Diophantine Equation for the Rice–Mele Model: Topological Aspect of Filling Numbers and Associated Spatial Pump

We introduce a long-period generic spatial modulation into typical model of the Thouless pump, namely, Rice--Mele (RM) model, to examine lattice analog fermion charge in quantum field theory. derive Diophantine equation relating and pumped charge, which leads one-dimensional (1D) Streda formula Hall effect (QHE). This implies that an adiabatic change periodicity yields pump such rightmost is right by Chern number compared with leftmost charge. causes length chain integer, thus providing opportunity for direct measurement 1D systems.