$p6$ chiral resonating valence bonds in the kagome antiferromagnet

The kagome Heisenberg antiferromagnet is mapped onto an effective Hamiltonian on the star superlattice by contractor renormalization. A comparison of ground-state energies on large lattices to density matrix renormalization group justifies truncation of effective interactions at range 3 (36 sites). Within our accuracy, magnetic and translational symmetries are not broken (i.e., a spin liquid ground state). However, we discover doublet spectral degeneracies which signal the onset of $p6$ chirality symmetry breaking. This is understood by a simple mean field analysis. Experimentally, the $p6$ chiral order parameter should split the optical phonon degeneracy near the zone center. The addition of weak next to nearest neighbor coupling is discussed.

Figure 1

The CORE-blocking scheme on the kagome lattice. $R$ and $L$ denote the two pinwheel ground states of the 12-site stars, and the arrows (pseudospins) denote the symmetrized Ising basis which spans the reduced Hilbert space of $H^{\text{CORE}}$ [see Eq. (2)].

Figure 2

(a) The mean field ground state of $H^{{\text{CORE}}_3}$ exhibiting $\langle {\sigma }^x\rangle >0$ order, which corresponds to two-dimensional chirality. (b) A typical singlet configuration in the corresponding ground state of the kagome lattice. Notice that there is no translational order, but that there are more pinwheel configurations of $|R\rangle$ than $|L\rangle$. (c) The two-dimer chiral order parameter defined in Eq. (8).

Figure 3

Kagome phonon spectra in the $p6m$ phase and $p6$ chiral phase calculated within a nearest neighbor spring constant model given in Ref. 28. In the top right, the dimer chiral correlations induce a linear coupling between excess dimer density $\delta {\rho }_d$ and chiral ionic displacements, as depicted by the arrows. This adds a chiral term to the dynamical matrix which splits the degeneracy of the optical phonons at the zone center.