Topological Constraints on Localized Modes: From Electrons in Crystals to Elastic Waves in Structured Media

The ability to construct spatially localized modes that respect the symmetries of a periodic structure is fundamental in both electronic and mechanical systems. In crystalline materials, these modes, known as Wannier functions, play a key role in modeling and simulations. However, in systems with nontrivial topological properties, such localized representations may not exist due to hidden global constraints imposed by the material’s symmetry and topology.

In this talk, we investigate when such obstructions arise and how they can be detected through a symmetry-aware mathematical framework. We demonstrate this process using the Haldane model, a well-known example of a topological insulator, and present a systematic, algorithmic method to construct symmetry-adapted, exponentially localized Wannier functions whenever possible.

Finally, we explore how these ideas extend beyond electronic systems. In particular, we show how similar symmetry and topology-based constraints govern the behavior of elastic waves in mechanical metamaterials, enabling the design of structures that support robust, localized vibrational modes. These insights pave the way for advanced control over wave propagation in engineered mechanical systems.