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Collective-density variables have proved to be a useful tool in theprediction and manipulation of how spatial patterns form in the classicalmany-body problem. Previous work has employed properties of collective-densityvariables along with a robust numerical optimization technique to find theclassical ground states of many-particle systems subject to radial pairpotentials in one, two and three dimensions. That work led to theidentification of ordered and disordered classical ground states. In thispaper, we extend these collective-coordinate studies by investigating theground states of directional pair potentials in two dimensions. Our studyfocuses on directional potentials whose Fourier representations are non-zero oncompact sets that are symmetric with respect to the origin and zero everywhereelse. We choose to focus on one representative set which has exoticground-state properties: two circles whose centers are separated by some fixeddistance. We obtain ground states for this "two-circle" potential that displaylarge void regions in the disordered regime. As more degrees of freedom areconstrained the ground states exhibit a collapse of dimensionalitycharacterized by the emergence of filamentary structures and linear chains.This collapse of dimensionality has not been observed before in relatedstudies.

Exotic Ground States of Directional Pair Potentials via Collective-Density Variables