Abstract
Berry's phase, which is associated with the slow cyclic motion with a finite period, looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the level crossing point in the parameter space in an exactly solvable model. This topology change of Berry's phase is visualized as a result of lensing effect; the monopole supposed to be located at the level crossing point appears at the displaced point when the variables of the model deviate from the precisely adiabatic movement. The effective magnetic field generated by Berry's phase is determined by a simple geometrical consideration of the magnetic flux coming from the displaced Dirac monopole.
Highlights
The notion of topology and topological phenomena have become common in various fields in physics
Berry’s phase, which is associated with the slow cyclic motion with a finite period, looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the level crossing point in the parameter space in an exactly solvable model
We here report on a more quantitative description of the magnetic field generated by Berry’s phase, which is essential to understanding the motion of a particle placed in the monopolelike field, together with a surprising connection of the topology change of Berry’s phase with the formal geometrical movement of Dirac’s monopole in the parameter space caused by the nonadiabatic variation of parameters
Summary
The notion of topology and topological phenomena have become common in various fields in physics. It is surprising that one encounters Dirac’s magnetic monopolelike topological phase [4] essentially at each level crossing point for the sufficiently slow cyclic motion in quantum mechanics [2,5]. We here report on a more quantitative description of the magnetic field generated by Berry’s phase, which is essential to understanding the motion of a particle placed in the monopolelike field, together with a surprising connection of the topology change of Berry’s phase with the formal geometrical movement of Dirac’s monopole in the parameter space caused by the nonadiabatic variation of parameters. The second term in the exponential of the exact solution (3) is customarily called Berry’s phase which is defined by a potential-like object (or connection) This potential describes an azimuthally symmetric static magnetic monopolelike object in the present case. The flux vanishes for η < 1 (i.e., B < μπT) and the object changes to a dipole [6]
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