Information about research on superconductivity and plasmas based on the Darwin Hamiltonian approach can be found in the following papers and links. The main result is that superconductivity is the phase transition caused by the Darwin (-Breit) term in the Hamiltonian, i.e. the attraction between parallel currents. Alternatively, the superconducitvity phase transition may be said to be caused by the energy lowering resulting from the correlation of electron momenta on the Fermi surface. Phonons only play a destructive role and partly determine the transition temperature by destroying the correlation of electron momenta at higher temperatures.
A short history of the idea: The famous russian physicist J. Frenkel (Yacov Ilich Frenkel, 1894-1952) originally suggested the magnetic explanation of superconductivity in 1933. The same year, 1933, Hans Bethe and H. Fr÷hlich published a paper claiming that Frenkel's explanation had to be wrong. Their argument is classical and based on a version of the Darwin Hamiltonian (Charles Galton Darwin, 1887-1962) derived by the authors, independently of Darwin (1920). Later the german physicist Heinrich Welker (1912-1981) again suggested a magnetic explanation of superconductivity (1939). He tried to get around the problems suggested by Bethe and Fr÷hlich by discussing quantum effects, in particular exchange. Welker also gave an elegant explanation of the Meissner effect (1938). Thus ended the early attempts at a magnetic theory of superconductivity.
As I see it the error in the Bethe-Fr÷hlich paper is the purely classical treatment. Electrons produce and interact with magnetic fields via the expectation value of their current density. The majority of electrons in a metal occupy states with real wave functions (zero current density) and thus neither produce nor interact with a magnetic field. The remaining few that do are the ones that are part of the superconducting condensate. I therefore think that Fr÷hlich and BCS are wrong; phonons do not cause superconductivity, they destroy it. Magnetic interaction is the true cause, but no one knows how to calculate accurately and quantum mechanically, the properties of systems of very many particles interacting magnetically. Order of magnitude estimates and simple approximations and idealizations, however, support the Frenkel-Welker-EssÚn theory.
Here is a list of my (Hanno EssÚn's) published papers on the subject. There are links to pdf-files of the papers (note that these are not exact versions of the published papers; they may differ in minor details).
A Study of Lattice
and Magnetic Interactions of Conduction Electrons
Physica Scripta 52 (1995) pp.388-394. (pdf-file)
The Darwin Magnetic Interaction Energy and its Macroscopic Consequences
Physical Review E 53 (1996) pp.5228-5239. (pdf-file)
Phase space energy of charged particles with negligible radiation; proof of spontaneous magnetic structures and new effective forces
Physical Review E 56 (1997) pp.5858-5865. (pdf-file)
Magnetism of Matter and Phase Space Energy of Charged Particle Systems
Journal of Physics A: Mathematical and General 32 (1999) pp.2297-2314. (pdf-file)
Circulating electrons, superconductivity, and the Darwin-Breit interaction
(in: http://xxx.lanl.gov/abs/cond-mat/0002096) or (also in: pdf-file)
Hamiltonian of a homogeneous two-component plasma (with A. Nordmark)
Physical Review E 69 (2004) pp.036404-1-036404-9. (pdf-file)
Electrodynamic model connecting superconducting response to magnetic field and to rotation
European Journal of Physics 26 (2005) pp.279 - 285. (pdf-file)
(also in: http://www.arxiv.org/abs/cond-mat/0407670)
Magnetic dynamics of simple collective modes in a two-sphere plasma model
Physics of Plasmas 12 (2005) 122101 (pdf-file)
Meissner effect, diamagnetism, and classical physics - a review (with: Miguel Fiolhais)
American Journal of Physics 80 (2012) pp.164 - 169.
Direct numerical simulation shows that the Darwin interaction can cause a phase transition. This is done in the following very relevant and important paper:
Vishal Mehra and Jayme De Luca
Long range magnetic order and the Darwin Lagrangian
Phys. Rev. E 61 (2000) pp.1199-1205.
Hanno EssÚn main page