In particular, I want to understand what fundamental (mathematical) structure gives rise to topological mechanical metamaterials, or topological protected states in general. According to a recent PNAS paper "Topological mechanics of gyroscopic metamaterials", gapped spectrum of excitations plays an important role:
A vast range of mechanical structures, including bridges, covalent glasses, and conventional metamaterials, can be ultimately modeled as networks of masses connected by springs. Recent studies have revealed that despite its apparent simplicity, this minimal setup is sufficient to construct topologically protected mechanical states that mimic the properties of their quantum analogs. This follows from the fact that, irrespective of its classic or quantum nature, a periodic material with a gapped spectrum of excitations can display topological behavior as a result of the nontrivial topology of its band structure.
Can anyone give me an intuitive or mathematical explanation of why this is the case? I would appreciate if the explanation doesn't rely too much on jargons from quantum mechanics or solid state physics, as I am no expert in those fields and using the language of a specific type of physics could sometimes obscure the underlying mathematical structure (but please feel free to give electronic or optical systems as examples). Thank you.