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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?

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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.

The Goldstone mode of SDW is a gapless excitation of spin system, which is similar to that of phonons (the elementary excitations of oscillating crystals).

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Kai Li
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