Unravelling the edge spectra of non-Hermitian Chern insulators
Unravelling the edge spectra of non-Hermitian Chern insulators
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DOI：
10.1103/physrevb.107.035101
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发表时间：
2023-01-03
影响因子：
3.7
通讯作者：
Zhao, Erhai
中科院分区：
文献类型：
--
作者：
Bartlett, James;Zhao, Erhai
非厄米陈氏绝缘子与厄米陈氏绝缘子有一个关键的区别：它们的边缘光谱非常丰富和混乱。例如，即使在主体光谱由两个陈氏数+/- 1的波段组成的简单情况下，根据模型参数的不同，平板几何形状中的边缘光谱可能在两个边缘上都有一个或两个边缘状态，或者只有一个边缘状态。这种对我们所熟悉的体-边对应关系的公然违背，使人们怀疑体陈恩数是否仍然是一个有用的拓扑不变量，并要求一个有效的理论，可以预测和解释从体哈密顿量中得到的无数边缘光谱，以恢复体-边对应关系。我们概述了如何建立这样的理论，以产生基于广义布里渊带（GBZ）的概念和块Toeplitz矩阵的渐近性质的边缘相图的透彻理解。通过求解和比较Qi-Wu-Zhang模型的三种非厄米推广来说明这一过程，Qi-Wu-Zhang模型是双带陈氏绝缘子的典型例子。我们惊奇地发现，在许多情况下，相边界和边缘状态的数量和位置可以解析得到。我们的分析还揭示了一个非厄米半金属相，其能量-动量谱形成了一个连续的膜，其边缘模式横过膜的孔或属。定义GBZ上的陈氏数的微妙之处，通常不是光滑流形，可能有奇点，用例子来证明。本文提出的方法可以推广到二维或三维的更复杂的非厄米绝缘体或半金属模型。
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 +/- 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.