Relation between Berry phases and entanglement besides convergence of levels for two entangled spin-1/2 particles in magnetic fields

We study the effects of an extra static magnetic field, coupling with one of the two entangled spin-1/2 particles, on Berry phases and entanglement of the adiabatic eigensatates of the system, while the other spin interacts with a rotating magnetic field satisfying the adiabatic condition. This static magnetic field can be used for controlling the Berry phases and the entangled state of the system. The relation of the Berry phases and entanglement to Dzyaloshinski-Moriya interaction, coupling coefficient, and the magnetic fields are also investigated. The results demonstrate a close relationship between the Berry phases and pairwise entanglement (as measured by the so-called entanglement of formation). We show that reversing the sign of the static magnetic field can cause exchanges of the Berry phases and entanglement between the adiabatic states. It is illustrated that the geometric phases and entanglement are good indicators to detect the convergence of the levels and each convergence of levels corresponds to abrupt changes in the Berry phases and entanglement