Quantum mechanics tells us particles are waves at the same time. One may then naively expect that the quantum state is extended, especially with translational invariance. However, it is not always true. The states can be localized and gapped. A typical example is a Haldane phase of the translationally invariant spin 1 chain. There can be various gapped states of quantum spins protected by some symmetries. Here introducing a periodic synthetic dimension as a time that breaks the symmetry and connects gapped phases, one can realize a topological pump with a non-trivial Chern number. Changing the closed cycle in the parameter space, the plateau transition characterized by the Chern number occurs; with boundaries, one can justify the bulk-edge correspondence that governs the emergence of boundary degrees of freedom in the gapped quantum spins. Have a look at our paper, ”Plateau transitions of a spin pump and bulk-edge correspondence” by Yoshihito Kuno and Yasuhiro Hatsugai in Phys. Rev. B 104, 045113 (2021) . Also arXiv:2102.09325.