分类 Dirac 哈密顿量中的体-边异常
Classifying bulk-edge anomalies in the Dirac Hamiltonian
AI总结：
本文研究了二维 Dirac 哈密顿量中体-边对应失效的异常现象，通过 Schubert 胞腔分解分类所有自伴边界条件，并利用无限能量处的异常绕数（ghost charge）实现了详尽的异常分类，发现异常普遍且稳定。
摘要：
我们研究了带质量项和大动量正则化的二维 Dirac 哈密顿量，并表明体-边对应失效。尽管存在明确定义的体拓扑指数——Chern 数——边缘模的数量却依赖于边界条件。这种异常的根源在于谱的无界性质。它通过散射理论中的 Levinson 定理被检测到，并通过无限能量处的异常绕数（被称为 ghost charge）进行量化。首先，我们利用 Grassmanian 的 Schubert 胞腔分解，对所有自伴边界条件进行了等价分类。然后，我们研究了哪些条件是异常的。我们在无限能量附近展开散射振幅，其中一个主导尺度捕获了渐近绕数。值得注意的是，这可以对每个自伴边界条件实现，从而得到详尽的异常分类。它表明异常是普遍存在且稳定的。在此过程中还揭示了具有 2 个 ghost charge 的边界条件。
英文摘要：
We study the Dirac Hamiltonian in dimension two with a mass term and a large momentum regularization, and show that bulk-edge correspondence fails. Despite a well defined bulk topological index --the Chern number--, the number of edge modes depends on the boundary condition. The origin of this anomaly is rooted in the unbounded nature of the spectrum. It is detected with Levinson's theorem from scattering theory and quantified via an anomalous winding number at infinite energy, dubbed ghost charge. First we classify, up to equivalence, all self-adjoint boundary conditions, using Schubert cell decomposition of a Grassmanian. Then, we investigate which ones are anomalous. We expand the scattering amplitude near infinite energy, for which a dominant scale captures the asymptotic winding number. Remarkably, this can be achieved for every self-adjoint boundary condition, leading to an exhaustive anomaly classification. It shows that anomalies are ubiquitous and stable. Boundary condition with a ghost charge of 2 is also revealed within the process.
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