A ug 1 99 6 Hall effect and geometric phases in Josephson junction arrays

Since effectively the local contact vortex velocity dependent part of the Mag-nus force in a Josephson junction array is zero in the classical limit, we predict zero classical Hall effect. In the quantum limit because of the geometric phases due to the finite superfluid density at superconductor grains, rich and complex Hall effect is found in this quantum regime due to the Thouless-Kohmoto-Nightingale-den-Nijs effect. 1 Since vortex dynamics is identical to that of an electron in the presence of a magnetic field, numerous models for the quantum Hall effect in both homogeneous and inhomogeneous superconductor films have been proposed. 1 2 Those models have fully explored the analogy to the quantum Hall effect in semiconductor heteorojunctions by treating vortices moving in a uniform magnetic field with a homogeneous background. While the treatment can be justified in a homogeneous superconductor film, it may not be so in inhomo-geneous cases such as in Josephson junction arrays, where both the (fraction part) number of magnetic flux, the frustration n, and the (fraction part) of the fictitious magnetic flux φ 0 per plaqutte are usually large, and the periodic potential for a vortex is strong. Therefore the simple contineous limit without