Topological pump of $SU(Q)$ quantum chain and Diophantine equation

A topological pump of the $SU(Q)$ quantum chain is proposed associated with a current due to a local $[U(1)]^{\otimes Q}$ gauge invariance of colored fermions. The $SU(Q)$ invariant dimer phases are characterized by the $Z_Q$ Berry phases as a topological order parameter with a $d$-dimensional twist space ($d=Q-1$) as a synthetic Brillouin zone. By inclusion of the symmetry breaking perturbation specified by a rational parameter $\Phi=P/Q$, the pump, that encloses around the phase boundary, is characterized by the $Q$ Chern numbers associated with the currents due to uniform infinitesimal twists. The analysis of the systems under the open/periodic/twisted boundary conditions clarifies the bulk-edge correspondence of the pump where the large gauge transformation generated by the center of mass (CoM) plays a central role. An explicit formula for the Chern number is given by using the Diophanine equation. Numerical demonstration by the exact diagonalization and the DMRG for finite systems ($Q=3,4$ and $5$) have been presented to confirm the general discussions for low energy spectra, edge states, CoM's, Chern numbers and the bulk-edge correspondence. A modified Lieb-Schultz-Mattis type argument for the general $SU(Q)$ quantum chain is also mentioned.