The geometric phase of a near-degenerate two-level system

For a degenerate or near degenerate system its adiabatic evolution does not satisfy the conventional adiabatic condition. In the present work the adiabatic evolution of a near degenerate two-level system has been studied and a common result for the geometric phase has been obtained. The geometric phase of the two level graphene is taken as an example to be numerically computed. The numerical results show that a periodic adiabatic evolution around the degenerate point (the Dirac points of wave vectors of graphene) does not give any geometric phase but only a conventional dynamic phase. Away from the Dirac point, however, a geometric phase accumulates in the system after a full cycle of adiabatic evolution around a Dirac point of graphene, and finally converges to the conventional Berry phase.