Mathematical and Physical Aspects of Topologically Protected States

On the Bulk-Boundary Correspondence Principle for Aperiodic Systems: A K-Theoretic Formulation

Emil Prodan

Yeshiva University


Concepts related with discrete aperiodic structures, such as the Hull, non-commutative Brillouin torus, etc., will be introduced and exemplified on simple cases, ranging from incommensurate lattices to disordered quasi-crystals. This will be followed by the introduction of the algebra of physical observables for systems with a boundary and of its ideal which determines the boundary conditions. Then the engine of the bulk-boundary principle can be expressed as an exact sequence between these algebras, which sets in motion a six-term exact sequence between their K-theories. From the expression of the connecting maps, one can then see explicitly when topological boundary states are to be expected. Examples will be concretely worked out.