Heisenberg antiferromagnet with anisotropic exchange on the kagomé lattice: Description of the magnetic properties of volborthite

Abstract

We study the properties of the Heisenberg antiferromagnet with spatially anisotropic nearest-neighbor exchange couplings on the kagom\'e net, i.e., with coupling $J$ in one lattice direction and couplings ${J}^{\ensuremath{'}}$ along the other two directions. For $J∕{J}^{\ensuremath{'}}\ensuremath{\gtrsim}1$, this model is believed to describe the magnetic properties of the mineral volborthite. In the classical limit, it exhibits two kinds of ground state: a ferrimagnetic state for $J∕{J}^{\ensuremath{'}}<1∕2$ and a large manifold of canted spin states for $J∕{J}^{\ensuremath{'}}>1∕2$. To include quantum effects self-consistently, we investigate the $\mathrm{Sp}(\mathcal{N})$ symmetric generalization of the original SU(2) symmetric model in the large-$\mathcal{N}$ limit. In addition to the dependence on the anisotropy, the $\mathrm{Sp}(\mathcal{N})$ symmetric model depends on a parameter $\ensuremath{\kappa}$ that measures the importance of quantum effects. Our numerical calculations reveal that, in the $\ensuremath{\kappa}\text{\ensuremath{-}}J∕{J}^{\ensuremath{'}}$ plane, the system shows a rich phase diagram containing a ferrimagnetic phase, an incommensurate phase, and a decoupled chain phase, the latter two with short- and long-range order. We corroborate these results by showing that the boundaries between the various phases and several other features of the $\mathrm{Sp}(\mathcal{N})$ phase diagram can be determined by analytical calculations. Finally, the application of a block-spin perturbation expansion to the trimerized version of the original spin-$1∕2$ model leads us to suggest that in the limit of strong anisotropy, $J∕{J}^{\ensuremath{'}}⪢1$, the ground state of the original model is a collinearly ordered antiferromagnet, which is separated from the incommensurate state by a quantum phase transition.