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This Letter reports measurements of the magnetic field dependence of the resistive critical temperature T c of a regular square network of superconducting aluminum. We find new effects of flux quantization corresonding to both integral (1,2,3,...) and fractional (T>T>T>T>T>T>T) numbers of flux quanta per unit cell of the network. The fractal fine structure of the upper critical-field line is identified as the edge of the Landau-level spectrum for a tight-binding problem on a square lattice.

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In a magnetic field, a wave function in a two-dimensional system is uniquely specified by the position of its nodes. We show that for high fields and a weak random potential, motion of the zeros of the wave function under smooth changes of the boundary conditions can be used to characterize the behavior of the one-electron states and distinguish between localized and extended states.PACS numbers: 72.15. Gd, 71.25.Pi, 71.50.+t Following the discovery of the quantized Hall effect 1 an argument for the quantization based on gauge invariance was given by Laughlin, 2 which also shows that changes in

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