Spin-orbit coupling in octamers in a spinel sulfide CuIr$_2$S$_4$: Competition between spin-singlet and quadrupolar states, and its relevance to remnant paramagnetism

摘要

We theoretically investigate magnetic properties in the low-temperature phasewith the formation of eight-site clusters, octamers, in the spinel compoundCuIr$_2$S$_4$. The octamer state was considered to be a spin-singlet stateinduced by a Peieirls instability through the strong anisotropy of $d$orbitals, the so-called orbital Peierls state. We reexamine this picture bytaking into account the spin-orbit coupling which was ignored in the previousstudy. We derive a low-energy effective model between $j_{\rm eff}=1/2$quasispins on Ir$^{4+}$ cations in an octamer from the multiorbital Hubbardmodel with the strong spin-orbit coupling by performing the perturbationexpansion from the strong correlation limit. The effective Hamiltonian is inthe form of the Kitaev-Heisenberg model but with an additional interaction, asymmetric off-diagonal exchange interaction originating from the perturbationprocess including both d-d and d-p-d hopping. Analyzing the effectiveHamiltonian on two sites and the octamer by the exact diagonalization, we findthat there is a competition between a spin-singlet state and a quadrupolarstate. The former singlet state is a conventional one, adiabatically connectedto the orbital-Peierls state. On the other hand, the latter quadrupolar stateis stabilized by the additional interaction, which consists of a linearcombination of different total spin momenta along the spin quantization axis.In the competing region, the model exhibits paramagnetic behavior with arenormalized small effective moment at low temperature. This peculiar remnantparamagnetism is not obtained in the Kitaev-Heisenberg model without theadditional interaction. Our results renew the picture of the octamer state andprovide a scenario for the intrinsic paramagnetic behavior recently observed ina muon spin rotation experiment [K. M. Kojima et al., Phys. Rev. Lett. 112,087203 (2014)].