Statistical mechanics of the lattice sphere packing problem

We present an efficient Monte Carlo method for the lattice sphere packing problem in $d$ dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond previous methods, not only in exploring higher dimensions but also in shedding light on the statistical mechanics underlying the problem in question. We observe evidence of a phase transition in the thermodynamic limit $d\to \infty$. In the dimensions explored in the present work, the results are consistent with a first-order crystallization transition but leave open the possibility that a glass transition is manifested in higher dimensions.

Figure 1

Traces of density as a function of reduced pressure in different runs in dimensions $d=11$ and $d=16$. In 11 dimensions, the simulation remains in equilibrium until around $p=4.8\times {10}^3$, where different runs continue along different branches, corresponding to basins of attractions of different extreme lattices. In 16 dimensions, all runs end up in the basin of the same lattice, but we observe that different runs make the transition into this basin at different pressures. It appears that at least one of the runs also spends some time in an intermediate state, presumably the basin of attraction of a different extreme lattice.

Figure 2

Traces of (normalized) density as a function of reduced pressure, averaged over all quasistatic compression runs that yielded the densest known lattice in each dimension $d=9,...,20$.