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.

arXiv comment15 pages, 3 figures

References in corpus (12)

• Topological Insulators
• Topological insulators and superconductors
• Non-Abelian Anyons and Topological Quantum Computation
• Classification of topological quantum matter with symmetries
• Topological phases of non-Hermitian systems
• Symmetry and Topology in Non-Hermitian Physics
• Bulk-boundary correspondence of topological insulators from their Green's functions
• Bulk-edge correspondence for two-dimensional topological insulators
• Bulk-edge correspondence for Chern topological phases: A viewpoint from a generalized index theorem
• Squaring the fermion: The threefold way and the fate of zero modes
• Topological Classification of Excitations in Quadratic Bosonic Systems
• Proof of Atiyah-Singer Index Theorem by Canonical Quantum Mechanics