We present a sufficient criterion for the emergence of cluster phases in an ensemble of interacting classical particles with repulsive two-body interactions. Through a zero-temperature analysis in the low density region we determine the relevant characteristics of the interaction potential that make the energy of a two-particle cluster-crystal become smaller than that of a simple triangular lattice in two dimensions. The method leads to a mathematical condition for the emergence of cluster crystals in terms of the sum of Fourier components of a regularized interaction potential, which can be in principle applied to any arbitrary shape of interactions. We apply the formalism to several examples of bounded and unbounded potentials with and without cluster-forming ability. In all cases, the emergence of self-assembled cluster crystals is well captured by the presented analytic criterion and verified with known results from molecular dynamics simulations at vanishingly temperatures. Our work generalises known results for bounded potentials to repulsive potentials of arbitrary shape.

中文翻译：

---

任意相互作用势的簇自组装条件

我们提出了一个相互排斥的经典粒子与排斥两体相互作用的集合中出现簇相的充分标准。通过在低密度区域的零温度分析，我们确定了相互作用电势的相关特征，该特征使两粒子簇晶体的能量在二维上小于简单三角晶格的能量。该方法根据正则化相互作用势的傅立叶分量之和得出了簇状晶体出现的数学条件，其原则上可以应用于任何任意形状的相互作用。我们将形式主义应用于具有和不具有成簇能力的有界和无界势的几个例子。在所有情况下，自组装团簇晶体的出现被提出的分析标准很好地捕获，并在消失的温度下用分子动力学模拟得到的已知结果进行了验证。我们的工作将已知势能推广到任意形状的排斥势。