Crystallization simulation

To simulate crystallization, large number (between 50 and 200) of density components $\rho_{q_i}$ were randomly seeded on a sphere. Their subsequent evloution was governed by GL energy minimization, both their position on the sphere as well as the amplitude. In the variationally predicted range of iQC stability the method consistently converges to iQC. Sometimes, due to randomness of initial conditions sometimes other states (with smaller number of momentum components, but higher energies) would appear. The parameters of simulations have to be chosen with reasonable care to avoid too large steps in evolution. With that, convergence to iQC seems to be pretty much guaranteed.

Among other (higher energy) states that sometimes appear is the 5-fold symmetric state, which is approximately the icosahedral state with 1 of 6 q's removed. It has not been carefully looked at variationally (to allow for deviation from ideal icosahedral angle) and hence it cannot be ruled out that it is optimal, somewhere. That would seem to correspond to a q2D quasicrystal, a la Penrose tiling.