Singularities of Berry connections inhibit the accuracy of the adiabatic approximation

Adiabatic approximation for quantum evolution is investigated addressing its dependence on the Berry connections that are functions of a slowly-varying parameter R  . When the Berry connections have singularities of type 1/Rσ1/Rσ with σ<1σ<1, the adiabatic fidelity converges to unit according to a power-law; When the singularity index σ becomes larger than one, adiabatic approximation breaks down. Two-level models are used to substantiate our theory.