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.

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