Minimality of the rock-salt structure for inverse-power and Gaussian interactions

Let $d eq 0$ and let $E_{f_1,f_2}$ be the charged-lattice energy defined by the source. For $\mathcal{V}>0$, let $\mathcal{V}^{1/d}\mathbb{Z}^d$ be the cubic lattice of volume $\mathcal{V}$, and let $varphi_pm$ be its alternating charge distribution. Let $p,q>d/2$ be the exponents in the inverse-power potentials, let $alpha,beta$ be the parameters in the Gaussian potentials, and let $V_L$ be the volume specified by the source.

Minimality of the rock-salt structure. There exist $\delta_0,\delta_1>0$, depending only on $d$, such that the global minimizer of $E_{f_1,f_2}$ is of the form

$$\big(\mathcal{V}^{1/d}\mathbb{Z}^d,\varphi_\pm\big)$$

for some $\mathcal{V}>0$ if either $f_1,f_2$ are the specified inverse-power potentials with $p,q>d/2$ and $p-q>\delta_0$, or they are the specified Gaussian potentials with $\beta-\alpha>\delta_1$. Moreover, in the inverse-power case, $\mathcal{V}=V_L$.

This conjecture is motivated by numerical comparisons in dimensions $2$, $3$, and $8$, where the rock-salt structure has the lowest observed energy among the tested competitors. The source does not provide a proof of the asserted global minimization statement.