In this chapter, we briefly introduce the evolution of symmetry as a mathematical concept applied to physical systems and lay the mathematical groundwork for discussion of topological physics. We explain how topological phases, like the Berry phase, can be obtained from a gauge symmetry of a quantum system. Also, we introduce numerical tools (e.g., Chern numbers, Wilson loops) for topological analysis of chemical solids based on the crystal structure and corresponding electronic structure.
Part of the book: Symmetry (Group Theory) and Mathematical Treatment in Chemistry