构造二维及高维周期和非周期绝缘体指数局部化广义Wannier函数的迭代投影位置算法
The Iterated Projected Position Algorithm for Constructing Exponentially Localized Generalized Wannier Functions for Periodic and Non-Periodic Insulators in Two Dimensions and Higher
摘要：
定域碱基在理解电子结构中起着重要的作用。在周期性绝缘体中，局域基的自然选择是由Wannier函数给出的，它依赖于被称为规范变换的幺正变换的选择。在过去的几十年里，已经有许多工作集中在优化规范的选择，使相应的Wannier函数最大限度地本地化或反映一些基本系统的对称性。在这项工作中，我们考虑完全非周期性的材料，通常的Wannier函数没有很好地定义和规范优化是不可能的。为了解决周期和非周期系统中指数定域广义Wannier函数的计算问题，我们讨论了迭代投影位置（IPP）算法。IPP算法基于矩阵对角化，因此与基于优化的方法不同，它不需要初始化，也不会陷入局部最小值。此外，IPP算法是保证了严格的分析，以产生指数本地化的功能，在一定的温和的假设。我们数值证明，IPP算法可以用来计算指数本地化基地的Halaldom模型，凯恩-梅勒模型（在两个$\mathbb{Z}_2$不变的偶和$\mathbb{Z}_2$不变的奇相位），和准晶格上的$p_x + i p_y$模型。
英文摘要：
Localized bases play an important role in understanding electronic structure. In periodic insulators, a natural choice of localized basis is given by the Wannier functions which depend a choice of unitary transform known as a gauge transformation. Over the past few decades, there have been many works which have focused on optimizing the choice of gauge so that the corresponding Wannier functions are maximally localized or reflect some symmetry of the underlying system. In this work, we consider fully non-periodic materials where the usual Wannier functions are not well defined and gauge optimization is impossible. To tackle the problem of calculating exponentially localized generalized Wannier functions in both periodic and non-periodic system we discuss the "Iterated Projected Position (IPP)" algorithm. The IPP algorithm is based on matrix diagonalization and therefore unlike optimization based approaches it does not require initialization and cannot get stuck at a local minimum. Furthermore, the IPP algorithm is guaranteed by a rigorous analysis to produce exponentially localized functions under certain mild assumptions. We numerically demonstrate that the IPP algorithm can be used to calculate exponentially localized bases for the Haldane model, the Kane-Mele model (in both $\mathbb{Z}_2$ invariant even and $\mathbb{Z}_2$ invariant odd phases), and the $p_x + i p_y$ model on a quasi-crystal lattice.
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