Spin cluster operator theory for the kagome lattice antiferromagnet

The spin-1/2 quantum antiferromagnet on the kagome lattice provides a quintessential example in strongly correlated electron physics in which the effects of both geometric frustration and quantum fluctuation are pushed to their limit. Among possible nonmagnetic ground states, the valence bond solid (VBS) with a 36-site unit cell is one of the most promising candidates. A natural theoretical framework for the analysis of such VBS order is to consider quantum states on a bond connecting the nearest-neighboring sites as fundamental quantum modes of the system and treat them as effectively independent ``bond particles.'' While correctly describing the VBS order in the ground state, this approach, known as the bond operator theory, significantly overestimates the lowest spin excitation energy. To overcome this problem, we take a next logical step in this paper to improve the bond operator theory and consider extended spin clusters as fundamental building blocks of the system. Depending on two possible configurations of the VBS order, various spin clusters are considered: (i) in the VBS order with staggered hexagonal resonance, we consider one spin cluster for a star of David shape and two spin clusters with each composed of a perfect hexagon and three attached dimers, and (ii) in the VBS order with uniform hexagonal resonance, we consider one spin cluster composed of a star of David and three attached dimers. It is shown that the majority of low-energy spin excitations are nearly or perfectly flat in energy. With most of its weight coming from the star of David, the lowest spin excitation has a gap much lower than the previous value obtained by the bond operator theory, narrowing the difference against exact diagonalization results. Despite the better agreement, however, the fact that the lowest spin excitation mostly comes from the star-of-David structure is at odds with the bond operator theory as well as previous series expansion studies. Expecting that the true low-energy spin excitations have contributions from both the star of David and its surroundings, we propose using two superclusters of spin in future work, treating spin excitations from both contributions on a more equal footing.