We learn in solid state physics that crystals cannot exist in one or two dimensions. The main enemy of low-dimensional crystals is the acoustic phonon (Goldstone mode). Due to its small energy in the long wavelength limit, it contributes signifiantly to the fluctuation of the atomic position and makes it diverge(infrared divergence). However, in real life, there are quasi low-dimensional periodic structures, such as graphene(2D) and polyacetylene(1D). Does this mean that the acoustic phonon is somehow gapped? I guess it can be made possible due to the relaxation in extra dimensions, such as ripples in graphene and bond distortions in polyacetylene. I think these relaxations may slightly break the translational symmetry so that long wavelength phonons get mixed with one another and become gapped. Maybe in field theory language, this amounts to the mass acquired by the Goldstone boson when the (spontaneously broken) symmetry is only approximate in the first place? Any opinions on my thoughts?