Effective Hamiltonians for large-S pyrochlore antiferromagnet

The pyrochlore lattice Heisenberg antiferromagnet has a massive classical ground state degeneracy. We summarize three approximation schemes, valid for large spin length $S$, to capture the (partial) lifting of this degeneracy when zero-point quantum fluctuations are taken into account; all three are related to analytic loop expansions. The first is harmonic order spin waves; at this order, there remains an infinite manifold of degenerate collinear ground states, related by a gauge-like symmetry. The second is anharmonic (quartic order) spin waves, using a self-consistent approximation; the harmonic-order degeneracy is split, but (within numerical precision) some degeneracy may remain, with entropy still of order $L$ in a system of $L^3$ sites. The third is a large-N approximation, a standard and convenient approach for frustrated antiferromagnets; however, the large-N result contradicts the harmonic order at $O(S)$ hence must be wrong (for large $S$).