Question

The phenomenon of high temperature superconductivity has been known for decades, particularly layered cuprate superconductors. We know the precise lattice structure of the materials. We know the band theory of electrons and how electronic orbitals mix. But yet, theoreticians still haven't solved high Tc superconductivity yet. What is the obstacle to solving it? What are we missing?

Answer 1

One problem is that band theory isn't everything! Crucially, band theory completely neglects the interactions between electrons. The fact that often one can do this and obtain near correct results is actually amazing, and worth several lecture courses to flesh out the reasons. However, it cannot always be correct. In many materials the electron-electron interaction dominates --- a good example is the so-called Mott insulator, where by band structure calculations you would think you get a half-filled band and so a conductor, but because the electrons repel each other so strongly you actually get a grid-locked lattice of electrons which cannot move, because moving any of them would put two electrons on top of each other! The cuprates are known to be Mott insulating when they are undoped; this is good evidence that interactions are very important. Unfortunately, without the massively simplifying assumption that electrons are independent (i.e. non-interacting) it is an almost intractable problem to describe their behaviour; indeed, we know from other strongly interacting systems such as fractional quantum hall systems that it's possible to end up with no electrons at all, but fractions of them --- the possibilities for novel electronic structure are really unimaginable.

The 2nd problem which has plagued the field is more technical, which simply that the materials don't behave in universal ways! Although we can point to many superficially similar aspects to many cuprates, it's actually not the case that quantitatively they are the same. For instance, the fabled "linear scaling" is actually incredibly hard to really get --- it's very sensitive on impurities, precise doping levels, etc. The flip-side to this is that if we just look at the qualitative features and ask "what theories predict these?" we actually have quite a few --- marginal Fermi liquid theories, quantum critical theories, strongly coupled gauge theories, Gutzwiller projection theory, etc. All of these will give a superconducting dome, with conducting behaviour at high doping, insulating at low, and some form of anomolous transport. However, experimental signatures are actually very hard to pin down without controversy about what really has been measured, so the debate continues.

In addition, the historically long argument has created some unpleasant sociology; some (many?) would claim that actually things are pretty settled, and that their favourite theory is clearly superior. This hasn't helped a consensus to form.