum), while in the spin density waves, it's the spin density $\rho_J$ (the number of carriers of spin per unit volume).

These are completely different things in the same sense as water is something else than the density of water.

In spin waves, typical in ferromagnets, a basis vector is a state that has almost all relevant electrons' spins "up" while one of them is "down". The expectation value of the number of spins at height $z$ that are "down" goes like $\cos kz$, for example.

In spin density waves, which is a different behavior of matter competing with (and equally important as) ferromagnets, it's the number of spin carriers per unit volume $\rho$, and not the spins $\vec J$ themselves, that is variable. The number of spin carriers may be highly variable if there are many states $N(E_F)$ near the Fermi energy, the highest occupied energy for electrons' states.

A precise analysis of the Fermi surfaces and their shapes is needed to determine whether a material with a high $N(E_F)$ will choose to be a ferromagnet, antiferromagnets, superconductors, or carriers of the spin-density (or charge-density) waves.

-

Spin waves are low-energy excitations seen in ferromagnets and antiferromagnets. This is a direct consequence of Goldstone's theorem whereby the original model broke a continuous symmetry while forming either of these two ordered states. The long-wavelength spin-wave excitation for the ferromagnet goes as $\omega(\mathbf{k}) \propto |\mathbf{k}|^2$, whereas in an antiferromagnet it goes as $\omega(\mathbf{k}) \propto |\mathbf{k}|$. Both are bosonic excitations and, as was pointed out, involves a single spin-flip in $\mathbf{r}$ space that is spread across the whole lattice.
Spin density wave is a state of matter, just as an antiferromagnet is one. In fact, it can be seen as a generalization of an antiferromagnet in the following sense: consider the Hubbard model with large nearest neighbour repulsion $U$ and small hopping amplitude $t$. From Anderson's superexchange ($2^{\textrm{nd}}$ order hopping processes), this may be mapped to an antiferromagnetic Heisenberg model with coupling $J = 4t^2/U$. For very large $U$, we recover the antiferromagnet; for intermediate $U$, a spin-density wave arises. This is a spatial modulation of the up-down spins on the lattice, each offset with respect to the other by a lattice spacing. But the total electron density $n = n_{\uparrow} + n_{\downarrow}$ is constant across the lattice.