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In textbooks for introducing Goldstone mode, people usually consider about phonon as a Goldstone mode emerging from translation symmetry breaking in lattice.

However, the rotation symmetry also simultaneously breaks in lattice but lacks of discussion.

What is the Goldstone mode when rotation symmetry breaks in lattice?

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In a lattice both translational and rotational symmetries are broken, which gives rise to longitudinal and transverse phonons which are the goldstone bosons of this symmetry-breaking. However you cannot link e.g. longitudinal modes to translational symmetry breaking and transverse to rotational : everything is mixed and the relationship between symmetry breaking and the different goldstone phonons depend on the symmetry of your lattice.

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    $\begingroup$ I have a question: In $d$ dimension, the symmetry of d-dim Euclidean group is broken, so in principle there are $d+d(d-1)/2=d(d+1)/2$ Goldstone boson. But in $d$ dimensional, there are $1$ longitudinal phonon and $d-1$ transverse phonons. What's up? $\endgroup$
    – 346699
    Commented Sep 5, 2016 at 0:22

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