We have measured the superconducting transition temperature (Tc(H)) as a function of magnetic field for a network of thin aluminum wires arranged in patterns from periodic to random and including the intermediate configurations of quasi-crystalline and incommensurate. We find sharp cusp-like dips in the Tc(H) curve for periodic, incommensurate (quasi-periodic) and quasi-crystalline networks reflecting the lock-in of the flux lattice with the underlying network. We find no similar fine structure for random arrays. The experiments therefore strongly suggest that there is a sense of commensurability with incommensurate structures. Can commensurate states exist on incommensurate lattices? To investigate this problem theoretically we introduce a simple model related to both the flux quantization problem as well as the problem of charged particles on a substrate and study the ground states by Monte Carlo simulated annealing. The model is one dimensional but allows for a wide variety of different patterns to be investigated including periodic systems with incommensurate areas, quasi-periodic systems with commensurate areas and various degrees of randomness corresponding to the patterns which can be readily produced by electron beam lithography. The Monte Carlo studies, as well as the experimental observations suggest that there are two types of energetically favorable ground states μ those which reflect the periodicity of the substrate (the configuration of the wire network) and those which reflect the inflation symmetry of the substrate, its self-similarity with change of length scale. This allows us to generalize the concept of commensurability to include incommensurate structures. In particular the commensurate states are those which have a Fourier spectrum consisting of Bragg spots which completely contain the Fourier spectrum of the substrate. However, for the quasi-crystalline substrates, we find that the states reflecting the inflation symmetry are more stable of robust than the periodic states.
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