  The notion of topological (Thouless) pumping in topological phases is
traditionally associated with Laughlin's pump argument for the quantization of
the Hall conductance in two-dimensional (2D) quantum Hall systems. It relies on
magnetic flux variations that thread the system of interest without penetrating
its bulk, in the spirit of Aharonov-Bohm effects. Here we explore a different
paradigm for topological pumping induced, instead, by magnetic flux variations
$\delta\chi$ inserted through the bulk of topological phases. We show that
$\delta\chi$ generically controls the analog of a topological pump, accompanied
by robust physical phenomena. We demonstrate this concept of bulk pumping in
two paradigmatic types of 2D topological phases: integer and fractional quantum
Hall systems and topological superconductors. We show, in particular, that bulk
pumping provides a unifying connection between seemingly distinct physical
effects such as density variations described by Streda's formula in quantum
Hall phases, and fractional Josephson currents in topological superconductors.
We discuss the generalization of bulk pumping to other types of topological
phases.
