Stability of quasi low-dimensional crystals?

[weejee]

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?

 

[eljose79]

As a scientist in the field of solid state physics, I agree with your thoughts on the existence of quasi low-dimensional periodic structures such as graphene and polyacetylene. These structures are able to exist due to the relaxation in extra dimensions, which breaks the translational symmetry and allows for the gap in the acoustic phonon.

In field theory language, this can be seen as the Goldstone boson acquiring mass due to the approximate symmetry in the first place. This is a common phenomenon in many condensed matter systems, where the symmetry is not exact and can be broken by various factors such as lattice distortions or impurities.

Overall, your understanding of the role of relaxation in extra dimensions in allowing for the existence of quasi low-dimensional crystals is correct. It is an important concept in solid state physics and has been extensively studied in the field. Keep up the good thinking and continue exploring the fascinating world of low-dimensional crystals!