标题： 准周期材料中声子的数学基础 显示英文标题

标题： Mathematical foundations of phonons in incommensurate materials

摘要： 在一些模型中，可以证明周期性配置在全局$\ell^2$或局部扰动下是稳定的。 这在非周期介质中并不成立。 我们感兴趣的非周期介质的特定类别来自于将两个二维周期晶体平行堆叠并相对扭转。 在周期介质中，声子是作用于$\ell^2$上的稳定性算子的广义特征向量，这些特征向量来源于机械能。 我们分析的目标是为给定的非周期介质类中的声子赋予意义。 正如在1D Frenkel-Kontorova模型中严格确立，并由其中一位作者之前应用过，我们假设可以通过连续堆叠周期函数来参数化最小化晶格变形相对于局部扰动，我们之前推导出了连续的能量密度泛函。 与离散能量泛函相比，这种（连续）能量密度在分析和计算上要容易得多。 为了转换为基于$\ell^2$的能量泛函，我们还研究了给定晶格变形下的偏移能量，在$\ell^1$扰动下。 我们的发现表明，在未变形的双层异质结构的情况下，虽然能量密度可以在假设单个层稳定的情况下被证明是稳定的，但在扭曲双层石墨烯的情况下，偏移能量则不稳定。 然后，我们建立了相对于弛豫晶格的偏移能量的稳定性和不稳定性条件。 最后，我们证明，在非共格双层同质结构的情况下，即两个相等的层，如果我们根据上述全局能量密度选择最小化变形，则在扭转角趋近于零的极限下，偏移能量是稳定的。 因此，在这种情况下，可以将声子定义为与偏移能量相关的稳定性算子的广义特征向量。 显示英文摘要

摘要： In some models, periodic configurations can be shown to be stable under, both, global $\ell^2$ or local perturbations. This is not the case for aperiodic media. The specific class of aperiodic media we are interested, in arise from taking two 2D periodic crystals and stacking them parallel at a relative twist. In periodic media, phonons are generalized eigenvectors for a stability operator acting on $\ell^2$, coming from a mechanical energy. The goal of our analysis is to provide phonons in the given class of aperiodic media with meaning. As rigorously established for the 1D Frenkel-Kontorova model and previously applied by one of the authors, we assume that we can parametrize minimizing lattice deformations w.r.t. local perturbations via continuous stacking-periodic functions, for which we previously derived a continuous energy density functional. Such (continuous) energy densities are analytically and computationally much better accessible compared to discrete energy functionals. In order to pass to an $\ell^2$-based energy functional, we also study the offset energy w.r.t. given lattice deformations, under $\ell^1$-perturbations. Our findings show that, in the case of an undeformed bilayer heterostructure, while the energy density can be shown to be stable under the assumption of stability of individual layers, the offset energy fails to be stable in the case of twisted bilayer graphene. We then establish conditions for stability and instability of the offset energy w.r.t. the relaxed lattice. Finally, we show that, in the case of incommensurate bilayer homostructures, i.e., two equal layers, if we choose minimizing deformations according to the global energy density above, the offset energy is stable in the limit of zero twist angle. Consequently, in this case, one can then define phonons as generalized eigenvectors w.r.t. the stability operator associated with the offset energy.