Crystalline ground states for classical particles

TL;DR

This paper demonstrates that certain pair interactions with specific Fourier transform properties lead to both periodic and aperiodic ground state configurations in any dimension, with unique structures at critical densities and degeneracy above them.

Contribution

It establishes the existence of periodic and aperiodic ground states for classical particles with specific Fourier transform conditions, extending understanding of ground state configurations in various dimensions.

Findings

Ground states exist for densities above certain thresholds.

Unique periodic ground states at critical densities.

Degeneracy of ground states increases above critical densities.

Abstract

Pair interactions whose Fourier transform is nonnegative and vanishes above a wave number K_0 are shown to give rise to periodic and aperiodic infinite volume ground state configurations (GSCs) in any dimension d. A typical three dimensional example is an interaction of asymptotic form cos(K_0 r)/r^4. The result is obtained for densities rho >= rho_d where rho_1=K_0/2pi, rho_2=(sqrt{3}/8)(K_0/pi)^2 and rho_3=(1/8sqrt{2})(K_0/pi)^3. At rho_d there is a unique periodic GSC which is the uniform chain, the triangular lattice and the bcc lattice for d=1,2,3, respectively. For rho>rho_d the GSC is nonunique and the degeneracy is continuous: Any periodic configuration of density rho with all reciprocal lattice vectors not smaller than K_0, and any union of such configurations, is a GSC. The fcc lattice is a GSC only for rho>=(1/6 sqrt{3})(K_0/pi)^3.