Goldstone modes of spin density wave

A spin density wave (SDW) is a phase in which a material suddenly shows a periodically modulated spin density $S_{\vec{q}}(\vec{r})$ below a certain critical tempereature $T_C$.

Obviously some kind of symmetry is broken when a SDW forms, however I'm not exactly sure which one. Maybe translational symmetry? However that is already broken by simply forming a crystal and I don't know whether there is such a thing as further breaking a symmetry. Which symmetry exactly is broken in the case of an SDW?

My second question is: When a continuous symmetry is broken, one can associate a Goldstone mode to it in the ordered phase. What is the Goldstone mode of a spin density wave? Also, is it always true that the Goldstone modes are the same as the elementary excitations of the solid?

An ordered SDW phase breaks both the continuous $SU(2)$ spin-rotation symmetry and the time-reversal symmetry (because the presence of either of these two symmetries would force the order parameter of SDW vanishing). It is the spontaneously broken of continuous spin-rotation symmetry that leads to the gapless Goldstone mode. Here is a related issue.
• I don't understand why you say there are only phason excitations in a SDW. A SDW takes the form $\vec{S}(x) = A \hat{n} \cos(kx - \delta)$. Since $A$, $\hat{n}$, and $\delta$ can all vary continuously, I think there are three types of SDW excitations: amplitudons (slow spatial variations in $A$), magnons (slow spatial variations in $\hat{n}$), and phasons (slow spatial variations in $\delta$). Amplitudons may be gapped as you say, but I think that magnons and phasons should both be gapless - magnons because they are the Goldstone mode for spontaneous SU(2) breaking ... – tparker Aug 5 '16 at 14:03