By Jorge Berger, Jacob Rubinstein

"The motto of connectivity and superconductivity is that the strategies of the Ginzburg-Landau equations are qualitatively encouraged through the topology of the bounds, as in multiply-connected samples. specific attentions is paid to the "zero set", the set of positions (usually referred to as "quantum vortices") the place the order parameter vanishes. the consequences thought of the following frequently develop into vital within the regime the place the coherence size is of the order of the size of the pattern. It takes the instinct of physicists and the attention of mathematicians to discover those new results. In Connectivity and Superconductivity, theoretical and experimental physicists are introduced including natural and utilized mathematicians to check those marvelous results.This quantity is meant to function a reference e-book for graduate scholars and researchers in physics or arithmetic attracted to superconductivity, or within the Schrodinger equation as a proscribing case of the Ginzburg-Landau equations."--BOOK JACKET. learn more... within the reminiscence of Shlomo Alexander / Pierre-Gilles de Gennes -- Topological concerns in superconductivity / Jacob Rubinstein -- The de Gennes-Alexander conception of superconducting micronetworks / José I. Castro, Arturo López -- Nodal units, multiplicity and superconductivity in non-simply attached domain names / Bernard Helffer ... [et al.] -- Connectivity and flux confinement phenomena in nanostructured superconductors / Victor V. Moshchalkov, very important Bruyndoncx, Lieve Van glance -- 0 set of the order parameter, specially in earrings / Jorge Berger -- chronic currents in Ginzburg-Landau versions / Luís Almeida, Fabrice Bethuel -- at the normal/superconducting section transition within the presence of huge magnetic fields / Peter Sternberg -- at the numerical answer of the time-dependent Ginzburg-Landau equations in multiply attached domain names / Gustavo C. Buscaglia, Carlos Bolech, Arturo López -- Formation of vortex-antivortex pairs / Sanatan Digal ... [et al.] -- The order parameter as a macroscopic quantum wavefunction / Antony J. Leggett -- The Ehrenberg-Siday-Aharonov-Bohm impression / Charles G. Kuper -- Connectivity and superconductivity in inhomogeneous constructions / man Deutscher

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When He does not vanish in Ω, there may still be a smooth transition between circulations along closed loops in Ω. But now the transition is mediated by vortices (see Chap. 5 for a detailed investigation of this eﬀect). We have shown that when M is a loop and µsc is simple, there is exactly one zero in the order parameter. e. assume uniform thickness, and consider a canonical version of the model in an arbitrary graph. Given a graph M we face two questions . First, is µsc simple? Then, if it is simple, what is the size and structure of the zero set?

Branch b ≡ (a, a ) with cross section area S and length Lb . The point x has curvilinear coordinate s level; the theory was devised to deal with randomly distributed ﬁlaments as a model for random heterogenous systems; these multiply connected structures are called “superconducting micronetworks”. Using the GL equations in the limit when ψ → 0, de Gennes studied the normal-superconductor second order phase transition boundary on the (He ,T ) plane; in this limit, He (the externally applied magnetic ﬁeld) can be identiﬁed with |∇×A| (the total local magnetic ﬁeld), because the induced superconducting current approaches zero: j →0.

For the left and right branches we have 36 J. I. Castro and A. L´ opez ψ0 (s) = exp(−iγe s/L) √ sin( µ0 L) ψ0 (s) = exp(iγe s/L) √ sin( µ0 L) sin √ √ sin µ0 (L − s) ± eiγe sin µ0 (L − s) ± e−iγe sin √ √ µ0 s (right branch) µ0 s (left branch) For the central branch: ψ0 (s) = exp(−iγe (s)) √ cos[ µ0 (L/2−s)] √ cos[ µ0 L/2] (central branch, m even) ψ0 (s) = exp(−iγe (s)) √ sin[ µ0 (L/2−s)] √ sin[ µ0 L/2] (central branch, m odd) where γe (s) = φe [sin(2πs/L)/2 + πs/L] 0≤s≤ φe [π(1 − s/L) − sin(2πs/L)/2] L 2 L 2 ≤s≤L It is interesting to note that [ψ0 (s)]left = (−1)m [ψ0 (L − s)]right , as it must be by the C2h symmetry in the diﬀerent modes, and that the order parameter is maximum at the center of the network for the even modes, and it vanishes at that point for the odd modes.

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