## 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.