Applications of K-Theory in Materials Science

Heisenberg formulation of quantum mechanics puts a focus on the physical observables which are organized as operator algebras. This approach and point of view has recently penetrated into metamaterials science. The goal of this talk is to explain this statement and to demonstrate the power of the theoretical tools that comes from the field of operator algebras, such as K-theory. As we shall see, the dynamical matrices of various classes of metamaterials generate specific topological algebras that can be computed explicitly. This will be exemplified using the class of periodic metamaterials, then that of quasi-periodic metamaterials and, lastly, that of metamaterials symmetric to a full space group. The talk will then introduce the notions of complete topological invariants and of stable homotopy. The equivalence class of a band projection with respect to the stable homotopy defines the complete topological invariant that can be associated with that projection and these equivalence classes lead to the K-group of the algebra of dynamical matrices. The last part of the talk exemplifies the K-groups for classes of metamaterials mentioned above. As we shall see, once the K-groups and their generators are known, one can produce a complete list of topological models which supply all topological phases supported by a particular class of metamaterials. Additional ways of using the K-groups will be presented.