Kagome´- and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons

The kagome´-lattice quantum Heisenberg antiferromagnet is studied by a large-N expansion based upon groups with symplectic Sp(N) symmetry. Two distinct types of ground states are found. (i) for large values of the ‘‘spin’’ the ground state has long-range magnetic order with the spins ordered in a √3 × √3 structure with 9 sites per unit cell. Quantum fluctutions are explicitly shown to select this structure from the large number of classically degenerate states. The only zero-energy excitations about the magnetically ordered state are shown to be the physical, infinite-wavelength, Goldstone spin waves; in contrast the naive semiclassical theory has zero-energy spin waves at all wave vectors. (ii) For small values of the ‘‘spin,’’ the ordered moment disappears and we obtain a quantum-disordered ground state with no broken symmetries. As in previous work on frustrated square-lattice antiferromagnets, this state is argued to possess unconfined, spin-1/2, bosonic, spinon excitations for all values of the underlying lattice spin. A similar, small-‘‘spin’’ quantum-disordered ground state with unconfined bosonic spinons is studied in the triangular-lattice quantum Heisenberg antiferromagnet by extending earlier results. A large N, Sp(N) theory of the classical kagome´ Heisenberg antiferromagnet at finite temperature is also presented: fluctuations of the √3 × √3 structure dominate, with a correlation length which diverges exponentially in the zero-temperature limit. The significance of these results for experimental kagome´-lattice systems is discussed.