Uniform Translation of a Lattice

[thatboi]

I'm currently reading through this brief review on symmetries, and on page 5, the following statement is made: "Why is there no energy cost for a uniform displacement? Well, there is a translational symmetry: moving all the atoms the same amount doesn’t change their interactions. But haven’t we broken that symmetry? That is precisely the point." Perhaps I just misunderstood the punchline, but what exactly did the author mean by the last sentence, "that is precisely the point"? The crystal lattice has broken continuous translational symmetry, but what does that have to do with uniform translations of the lattice and the action costing 0 energy?

 

[Haborix]

This is just a cryptic way of stating Goldstone's theorem. Associated with the spontaneous breaking of a continuous symmetry is the introduction of long wavelength modes. The uniform translation of the lattice is the long wavelength mode.