Fermions have received more attention in condensed matter physics for the simple reason that electrons, that are ubiquitous in solids and determine the electrical, optical and elastic properties of matter, are fermions. The experimental progress in trapping and cooling cold atoms however means people can now prepare artificial lattices, and populate them with their choice of bosons and fermions, or combinations of them. This has predictably renewed interest in bosons and their properties, many of which were studied in the light of phenomena in liquid helium, the only interesting bosonic system that people were able to create in the lab for nearly a century.
The book on Bose liquids by Pines and Nozieres ("The theory of quantum liquids, vol. 2", Addison-Wesley, 1990) summarizes the following open questions :
1) a microscopic theory of the ground state and elementary excitation spectrum of superfluid Helium,
2) extension of the simple theory for a dilute Bose gas to a system at finite temperature or out of equilibrium,
3) a clear understanding of a number of "vortex" properties, including the critical current for superflow in capillaries,
and
4) a theory for the logarithmic singularity observed in the immediate vicinity of the lambda point.
These questions are from before the age of dilute BECs in traps and so focus a bit more on helium. I am curious to know if some of these questions have been answered, at least in part, since then, and if newer interesting questions have arisen, that could be added to this list. One that I can think of and will proceed to add to do this list is the following :
5) is it possible to have a "supersolid" phase of matter where crystalline and superfluid order can coexist?
This question is central to my PhD thesis. It's an old question but interest in it was revived recently by some encouraging experiments, which then were explained away as an effect produced by crystal defects. More on that later. The other question in this list that I have been interested in lately is that of vortex dynamics in a superfluid. This is a complicated problem, because a vortex is a mathematical singularity, there are associated problems of treating it as a single particle (it clearly is not, but an action can be written down for it where the vortex is like a charged particle in a magnetic field) and it has non-trivial couplings with the normal and superfluid components of the ambient fluid. I will provide more details on this vortex problem in a later blogpost.
