User talk:M.A.Sarmento/sandbox

= Intertype superconductivity =

In the conventional Ginzburg-Landau theory there are two superconducting types, I and II, and the difference between them is evaluated through the Ginzburg-Landau parameter $$\kappa = \lambda_{L}/\xi $$, in which $$ \lambda_{L} $$ is the London penetration depth and $$\xi$$ is the coherence length. The change between types occur when $$\kappa$$ crosses a critical value $$\kappa_{0}=1/\sqrt{2}$$: for $$\kappa>\kappa_{0}$$ there is type II and for $$\kappa<\kappa_{0}$$, type I. At $$\kappa=\kappa_{0}$$, the Ginzburg-Landau equations are written as self-dual Bogomolnyi equations.

$$(\partial_{y}+i\partial_{x}) \Psi = (A_{x} - i A_{y}) $$

$$\mathcal{B} = 1 - |\Psi|^{2}, $$

with $$A$$ the potential vector and $$\mathcal{B}$$ the magnetic induction. The Bogomolnyi point ("B-point") $$\kappa=\kappa_{0}$$ is a superconducting state infinitely degenerate. However, the full microscopic theory is not degenerate as there are corrections of higher orders to the Ginzburg-Landau theory. As a result, there is a hierarchy of stability to configurations in a domain between the conventional types I e II in the phase diagram (coined "intertype").

To describe the remotion of degeneracy the simpler form is to produce a perturbation theory in deviations from the critical temperature. To understand the stability of configurations within the intertype domain it suffices a correction of a single order to the Ginzburg-Landau theory. As the interest is the stability in the neighborhood of $$\kappa=\kappa_{0}$$ in a $$(\kappa,T)$$ diagram, an expansion in deviation $$\delta \kappa=\kappa-\kappa_{0}$$. The result of Gibbs free energy relative to the Meissner state is provided through

$$ \mathcal{G} = \tau^{2} (-\sqrt{2} \mathcal{I} \delta k + \tau[ C_{1} \mathcal{I}  + C_{2} \mathcal{J}]) $$

with the definitions

$$\mathcal{I}= \int |\Psi|^{2} (1-|\Psi|^{2})d \vec{x} $$ and $$\mathcal{J}= \int |\Psi|^{4} (1-|\Psi|^{2})d \vec{x}$$

with $$\Psi$$ solution to Bogomolnyi. $$C_{1}$$ and $$C_{2}$$ dependent in the band structure of material and universal constants in the case of single-band conventional superconductor. For instance, $$C_{1}=-0.41$$ and $$C_{2}=0.68$$ for Fermi spherical surfaces in the single-band conventional superconductor. Therefore, the Bogomolnyi equation is a key equation in the treatment of solution stability in the vicinity of $$\kappa=\kappa_{0}$$, as it provides the information of stability concerning many superconducting configurations.[citation to my work, wilmer's,etc]

Stripe, droplet, donuts and vortex
The first Bogomolnyi equation is remarkably reduced to a modified Liouville equation for the order parameter [citation to my work]

$$ \nabla^{2} \log \Psi = |\Psi|^{2}-1 $$

Several localized solutions can be obtained from this representation. For instance, a localized solution to the unidimensional equation is represented through the first-order differential equation

$$ \Psi' = \pm \Psi \sqrt{\Psi^{2}-2\ln [c |\Psi|]} $$

joining each solution branch at the point of maximum. Considering the bidimensional topology, a set of solutions with rotational symmetry is obtained through the below representation for different choices of the sign. It is convenient to represent bidimensional solutions with rotational symmetry as a system of first-order

$$\psi_{N}' =\pm \frac{1}{r}\psi_{N} \phi_{N} $$

$$\phi_{N}'= \pm \frac{r}{N}(|\psi_{N}|^{2N}-1) $$

for the initial value problem $$\lim_{r_{0} \to 0} \psi_{N}(r_{0}) =\psi_{N,0}$$ e $$\lim_{r_{0} \to 0} \phi_{N}(r_{0})=\phi_{N,0}$$ with $$\Psi_{N} = \psi_{N}^{1/N}$$. For obtaining both vortex and donuts, $$\psi_{N,0} \to 0$$; for donuts $$\psi_{N,0} \to 1 $$. For a positive choice of sign, there are donuts and vortex. With the negative choice for the sign, when $$\phi_{0, N}$$ trespass a critical value the solution to droplets is obtained. For droplets, the slope of the condensate at the origin is zero and the way it reaches infinity is controlled through $$N$$. For the mentioned solutions, $$N$$ is the vorticity, in such a way that not only vortex carries this feature, but also other structures such as droplets and donuts. [citation to my work] --M.A.Sarmento (talk) 15:47, 31 December 2019 (UTC)