非公度材料中声子的数学基础
Mathematical foundations of phonons in incommensurate materials
AI总结：
本文研究扭转双层等非公度材料中声子的严格定义，通过连续堆叠周期能量密度与 ℓ¹ 扰动下的偏移能量分析稳定性条件，并证明零扭转角同质双层可定义广义特征向量型声子。
摘要：
在某些模型中，可以证明周期构型在全局 ℓ² 扰动或局部扰动下均保持稳定。非周期介质则并非如此。我们感兴趣的特定一类非周期介质，来自将两个二维周期晶体平行堆叠并施加相对扭转。在周期介质中，声子是作用于 ℓ² 上的稳定性算子的广义特征向量，该算子源于机械能。我们分析的目标是为给定类别的非周期介质中的声子赋予意义。正如针对一维 Frenkel–Kontorova 模型严格建立、并由其中一位作者此前应用的那样，我们假设可以通过连续的堆叠周期函数来参数化相对于局部扰动的最小化晶格形变，而且我们此前已为这类函数推导了连续能量密度泛函。与离散能量泛函相比，此类（连续）能量密度在解析和计算上都更容易处理。为了过渡到基于 ℓ² 的能量泛函，我们还研究了在 ℓ¹ 扰动下相对于给定晶格形变的偏移能量。研究结果表明，在未形变双层异质结构的情形中，虽然在各单层稳定的假设下可以证明能量密度稳定，但在扭转双层石墨烯中，偏移能量并不稳定。随后，我们建立了相对于弛豫晶格的偏移能量稳定与不稳定的条件。最后，我们证明，在非公度双层同质结构（即两个相同层）的情形中，如果按照上述全局能量密度选择最小化形变，则在零扭转角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。
英文摘要：
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.
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