The classical nearest-neighbor Heisenberg antiferromagnet on the pyrochlore lattice is very unusual in that it does not order magnetically at any non-zero temperature. This is a result of the very large classical degeneracy of ground states. One mechanism that can be important for small spin s in splitting this degeneracy is quantum fluctuations. The same large degeneracy which prevents order classically also makes theory of quantum fluctuations very tricky. Pioneering and tour-de-force work on the subject has been done by Chris Henley and collaborators (see this paper ). Unfortunately it turns out that it is necessary to go beyond the leading order (spin qave zero point energy) to fully split the ground state degeneracy, a task which at the time of writing (September 2006) is still not complete. We have taken an alternative approach, in which rather than assume s is large (as in Henley's calculations), we instead assume spin fluctuations transverse to some preferred Ising axis are small. The approach has the advantage that it lets one treat arbitrary s, and includes the effects of quantum tunneling, which are non-perturbative in the other method. The two techniques can be compared in an overlapping limit of validity and are found (fortunately!) to agree. The result of our calculations is an effective "quantum Ising" Hamiltonian that describes the low energy states, and can usually be much more readily studied than the original Heisenberg model. Our approach is technically pretty tough (as is Henley's), and for the pyrochlore lattice requires carrying out 6^th order degenerate perturbation theory!

You can read a detailed account and analysis for the pyrochlore lattice in cond-mat/0607210 . A gentler version is in cond-mat/0608131, which also contains applications of the same technique to other lattices. Some of the findings are shown here for the half-magnetized states (corresponding to the plateaus seen in the spinel chromites). The left-hand figure shows the magnetic unit of the trigonal₇ state, which is found to be the ground state for s>3/2. The colored dots are the minority spins aligned anti-parallel to the field (parallel spins are not shown). The right-hand figure shows the magnetic unit cell of the R state, in the same convention.