Topological classification of adiabatic processes

Certain band insulators allow for the adiabatic pumping of quantized charge or spin for special time dependences of the Hamiltonian. These topological pumps are closely related to two-dimensional topological insulating phases of matter upon rolling the insulator up to a cylinder and threading it with a time-dependent flux. In this paper we extend the classification of topological pumps to the Wigner-Dyson and chiral classes, coupled to multichannel leads. The topological index distinguishing different topological classes is formulated in terms of the scattering matrix of the system. We argue that similar to topologically nontrivial insulators, topological pumps are characterized by the appearance of protected gapless end states during the course of a pumping cycle. We show that this property allows for the pumping of quantized charge or spin in the weak-coupling limit. Our results may also be applied to two-dimensional topological insulators, where they give a physically transparent interpretation of the topologically nontrivial phases in terms of scattering matrices.