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.
翻译:光源化地方化 Wannier 函数( ELWFs ) 是Fermi 投射由各种功能组成的材料的异常基础, 这些材料的衰减速度快于其峰值。 当材料是绝缘和晶状时, 保证ELFs在一、 二和三维维上存在的条件是众所周知的, 并且建造ELFS的方法在数字上是相当发达的。 我们认为, 材料是隔热的, 但不一定是晶状的, 更不为人所知。 在一个空间层面, Kivelson 和Nenciu- Nenciu- Nenciu 证明ELFs 能够作为在Fermi 投影上自行联合操作者的一种功能。 在这项工作中, 我们确定了一个假设, 我们可以将Kevilson- Nenciu- Nenciu- Nenciu 数字结果概括化为两个层面以上。 根据这个假设, 我们证明, ELFS 和NEFFSFs 的自动化操作者序列的序列功能可以作为在FFFFFFFS 的顶级模型中的一种结构, 我们推测, 我们的顶层的模型的模型的模型可以用来模拟的模型的模型可以用来模拟。