Quantum interference in superconducting wire networks and Josephson junction arrays: An analytical approach based on multiple-loop Aharonov-Bohm Feynman path integrals

We investigate analytically and numerically the mean-field superconducting-normal phase boundaries of two-dimensional superconducting wire networks and Josephson junction arrays immersed in a transverse magnetic field. The geometries we consider include square, honeycomb, triangular, and kagomé lattices. Our approach is based on an analytical study of multiple-loop Aharonov-Bohm effects: the quantum interference between different electron closed paths where each one of them encloses a net magnetic flux. Specifically, we compute exactly the sums of magnetic phase factors, i.e., the lattice path integrals, on all closed lattice paths of different lengths. A very large number, e.g., up to ${10}^{81}$ for the square lattice, of exact lattice path integrals are obtained. Analytic results of these lattice path integrals then enable us to obtain the resistive transition temperature as a continuous function of the field. In particular, we can analyze measurable effects on the superconducting transition temperature $T_c\mathrm{}\left(B\right)\mathrm{}$ as a function of the magnetic field B, originating from the electron trajectories over loops of various lengths. In addition to systematically deriving previously observed features and understanding the physical origin of the dips in $T_c\mathrm{}\left(B\right)\mathrm{}$ as a result of multiple-loop quantum interference effects, we also find novel results. In particular, we explicitly derive the self-similarity in the phase diagram of square networks. Our approach allows us to analyze the complex structure present in the phase boundaries from the viewpoint of quantum interference effects due to the electron motion on the underlying lattices. The physical origin of the structures in the phase diagrams is derived in terms of the size of regions of the lattice explored by the electrons. Namely, the larger the region of the sample the electrons can explore (and thus the larger the number of paths the electron can take), the finer and sharper structure appears in the phase boundary. Our results for kagomé and honeycomb lattices compare very well with recent experimental measurements by Xiao et al. [preceding paper, Phys. Rev. B 65, 214503 (2001)].