Spin-orbit coupling in octamers in the spinel sulfide CuIr2 S4: Competition between spin-singlet and quadrupolar states and its relevance to remnant paramagnetism

We theoretically investigate magnetic properties in the low-temperature phase with the formation of eight-site clusters, octamers, in the spinel compound CuIr2S4. The octamer state was considered to be a spin-singlet state induced by a Peierls instability through the strong anisotropy of d orbitals, the so-called orbital Peierls state. We reexamine this picture by taking into account the spin-orbit coupling, which was ignored in the previous study. We derive a low-energy effective model between jeff=1/2 quasispins on Ir4+ cations in an octamer from the multiorbital Hubbard model with the strong spin-orbit coupling by performing the perturbation expansion from the strong correlation limit. The effective Hamiltonian is in the form of the Kitaev-Heisenberg model but with an additional interaction, a symmetric off-diagonal exchange interaction originating from the perturbation process including both d-d and d-p-d hoppings. Analyzing the effective Hamiltonian on two sites and the octamer by the exact diagonalization, we find that there is competition between a spin-singlet state and a quadrupolar state. The former singlet state is a conventional one, adiabatically connected to the orbital Peierls state. On the other hand, the latter quadrupolar state is stabilized by the additional interaction, which consists of a linear combination of different total spin momenta along the spin quantization axis. In the competing region, the model exhibits paramagnetic behavior with a renormalized small effective moment at low temperature. This peculiar remnant paramagnetism is not obtained in the Kitaev-Heisenberg model without the additional interaction. Our results renew the picture of the octamer state and provide a scenario for the intrinsic paramagnetic behavior recently observed in a muon spin rotation experiment.