Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher
Jianfeng Lu, Kevin D. Stubbs, Alexander B. Watson
备注：
64 pages, 15 figures. Many revisions: (1) Proof generalized WFs we construct are (a) WFs when system periodic (b) respectful of time-reversal, so gap conditions imply topological triviality (2) Proofs streamlined using integral kernel of P; generalized to more general position operators (3) Fixed error in constructive proof that PXP gaps implies analytic and periodic Bloch frame in 2d
Exponentially-localized Wannier functions (ELWFs) are an orthonormal basis of the Fermi projection of a material consisting of functions which decay exponentially fast away from their maxima. When the material is insulating and crystalline, conditions which guarantee existence of ELWFs in dimensions one, two, and three are well-known, and methods for constructing the ELWFs numerically are well-developed. We consider the case where the material is insulating but not necessarily crystalline, where much less is known. In one spatial dimension, Kivelson and Nenciu-Nenciu have proved ELWFs can be constructed as the eigenfunctions of a self-adjoint operator acting on the Fermi projection. In this work, we identify an assumption under which we can generalize the Kivelson-Nenciu-Nenciu result to two dimensions and higher. Under this assumption, we prove that ELWFs can be constructed as the eigenfunctions of a sequence of self-adjoint operators acting on the Fermi projection. We conjecture that the assumption we make is equivalent to vanishing of topological obstructions to the existence of ELWFs in the special case where the material is crystalline. We numerically verify that our construction yields ELWFs in various cases where our assumption holds and provide numerical evidence for our conjecture.
The Iterated Projected Position Algorithm for Constructing Exponentially Localized Generalized Wannier Functions for Periodic and Non-Periodic Insulators in Two Dimensions and Higher
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。
