Cracking the code of topological boundaries: Exact solutions for zero modes

What is it about?

Topological physics uncovers how global properties of materials lead to remarkable boundary phenomena, such as the appearance of edge modes in topological insulators or Majorana zero modes in superconductors. These boundary modes promise revolutionary applications in quantum computing and beyond. While topology guarantees the existence of these modes, it says little about their precise physical properties and their spatial structure. This is where differential equations come into play: However, solving them for realistic systems has been a daunting task. In this study, we go beyond approximations and present exact analytical solutions to the differential equations describing zero modes at smooth domain walls, where the transitions between phases are gradual rather than sharp. These solutions reveal how the spatial behavior of the boundary modes is governed by three measurable length scales: the domain wall width, the decay length, and the oscillation wavelength. Depending on these parameters, the modes transition from being "featureless" (having no hair) to exhibiting intricate spatial features (having "short" or "long hair"). The paper also unearths an unexpected duality between topological zero modes and nontopological Shockley modes, linking two seemingly distinct phenomena through a unified mathematical framework. This discovery refines our understanding of the interplay between topology and spatial localization, offering new insights into edge modes in topological insulators and superconductors. By solving these equations exactly, we also provide practical tools for experimentalists to probe and control these modes and for their potential technological applications.

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Photo by Ben Wicks on Unsplash

Photo by Ben Wicks on Unsplash

Why is it important?

This work is significant because it provides exact analytical solutions for zero modes at smooth domain walls, bridging the gap between theoretical predictions and realistic systems. By identifying universal relations between measurable quantities like domain wall width, decay rate, and oscillation momentum, it quantifies the bulk-boundary correspondence in practical terms. The discovery of a duality between topological zero modes and nontopological Shockley modes unifies two distinct phenomena, offering new conceptual insights. These findings advance our understanding of localized modes in topological systems and lay the groundwork for optimizing technologies like quantum computing, where zero modes play a key role.