Spin-singlet order in a pyrochlore antiferromagnet

The ground state of an $S=\frac{1}{2}$ quantum antiferromagnet on a pyrochlore lattice is investigated based on an effective low-energy Hamiltonian in the spin-singlet sector. In the level of the mean-field approximation, it is known that a majority part of the spins shows a long-range order of spin-singlet dimers, with a coexisting nonordered part in the singlet sector. In this paper, I first discuss elementary excitations in the ordered part and their coupling to the remaining nonordered part. Next, I derive a coupling-mediated effective interaction in the nonordered part, and examine the possibility of symmetry breaking. I find that the remaining part turns out to show a $\mathbf{q}=0$ order of either dimers or tetramers. I also find that this symmetry breaking leads to low-energy singlet excitations with linear dispersions, within the harmonic approximation for quantum fluctuations. The system has a hierarchical structure of relevant energy scales, and this may be a general feature of geometrically frustrated magnets.