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

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## Abstract

Collective-density variables have proved to be a useful tool in the prediction and manipulation of how spatial patterns form in the classical many-body problem. Previous work has employed properties of collective-density variables along with a robust numerical optimization technique to find the classical ground states of many-particle systems subject to radial pair potentials in one, two and three dimensions. That work led to the identification of ordered and disordered classical ground states. In this paper, we extend these collective-coordinate studies by investigating the ground states of directional pair potentials in two dimensions. Our study focuses on directional potentials whose Fourier representations are non-zero on compact sets that are symmetric with respect to the origin and zero everywhere else. We choose to focus on one representative set that has exotic ground-state properties: two circles whose centers are separated by some fixed distance. We obtain ground states for this “two-circle” potential that display large void regions in the disordered regime. As more degrees of freedom are constrained the ground states exhibit a collapse of dimensionality characterized by the emergence of filamentary structures and linear chains. This collapse of dimensionality has not been observed before in related studies.

## Keywords

Ground states Collective coordinates Directional potentials## Notes

### Acknowledgements

We are pleased to offer this contribution to honor M. Fisher, J. Percus, and B. Widom whose own remarkable contributions have vividly demonstrated the originality intrinsic to statistical mechanics. This work was supported by the Office of Basic Energy Sciences, U.S. Department of Energy, under Grant No. DE-FG02-04-ER46108.

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