Crystal formation as spontaneous broken symmetry

Crystal formation is an example of the Anderson "More is different" Higgs-Goldstone macro-quantum coherent Glauber state effect in condensed matter physics.

What is the order parameter? For simplicity consider a 1-dim periodic lattice spacing a.

We have spatial variation of properties in the static limit in the ground state (neglecting zero point jiggle as an approximation. It is the residual micro-quantum incoherent noise on the macro-quantum coherent signal.).

This is a Glauber coherent state of "tachyonic" virtual off-shell phonons - where the speed of sound cs is the analog of the speed of light c in quantum field theory.

i.e. real phonons on-shell obey

f = csk

In contrast, the virtual phonons off the above mass shell (dispersion curve) act as if they had dispersion

f^2 = (csk)^2 - (cs/a)^2

the quasi imaginary rest mass = ihcs/a

in the Landau-Ginzburg description

Or we can just say that the 1-dim crystal order parameter is a kind of macro-quantum coherent Bose-Einstein condensate of virtual phonons of zero frequency and wave vector k = 1/a analogous to the virtual photons making the Coulomb potential. The near EM fields are also coherent Glauber states of off-shell virtual photons.