The emergence of long range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in physics. To address this challenge we study a discrete microscopic model for an elastic crystal with dislocations in three dimensions, previously introduced by Ariza and Ortiz. The model is rich enough to support some realistic features of three-dimensional dislocation theory, most notably grains and the Read– Shockley law for grain boundaries, which we rigorously derive in a simple, explicit geometry. We analyze the model at positive temperatures, in terms of a Gibbs distribution with energy function given by the Ariza–Ortiz Hamiltonian plus a contribution from the dislocation cores. Our main result is that the model exhibits long range positional order at low temperatures. The proof is based on the tools of discrete exterior calculus, together with cluster expansion techniques.

中文翻译：

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固体原子模型中的长程有序

在具有连续对称性的原子系统中，低温下长程有序的出现是物理学中一个基本但知之甚少的现象。为了应对这一挑战，我们研究了具有三维位错的弹性晶体的离散微观模型，该模型先前由 Ariza 和 Ortiz 引入。该模型足够丰富，足以支持 3 维位错理论的一些现实特征，最显着的是晶粒和晶界的 Read-Shockley 定律，我们在简单、明确的几何中严格推导出。我们根据 Ariza-Ortiz Hamiltonian 给出的能量函数加上位错核心的贡献的吉布斯分布来分析正温度下的模型。我们的主要结果是该模型在低温下表现出长程位置顺序。