Topology change from a monopole to a dipole in Berry’s phase

Abstract

The smooth topology change of Berry's phase from a Dirac monopole-like configuration to a dipole configuration, when one approaches the monopole position in the parameter space, is analyzed in an exactly solvable model. A novel aspect of Berry's connection A k is that the geometrical center of the monopole-like configuration and the origin of the Dirac string are displaced in the parameter space. Gauss' theorem S (∇ × A) · d S = V ∇ · (∇ × A)dV = 0 for a volume V which is free of singularities shows that a combination of the monopole-like configuration and the Dirac string is effectively a dipole. The smooth topology change from a dipole to a monopole with a quantized magnetic charge e M = 2π takes place when one regards the Dirac string as unobservable if it satisfies the Wu-Yang gauge invariance condition. In the transitional region from a dipole to a monopole, a half-monopole appears with an observable Dirac string, which is analogous to the Aharonov-Bohm phase of an electron for the magnetic flux generated by the Cooper pair condensation. The main topological features of an exactly solvable model are shown to be supported by a generic model of Berry's phase.

A new magnetic monopole inspired by Berry's phase

“…In the analysis of topology change, the transition from η > 1 to η < 1 through the critical value η = 1 is important. In Fig.2, we thus show the relation between α(θ, η) and θ at the transition region near η = 1 given by (3). For the parameters η = 1 ± ǫ with a small positive ǫ, the value α(θ, η) departs from the common value 1 2 θ assumed at around θ = 0 and splits into two branches for the values of the parameter θ close to θ = π.…”

“…We start with the analysis of the parameter α(θ, η). In Fig.1, we show the relation between θ and tan α(θ, η) for the case 0 ≤ η < 1 given by (3). For this parameter range, we have a singularity at cos θ 0 = −η in the denominator of (3).…”

“…and A θ = A r = 0 with a magnetic charge e M . This is a translation of a monopolelike object associated with Berry's phase in the parameter space [3] to the one in the real space. The variable Θ(θ, η) is defined by…”

A path integral derivation of the equations of anomalous Hall effect

“…This general parametrization of Berry's phase(3.23) is shown to be valid for an exactly solvable model of Berry's phase of a two-level crossing problem[24].…”

“…The trivial Berry's phase (3.21) also implies that Berry's phase has no singularity at the origin | p| → 0 with fixed T , although a genuine Dirac monopole may suggest such a singularity. Rather, it is shown that the exactly solvable model gives a vanishing Berry's phase there [24].…”