Atomic Bose Condensation and the Lattice

This is a lattice talk, but at first it won’t seem like it; it will be about atomic systems, and effective field theories. But studying the effective description of the atomic system will lead to a question suited to lattice techniques. I am summarizing 3 papers [1–3], all with Peter Arnold; the middle one contains all the lattice stuff. Many systems exhibit Bose condensation; for instance, liquid Helium, Cooper pairs in superconductors, and arguably nucleon Cooper pairs in large nuclei. These examples condense because of interactions; none exhibit Bose condensation as originally proposed by Bose and Einstein, where condensation occurs exclusively because the temperature is so low that the only place to store the bosons is in the ground state. The interest in atomic Bose condensation mainly arises because this is what does happen here; mutual interactions are not that important, the dominant reason for condensation is that there is not enough heat to store the density of atoms present in the excited states. In fact, interactions must be weak in an atomic Bose condensate; otherwise it would condense into a solid. The beauty of small interactions is it means we can calculate; the coupling is weak enough that perturbatively based “field theorist” techniques, like effective field theories and loopwise expansions, can be used. The question I will address in this talk is, how do interactions modify the relation between density and the temperature of the Bose condensation transition? For simplicity, and to follow the literature, I consider a homogeneous system. (This also makes the question well posed.) It means that the problem addressed is a little artificial and has less contact with experiment than we would like; but nevertheless since there is already a surprisingly big literature on the problem as posed, it is worth addressing.