Unified bulk-boundary correspondence for band insulators

TL;DR

This paper introduces a unified framework for bulk-boundary correspondence in insulators, defining new bulk and boundary topological invariants that accurately predict boundary modes across various systems, including challenging cases.

Contribution

The authors develop a general methodology for bulk-boundary correspondence in 1D and 2D insulators, introducing bulk numbers and the pole winding number that work universally regardless of symmetries.

Findings

Exact in-gap mode counting in 1D systems

Robust surface bands characterized by the pole winding number

Application to graphene edges and prediction of boundary modes

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.