  An extended Bose-Hubbard model (EBHM) with three- and four-body constraints
can be feasible in cold atoms in an optical lattice. A rich phase structure
including various symmetry-protected topological (SPT) phases is obtained
numerically with suitable parameter settings and particle filling. The SPT
phase is characterized by the Berry phase as a local topological order
parameter and the structure of the entanglement spectrum (ES). Based on the
presence of various topological phases, separated by gapless phase boundaries,
the EBHM exhibits various bosonic topological pumps, which are constructed by
connecting the different SPT phases without gap closing. The bulk topological
pumps exhibit the plateau transitions characterized by many-body Chern numbers.
For the system with boundary, the center of mass (CoM) under grand canonical
ensemble elucidates the contributions of multiple edge states and reveals the
topology of the system. We demonstrate that the interacting bosonic pumps obey
the bulk-edge-correspondence.
