Figure 1: (a) A diagram illustrating parallel transport of a vector constrained to stay tangential to the surface of a sphere: if the red arrow is transported around a closed loop from A → B → C, this constraint causes it to undergo a rotation. The angle of rotation arises from the geometry of the vectors path and is an example of a Berry phase. (b) A schematic of the Aharonov-Bohm effect: an electron processing around an enclosed magnetic field via a loop, C, acquires a Berry phase, a (=), due to coupling of its wavefunction to the magnetic vector potential, despite there being a negligible magnetic field along the path of the electron. The effect is a consequence of the requirement that the electromagnetic potential be invariant with respect to a fixed gauge.