Proof of bulk-edge correspondence for band topology by Toeplitz algebra

Abstract

We rigorously yet concisely prove the bulk-edge correspondence for general $d$-dimensional ($d$D) topological insulators in complex Altland-Zirnbauer classes, which states that the bulk topological number equals to the edge-mode index. Specifically, an essential formula is discovered that links the quantity expressed by Toeplitz algebra, i.e., hopping terms on the lattice with an edge, to the Fourier series on the bulk Brillouin zone. We then apply it to chiral models and utilize exterior differential calculations, instead of the sophisticated \emph{K}-theory, to show that the winding number of bulk system equals to the Fredholm index of 1D edge Hamiltonian, or to the sum of edge winding numbers for higher odd dimensions. Moreover, this result is inherited to the even-dimensional Chern insulators as each of them can be mapped to an odd-dimensional chiral model. It is revealed that the Chern number of bulk system is identical to the spectral flow of 2D edge Hamiltonian, or to the negative sum of edge Chern numbers for higher even dimensions. Our methods and conclusions are friendly to physicists and could be easily extended to other physical scenarios.

Figures (3)

• Figure 1: Zero-mode domain of the edge Hamiltonian for $\left(2n+1\right)$D chiral models ($n\geq1$). The domain of zero modes (red) in BZ $T^{2n}$ consists of smooth closed submanifolds (or discrete points) $D_{i}$ contained in a neighborhood $B\left(D_{i};\rho\right)$ (blue), with $\rho$ the distance between the boundary of neighborhood $\partial B\left(D_{i};\rho\right)$ (green) and $D_{i}$. Here $\partial B\left(D_{i};\rho\right)$ is a $\left(2n-1\right)$D closed manifold.
• Figure 2: Treatment of Eq. (\ref{['eq:ind_oddD']}) for overlapping $D_{i}$. In this case, the Stokes' formula gives $C_{n-1}\text{Tr}\int_{T^{2n}\backslash D}\left(Q^{-1}\text{d}Q\right)^{2n}=\sum_{i}C_{n-1}\text{Tr}\int_{X_{i}}\left(Q^{-1}\text{d}Q\right)^{2n-1}$. Due to Eq. (\ref{['eq:limit']}), $\text{Tr}\left(Q^{-1}\text{d}Q\right)^{2n-1}=\text{tr}\left(g_{i}^{-1}\text{d}g_{i}\right)^{2n-1}$ for the most area of $X_{i}$. Besides, $X_{i}$ covers the most area of $\partial B\left(D_{i};\rho\right)$. Therefore, under the limit $\rho\rightarrow0$, $\sum_{i}C_{n-1}\text{Tr}\int_{X_{i}}\left(Q^{-1}\text{d}Q\right)^{2n-1}$ could be replaced by $\sum_{i}C_{n-1}\int_{\partial B\left(D_{i};\rho\right)}\text{tr}\left(g_{i}^{-1}\text{d}g_{i}\right)^{2n-1}=\sum_{i}\deg_{i}=\text{ind}$, which means that Eq. (\ref{['eq:ind_oddD']}) still holds.
• Figure 3: Zero-mode domains of the edge Hamiltonian for $\left(2n\right)$D Chern insulators and the constructed $\left(2n+1\right)$D chiral models. The zero-mode domain of $H(\tilde{\bm{k}})$ in BZ $T^{2n-1}$ consists of smooth closed manifolds $D_{i}$ (red) contained in a $\left(2n-1\right)$D neighborhood $B\left(D_{i};\rho\right)$ (dark blue). Correspondingly the zero-mode domain of $Q(\tilde{\bm{k}},\omega)=H(\tilde{\bm{k}})-{\rm i}\omega I$ in $T^{2n-1}\times S^{1}$ consists of $D_{i}^{\prime}=\left(D_{i},0\right)$ contained in a $\left(2n\right)$D neighborhood $B^{\prime}\left(D_{i}^{\prime};\rho\right)$ (light blue). Here $\partial B\left(D_{i};\rho\right)$ and $\partial B^{\prime}\left(D_{i};\rho\right)$ are a $\left(2n-2\right)$D and a $\left(2n-1\right)$D closed manifold, respectively.