# Connectivity and superconductivity by Jorge Berger, Jacob Rubinstein

By Jorge Berger, Jacob Rubinstein

"The motto of connectivity and superconductivity is that the suggestions of the Ginzburg-Landau equations are qualitatively inspired by means of the topology of the bounds, as in multiply-connected samples. designated attentions is paid to the "zero set", the set of positions (usually referred to as "quantum vortices") the place the order parameter vanishes. the results thought of right here often turn 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 study those superb results.This quantity is meant to function a reference booklet for graduate scholars and researchers in physics or arithmetic attracted to superconductivity, or within the Schrodinger equation as a restricting 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 thought 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, important Bruyndoncx, Lieve Van glance -- 0 set of the order parameter, in particular in jewelry / Jorge Berger -- continual currents in Ginzburg-Landau types / Luís Almeida, Fabrice Bethuel -- at the normal/superconducting part transition within the presence of enormous 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 impact / Charles G. Kuper -- Connectivity and superconductivity in inhomogeneous constructions / man Deutscher

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**Example text**

7 shows the thermodynamical phase diagram resulting from the solutions a), b) and c); the solid line breaks at certain “critical” magnetic ﬂux values where the mode determining the N-S phase boundary changes . 784, and so on). 22). The central branches carry no currents; the peripheral branches support the same current √ µ0 L m 2 j0 = −J |ψ0 (1)| sin 2π(φe − ) . √ sin( µ0 L) 4 The number m takes the value 0 for both Ag modes; m = 2 for the mode Bg and m is 1or 3 for the modes Eg . The current j0 as a function of φe for the diﬀerent modes shows inversions and jumps that allows for the ﬂuxoid quantization condition to be satisﬁed.

45) where Λ is a diagonal matrix with diagonal entries belonging to (0, +∞]. 29). Then min H(ψ), min K(y, ξ + πZ(y), Λ(y)), ψ ∈ H 1 (M ; C) = y ∈ H 1 (M ), ξ ∈ 2πZ |C| − Φ . 46) 16 Jacob Rubinstein Thus the phase equations were integrated up to a set of integers {Nl , l = 1, . . , |C|}. 45) we see at once that the minimizers are periodic with respect to each Φl with a (non-dimensional) period of 2π. This is of course well known experimentally. If the matrix (B T ΛB)−1 is diagonal, then when there are no zeros, for any value of Λ the energy is minimized when every ξl2 is minimized.

It is again convenient to use the magnetic vector potential in the symmetric gauge Ae = −(He r/2)eθ . The ﬂux across the basic loop is Φe = σ He · dσ = Ae · dr =He (πR2 /4) . 2 The de Gennes–Alexander Theory of Superconducting Micronetworks 37 Fig. 6. Geometry of the “double yin-yang” (2YY). This network has the same topology and, under the external magnetic ﬁeld, the same symmetry C4h as the network of Fig. 2 From the symmetry of the network it is easy to see that the circulation of Ae along the branches takes the values γe = 2π Φ0 2 1 Ae · dr = 2πΦe 2π = 2πφe , γe = Φ0 Φ0 0 1 Ae · dr = πΦe γe = πφe = Φ0 2 With L = πR/2, the nodal equations are 4 √ 4 cos( µ0 L) ψ0 (0) − e−iγe /2 ψ0 (a) = 0 (central node 0) a=1 √ 3 cos( µ0 L) ψ0 (a) − eiγe ψ0 (a + 1) − e−iγe ψ0 (a − 1) − eiγe /2 ψ0 (0) = 0 (a = 1, 2, 3, 4) for which there are ﬁve solutions: a) Two of type Ag with √ exp(iπφe ) cos( µ0 L) ψ0 (0) = ψ0 (2) = ψ0 (3) = ψ0 (4) = ψ0 (1).