Motivated by recent experiments demonstrating the creation of atomically sharp interfaces between hexagonal sapphire and cubic ${\mathrm{SrTiO}}_3$ with finite twist, we here develop and study a general electronic band theory for this novel class of moiré heterostructures. We take into account the three-dimensional nature of the two crystals, allow for arbitrary combinations of Bravais lattices, finite twist angles, and different locations in momentum space of the low-energy electronic bands of the constituent materials. We analyze the general condition for a well-defined crystalline limit in the interface electron system and classify the associated “crystalline reference points.'' We discuss this in detail for the example of the two-dimensional lattice planes being square and triangular lattices on the two sides of the interface; this reveals nontrivial reference points at finite twist angle and lattice mismatch, leading to a unique form of magic angles, which we refer to as “geometric magic angles.” We further show that band structures of mixed dimensionality naturally emerge, where quasi-one- and two-dimensional pockets coexist. Explicit computations for different bulk Bloch Hamiltonians yield a collection of interesting features, such as isolated bands localized at interfaces of nontopological insulators, Dirac cones, van Hove singularities, a nontrivial evolution of the band structures with Zeeman field, and topological interface bands. Our work illustrates the potential of these heterostructures and is anticipated to provide the foundation for moiré interface design and for the analysis of correlated physics in these systems.

中文翻译：

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界面超晶格异质结构的能带理论


受到近期实验的启发，该实验证明了六方蓝宝石和立方蓝宝石之间形成了原子级锐利界面 ${\mathrm{SrTiO}}_3$ 通过有限扭曲，我们在这里开发和研究这种新型莫尔异质结构的通用电子能带理论。我们考虑到两种晶体的三维性质，允许布拉维晶格、有限扭转角以及组成材料的低能电子带动量空间中的不同位置的任意组合。我们分析了界面电子系统中明确的晶体极限的一般条件，并对相关的“晶体参考点”进行了分类。我们以二维晶格平面为正方形和三角形晶格的示例详细讨论了这一点。接口的两侧；这揭示了有限扭转角和晶格失配处的重要参考点，从而产生了独特形式的魔角，我们将其称为“几何魔角”。我们进一步表明，混合维度的能带结构自然出现，其中准一维和二维口袋共存。对不同块体布洛赫哈密顿量的显式计算产生了一系列有趣的特征，例如位于非拓扑绝缘体、狄拉克锥、范霍夫奇点的界面处的孤立能带、塞曼场能带结构的非平凡演化以及拓扑界面能带。我们的工作说明了这些异质结构的潜力，并有望为莫尔界面设计和这些系统中相关物理分析提供基础。