Traditional physics classifies phases of matter (like liquids freezing into solids or magnets aligning) using the Landau-Ginzburg paradigm, which relies on spontaneous symmetry breaking and local order parameters. Topological phases of matter break this paradigm. They do not rely on symmetry breaking; instead, they are characterized by global topological invariants, robust ground-state degeneracy, and long-range quantum entanglement.

The main categories of topological materials include:

• Topological Insulators (TIs): These materials act as electrical insulators in their interior (bulk) but feature highly conductive states on their boundaries (surfaces or edges). These surface states are protected by mathematical topology and specific symmetries (like time-reversal), making the electron flow remarkably robust against scattering by  ...