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.

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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  • $\begingroup$ Thanks for your answer, however I'm still a little confused. I know that the Goldstone mode corresponds to an excitation of the spin system, but - and correct me if I'm wrong - I don't think you answered my question what exactly his excitation is? For example in the charge density wave case one can identify Amplitudons and Phasons as the elementary excitations of the ground state, i.e. either letting the amplitude of the wave oscillate or change the phase of the wave. Is there no similiar concept with SDWs? $\endgroup$ – user17574 Jan 22 '15 at 13:10
  • $\begingroup$ @user17574 In my understanding, the Goldstone mode always corresponds to the fluctuation of the "phase" while the fluctuation of "amplitude" may be called Higgs mode. So when we talk about the gapless Goldstone mode in SDW, it should relates to the fluctuation of spin directions rather than spin length. Thus I think there is only "Phason" excitations in SDW, but conventionally we call the excitations "magnons" or spin-wave (classical partner). I hope this comment may be helpful to you. $\endgroup$ – Kai Li Jan 22 '15 at 13:54
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    $\begingroup$ 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 ... $\endgroup$ – tparker Aug 5 '16 at 14:03
  • $\begingroup$ ... and phasons for the reasons discussed at physics.stackexchange.com/questions/252035/…. $\endgroup$ – tparker Aug 5 '16 at 14:03
  • $\begingroup$ @tparker Your comments are very helpful. Is there any reason that amplitudon should be gapped? $\endgroup$ – xiaohuamao Mar 12 '19 at 1:05

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