## Order of Superconductive Phase Transition and Tricritical Point

A second-order nontrivial critical point and a tricritical point
was found for the first time in 1982 in

H. Kleinert,
Disorder Version of the Abelian Higgs Model
and the Order of the Superconductive Phase Transition,

Lett. Nuovo Cimento 35, 405 (1982)
.

He found this by converting the GL model into a Disorder Field Theory,
whose Feynman diagrams describe fluctuating vortex lines rather than the particle orbits of Cooper pairs.
A detailed explanation is contained in Chapter 13 of the textbook

H. Kleinert, Gauge Fields in Condensed Matter
Vol. I Superflow and Vortex Lines,
World Scientific, Singapore 1989, pp. 1--744.

which can be read in full on the internet starting from here!

Kleinert predicted the tricritical value of the
Ginzburg parameter kappa to be approximately 0.8/\sqrt{2}$. This value was confirmed 20 years later by Monte Carlo simulations (see also here!) The tricritial lies close to the mean field value kappa=1/\sqrt{2}$
where type II superconductors become type I,
the latter undergoing a first-order phase transition.
The physical reason for the tricritical point is
the opposite sign of the average short-range interaction
between vortex lines, when crossing from type I to type II,
which in the disorder field theory is represented by a
sign change of the fourth-order interaction term in the disorder field.

In four spacetime dimensions the question of a first-order trasition and a possible tricrititical value of kappa is supposedly ruled by the Coleman-Weinberg theorem. See the discussion here.