Packing hard spheres with short-range attraction in infinite dimension: Phase structure and algorithmic implications

TL;DR

This paper investigates the phase behavior and packing efficiency of high-dimensional hard spheres with short-range attraction using the replica method, revealing complex phase transitions and implications for polynomial-time packing algorithms.

Contribution

It extends the understanding of sphere packing in infinite dimensions by analyzing the phase diagram with attractive interactions and suggests new polynomial-time packing strategies.

Findings

Reentrant liquid-glass transition line identified

Existence of two distinct glass states and a glass-to-glass transition

Potential for more compact packings than pure hard-spheres at certain densities

Abstract

We study, via the replica method of disordered systems, the packing problem of hard-spheres with a square-well attractive potential when the space dimensionality, d, becomes infinitely large. The phase diagram of the system exhibits reentrancy of the liquid-glass transition line, two distinct glass states and a glass-to-glass transition, much similar to what has been previously obtained by Mode-Coupling Theory, numerical simulations and experiments. The presence of the phase reentrance implies that for a suitable choice of the intensity and attraction range, high-density sphere packings more compact than the one corresponding to pure hard-spheres can be constructed in polynomial time in the number of particles (at fixed, large d) for packing fractions smaller than 6.5 d 2^{-d}. Although our derivation is not a formal mathematical proof, we believe it meets the standards of rigor of theoretical physics, and at this level of rigor it provides a small improvement of the lower bound on the sphere packing problem.