## Connectivity and SuperconductivityThe motto of connectivity and superconductivity is that the solutions of the Ginzburg–Landau equations are qualitatively in?uenced by the topology of the boundaries. Special attention is given to the “zero set”,the set of the positions (usually known as “quantum vortices”) where the order parameter vanishes. The paradigm of connectivity and superconductivity is the Little– Parks e?ect,discussed in most textbooks on superconductivity. This volume is intended to serve as a reference book for graduate students and researchers in physics or mathematics interested in superconductivity, or in the Schr ̈ odinger equation as a limiting case of the Ginzburg–Landau equations. The e?ects considered here usually become important in the regime where the coherence length is of the order of the dimensions of the sample. While in the Little–Parks days a lot of ingenuity was required to achieve this regime, present microelectronic techniques have transformed it into a routine. Mo- over,measurement and visualization techniques are developing at a pace which makes it reasonable to expect veri?cation of distributions,and not only of global properties. Activity in the ?eld has grown and diversi?ed substantially in recent years. We have therefore invited experts ranging from experimental and theoretical physicists to pure and applied mathematicians to contribute articles for this book. While the skeleton of the book deals with superconductivity,micron- works and generalizations of the Little–Parks situation,there are also articles which deal with applications of the Ginzburg–Landau formalism to several fundamental topics,such as quantum coherence,cosmology,and questions in materials science. |

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### Contents

Table of Contents | 3 |

The de GennesAlexander Theoryof Superconducting Micronetworks | 23 |

Nodal Sets Multiplicity and Superconductivity in Nonsimply Connected Domains12 | 63 |

Connectivity and Flux Confinement Phenomena in Nanostructured Superconductors | 87 |

Zero Set of the Order Parameter Especially in Rings | 138 |

Persistent Currents in GinzburgLandau Models | 174 |

On the NormalSuperconducting Phase Transition in the Presence of Large Magnetic Fields1 | 188 |

On the Numerical Solution of the TimeDependent GinzburgLandau Equations in Multiply Connected Domains | 200 |

Formation of VortexAntivortex Pairs | 215 |

The Order Parameter as a Macroscopic Quantum Wavefunction | 230 |

The EhrenbergSidayAharonovBohm Effect | 239 |

Connectivity and Superconductivity in Inhomogeneous Structures | 248 |

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### Common terms and phrases

Abrikosov Abrikosov vortex antidot applied field behavior Berger boundary conditions branch coherence length connectivity consider Cooper pairs corresponding critical field curve defect density defined denote dependence dimensional disk domain effect eigenfunction electron equations experimental external finite flux regime fluxoid free energy function gauge Gennes Ginzburg-Landau Ginzburg-Landau theory groundstate Helffer hole integer interface Landau lattice Lett linear loop magnetic field magnetic flux magnetic potential Math Meissner Meissner effect mesoscopic minimizer modes nodal set nodes normal nucleation obtained order parameter oscillations pair parabolic phase boundary phase diagram phase transition Phys problem quantization radius region rhombic Rubinstein sample Schrödinger operator solution square structure supercurrent superfluid symmetry Tc(H temperature theorem theory topological defects triangular ladder V.V. Moshchalkov vanishes vector potential vortices width winding number zero set