Krylov complexity and Berry phase in quantum adiabatic dynamics

The connection between Krylov complexity and Berry phase in adiabatic dynamics is investigated under the instantaneous eigenstate basis of a slowly evolving spin system. Adiabatic dynamics force the Krylov complexity to vanish if the initial Krylov basis is an instantaneous eigenstate of the Hamiltonian. Nevertheless, we demonstrate that the Krylov complexity will be nonvanishing if the initial Krylov basis is a superposition state rather than an eigenstate. Time evolution of Krylov complexity will behave periodically or quasi-periodically depending on the controlling parameters. In particular, for a single qubit system with constant parameters, the Krylov complexity will oscillate harmonically in time with the frequency relevant to the combination of the strength of the external field and the geometric Berry phase.