论文标题
揭开非赫米特·克恩绝缘子的边缘光谱
Unravelling the edge spectra of non-Hermitian Chern insulators
论文作者
论文摘要
非热的Chern绝缘子在一个关键方面与其遗产表亲不同：它们的边缘光谱令人难以置信和混乱。例如，即使在简单的情况下，整体频谱由两个带有Chern Number $ \ pm 1 $的频段组成，平板几何形状中的边缘频谱在两个边缘上也可能具有一个或两个边缘状态，或者仅在其中一个边缘，取决于模型参数。这种公然违反熟悉的散装对应关系对散装Chern数量是否仍然是一个有用的拓扑不变的怀疑，并需要一种工作理论，可以预测和解释批量汉密尔顿的无数边缘光谱，以恢复散装散装对应关系。我们概述了如何建立这种理论，以根据广义布里鲁因区（GBZ）的概念（GBZ）和块Toeplitz矩阵的渐近性能对边缘相图产生透彻的理解。通过求解和比较Qi-wu-Zhang模型的三个非铁概括（两波段Chern绝缘子的规范示例）来说明该过程。我们发现，令人惊讶的是，在许多情况下，可以通过分析获得相边界和边缘状态的数量和位置。我们的分析还揭示了一个非暖半阶段，其能量光谱形成了连续的膜，边缘模式横向膜的孔或属。使用示例证明了Chern数在GBZ上定义Chern数字的微妙之处。此处介绍的方法可以推广到两个或三个维度中的非铁绝缘子或半法的更复杂模型。
Non-Hermitian Chern insulators differ from their Hermitian cousins in one key aspect: their edge spectra are incredibly rich and confounding. For example, even in the simple case where the bulk spectrum consists of two bands with Chern number $\pm 1$, the edge spectrum in the slab geometry may have one or two edge states on both edges, or only at one of the edges, depending on the model parameters. This blatant violation of the familiar bulk-edge correspondence casts doubt on whether the bulk Chern number can still be a useful topological invariant, and demands a working theory that can predict and explain the myriad of edge spectra from the bulk Hamiltonian to restore the bulk-edge correspondence. We outline how such a theory can be set up to yield a thorough understanding of the edge phase diagram based on the notion of the generalized Brillouin zone (GBZ) and the asymptotic properties of block Toeplitz matrices. The procedure is illustrated by solving and comparing three non-Hermitian generalizations of the Qi-Wu-Zhang model, a canonical example of two-band Chern insulators. We find that, surprisingly, in many cases the phase boundaries and the number and location of the edge states can be obtained analytically. Our analysis also reveals a non-Hermitian semimetal phase whose energy-momentum spectrum forms a continuous membrane with the edge modes transversing the hole, or genus, of the membrane. Subtleties in defining the Chern number over GBZ, which in general is not a smooth manifold and may have singularities, are demonstrated using examples. The approach presented here can be generalized to more complicated models of non-Hermitian insulators or semimetals in two or three dimensions.
