Counterintuitive ground states in soft-core models

Abstract

It is well known that statistical mechanics systems exhibit subtle behavior in high dimensions. In this paper, we show that certain natural soft-core models, such as the Gaussian core model, have unexpectedly complex ground states even in relatively low dimensions. Specifically, we disprove a conjecture of Torquato and Stillinger, who predicted that dilute ground states of the Gaussian core model in dimensions 2 through 8 would be Bravais lattices. We show that in dimensions 5 and 7, there are in fact lower-energy non-Bravais lattices. (The nearest three-dimensional analog is the hexagonal close-packing, but it has higher energy than the face-centered cubic lattice.) We believe these phenomena are in fact quite widespread, and we relate them to decorrelation in high dimensions.

Dimension reduction techniques for the minimization of theta functions on lattices

“…This theorem is, so far as we know, the only general-framework analogue of the special case considerations done in [17]. A similar classification in dimension 3 seems to be an interesting future direction of research.…”

“…In this section we consider our Theorem 1.1 within the theory of fibered packings, as introduced by Conway and Sloane [19] and extended by Cohn and Kumar [17]. The basic definition is the following: Definition 2.9 (fibering configurations, cf.…”

“…In fact the configurations considered in [19] and [17] are of a more special type, as described below. Note that the definition (2.10) is a special case of the above definition for Λ 1 = τ Z.…”

Generating dense packings of hard spheres by soft interaction design

“…This would also shed some light on the issue of the number of operations required to construct a dense packing, which appears still quite prohibitive as d increases. For some related works in this direction on soft-core models, see [79,80].…”

Self-assembled structures of Gaussian nematic particles

“…Accordingly, a powerful and elegant proof on the ground-state structures of certain classes of bounded pair interactions has been established in the work of Sütő [16][17][18]. Further, Torquato and Stillinger have recently derived generalized duality relations, which act as guides in the search for ground states [19], and they also analyzed the phase behavior of the Gaussian model in high spatial dimensions [20], for which Cohn and Kumar subsequently showed that its ground states are non-Bravais lattices at spatial dimension 5 and 7 [21]. For diverging potentials, Theil has given a rigorous proof of the crystallization of LennardJones particles in two dimensions in a triangular lattice [22].…”