Topological indexes in symmetry preserving dynamics

The quench dynamics of topological phases have received intensive investigations in recent years. In this work, we prove exactly that the topological invariants for both $\mathbb{Z}$ and $\mathbb{Z}_2$ indexes are independent of time in symmetry preserving dynamics. We first reach this conclusion by a direct relation between the time derivative of Berry connection and the Hamiltonian energy based on the time dependent Hellman-Feynman theorem, with which we show exactly that the topological indexes for systems without and with time reversal symmetry are unchanged during evolution. In contrast, the geometry phase without symmetry protection in a closed parameter space can change dramtically with time, as revealed from the parameterized Landau-Zener model. Then we interpret this result by showing that the time dependent wave function is essentially the eigenvector of an auxiliary Hamiltonian, which has exactly the same spectra and symmetries as the original Hamiltonian. For this reason, the adiabatic evolution between the original and auxiliary Hamiltonian will not lead to gap closing and reopening, thus the topological indexes are independent of time. This result has generality and can be applied to models with other symmetries and dimensions, and may even be applied to gapless phases. Finally, possible ways to outreach this rigorous result are discussed.