论文标题
紧凑的支持Wannier功能和严格的本地投影仪
Compactly Supported Wannier Functions and Strictly Local Projectors
论文作者
论文摘要
最大程度地局部的Wannier功能有助于理解晶体材料的许多特性。在没有拓扑障碍物的情况下，它们至少是指数性的。在某些情况下，例如平坦的汉密尔顿人，可以构建更具本地化的威尼尔功能，从而使它们得到紧凑的支持，从而在相应的位置之外具有零支持。在哪些一般条件下，可以构建紧凑的Wannier功能？我们在本文中回答了这个问题。具体而言，我们表明，在一维非相互作用的紧密结合模型中，投影操作员的严格位置是一个必要且充分的条件，可以通过紧凑的正交基础来跨越子空间，而与晶格翻译对称性无关。对于任何严格的本地投影仪，我们提供了获得此类基础的程序。对于更高的维系统，我们讨论了一些其他条件，在这些条件下，占领子空间由紧凑的正交基础跨越，并表明在许多情况下，相应的投影仪在拓扑上是微不足道的。我们还表明，当且仅当对于任何选择的轴时，任意维度的投影仪是严格的局部局部性，其图像是由沿该轴紧凑支撑的混合Wannier函数跨越的。
Wannier functions that are maximally localized help in understanding many properties of crystalline materials. In the absence of topological obstructions, they are at least exponentially localized. In some cases such as flat-band Hamiltonians, it is possible to construct Wannier functions that are even more localized, so that they are compactly supported thus having zero support outside their corresponding locations. Under what general conditions is it possible to construct compactly supported Wannier functions? We answer this question in this paper. Specifically, we show that in 1d non-interacting tight-binding models, strict locality of the projection operator is a necessary and sufficient condition for a subspace to be spanned by a compactly supported orthogonal basis, independent of lattice translation symmetry. For any strictly local projector, we provide a procedure for obtaining such a basis. For higher dimensional systems, we discuss some additional conditions under which an occupied subspace is spanned by a compactly supported orthogonal basis, and show that the corresponding projectors are topologically trivial in many cases. We also show that a projector in arbitrary dimensions is strictly local if and only if for any chosen axis, its image is spanned by hybrid Wannier functions that are compactly supported along that axis.
