Unified bulk-boundary correspondence for band insulators

Abstract

The bulk-boundary correspondence, a topic of intensive research interest over the past decades, is one of the quintessential ideas in the physics of topological quantum matter. Nevertheless, it has not been proven in all generality and has in certain scenarios even been shown to fail, depending on the boundary profiles of the terminated system. Here, we introduce bulk numbers that capture the exact number of in-gap modes, without any such subtleties in one spatial dimension. Similarly, based on these 1D bulk numbers, we define a new 2D winding number, which we call the pole winding number, that specifies the number of robust metallic surface bands in the gap as well as their topological character. The underlying general methodology relies on a simple continuous extrapolation from the bulk to the boundary, while tracking the evolution of Green's function's poles in the vicinity of the bulk band edges. As a main result we find that all the obtained numbers can be applied to the known insulating phases in a unified manner regardless of the specific symmetries. Additionally, from a computational point of view, these numbers can be effectively evaluated without any gauge fixing problems. In particular, we directly apply our bulk-boundary correspondence construction to various systems, including 1D examples without a traditional bulk-boundary correspondence, and predict the existence of boundary modes on various experimentally studied graphene edges, such as open boundaries and grain boundaries. Finally, we sketch the 3D generalization of the pole winding number by in the context of topological insulators.

19 pages, 11 figures

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• Subdimensional topologies, indicators and higher order phases
• The interacting Rice-Mele model: bulk and boundaries
• Odd-Frequency Pairs in Chiral Symmetric Systems: Spectral Bulk-Boundary Correspondence and Topological Criticality
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• Surface Green's functions and boundary modes using impurities: Weyl semimetals and topological insulators
• Extended Bloch theorem for topological lattice models with open boundaries
• Exact edge, bulk and bound states of finite topological systems
• Photo-induced electronic and spin topological phase transitions in monolayer bismuth
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• Robust edge states induced by electron-phonon interaction in graphene nanoribbons
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• Higher-order topological superconductors characterized by Fermi level crossings
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• An elementary proof and detailed investigation of the bulk-boundary correspondence in the generic two-band model of Chern insulators
• Suppressing backscattering of helical edge modes with a spin bath
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• Robust boundary Luttinger surfaces in topological band structures