# Topological indexes in symmetry preserving dynamics. (arXiv:1802.02931v1 [quant-ph])

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.