二维自伴边界与界面问题的体—边对应
A bulk--edge correspondence for two-dimensional self-adjoint boundary and interface problems
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中文总结 AI 辅助
本文针对二维拓扑绝缘体边界或界面问题，证明量子化输运依赖于自伴实现选择，建立了体—边对应公式，并推广到任意透射条件，通过八个例子验证理论。
中文摘要 AI 辅助
本文分析了占据半平面的二维拓扑绝缘体边界处或分隔两个绝缘体的界面处的量子化非对称输运。与在欧几里得平面上提出的问题不同，终止样品需要选择自伴实现 $H_L$，我们证明量子化输运依赖于该选择。模拟非对称输运的物理可观测量是边缘或界面电流 $\sigma_I[H_L]=\Tr\\,i[H_L,P]\varphi'(H_L)$。我们通过两条路径证明其良定义性。对于 Shapiro--Lopatinski 椭圆实现，我们使用 Boutet de Monvel 微积分的估计证明电流可观测量是良定义的迹，$\mathcal I=2\pi\sigma_I[H_L]$ 是 Fredholm 指标，并且它在形变以及算符及其终止的紧支撑改变下稳定。对于沿边缘平移不变的算符，当边缘谱在大动量下逃逸体隙时，谱分析定义 $\mathcal I$。此时 $\mathcal I$ 也是谱流和 Maslov 指标。我们的主要定理是体—边对应 \\[ \mathcal I=\cR-\cR_L+\eta\bigl(L_\infty,\Lm^\Phi(+)\bigr)-\eta\bigl(L_\infty,\Lm^\Phi(-)\bigr), \\] 我们确定了 $\mathcal I$ 的范围。我们将公式推广到具有任意透射条件的界面。在透明条件下，该不变量等于欧几里得平面的体差不变量。八个例子说明了理论发现。
英文摘要
This paper analyzes the quantized asymmetric transport along the boundary of a two-dimensional topological insulator occupying a half plane, or along the interface separating two insulators. In contrast with problems posed on the Euclidean plane, terminating a sample requires a choice of self-adjointrealization, $H_L$, and we show that a quantized transport depends on that choice. The physical observable modeling the asymmetric transport is the edge, or interface, current $σ_I[H_L]=\Tr\,i[H_L,P]φ'(H_L)$. It is shown to be well defined along two routes. For Shapiro--Lopatinski elliptic realizations, we use estimates from Boutet de Monvel's calculus to show that the current observable is a well-defined trace, that $\mathcal I=2πσ_I[H_L]$ is a Fredholm index, and that it is stable under deformations and compactly supported changes of the operator and of its termination. For translation invariant operators along the edge, a spectral analysis defines $\mathcal I$ when the edge spectrum escapes a bulk gap at large momenta. $\mathcal I$ is then also a spectral flow and a Maslov index. Our main theorem is a bulk--edge correspondence \[ \mathcal I=\cR-\cR_L+η\bigl(L_\infty,\Lm^Φ(+)\bigr)-η\bigl(L_\infty,\Lm^Φ(-)\bigr), \] We determine the range of $\mathcal I$. We extend the formula to interfaces with arbitrary transmission conditions. At the transparent condition, the invariant equals the bulk-difference invariant of the Euclidean plane. Eight examples illustrate the theoretical findings.
发表机构
- University of Chicago（芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。
