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We theoretically investigate the Cooper-pair symmetry to be realized in hole-doped monolayer MoS2 by solving linearized BCS gap equations on the three-orbital attractive Hubbard-like model in the presence of the atomic spin-orbit coupling. In hole-doped monolayer MoS2, both spin-orbit coupling and the multi-orbital effects are more prominent than those of electron-doped system. Near the valence band edge, the Fermi surfaces are composed of three different types of hole pockets, namely, one mainly consisting of the almost spin-degenerate |d z 2 orbital near Γ point, and the others of the spin-split upper and lower bands near K and K ′ points arising from the |d x 2 −y 2 and |dxy orbitals. The number of relevant Fermi pockets increases with increase of the doping. At very low doping, the upper split bands of |d x 2 −y 2 and |dxy are concerned, yielding extremely low Tc due to small density of states of the split bands. For further doping, the conventional spin-singlet state (SS) appears in the Γ pocket, which has a mixture of the spin-triplet (orbital-singlet) (ST-OS) and spinsinglet (orbital-triplet) (SS-OT) states in the K and K ′ pockets. The ratio of the mixture depends on the relative strength of the interactions, and the sign of the exchange interactions. Moderately strong ferromagnetic exchange interactions even lead to the pairing state with the dominant ST-OS state over the conventional SS one. With these observations, we expect that the fascinating pairing with relatively high Tc emerges at high doping that involves all the three Fermi pockets.

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We theoretically investigate electronic orderings with the electric axial moment without breakings of both spatial inversion and time-reversal symmetries in the zigzag-chain system. Especially, we elucidate the role of the local odd-parity hybridization arising from locally noncentrosymmetric lattice structures based on symmetry and microscopic model analyses. We show that the odd-parity crystalline electric field gives rise to an effective cross-product coupling between the electric dipole and electric toroidal dipole, the latter of which corresponds to the electric axial moment. As a result, the staggered component of the electric axial moment is induced by applying an external electric field, while its uniform component is induced via the appearance of staggered electric dipole ordering. We also show that uniform electric quadrupole ordering accompanies uniform electric axial moment. Furthermore, we discuss transverse magnetization as a consequence of the orderings with the uniform electric axial moment. Our results extend the scope of materials exhibiting electric axial ordering to those with locally noncentrosymmetric lattice structures.

We investigate the nature of the time-reversal breaking pairing state in the hole-doped monolayer MoS 2 on the basis of the realistic three-orbital attractive Hubbard-like model with the atomic spin-orbit coupling. Due to the multi-band features arising from the Mo d orbitals in the noncentrosymmetric crystal structure, the Lifshitz transition takes place upon hole doping. Across the Lifshitz transition point, the sign of the relative phase between the Cooper-pair components drastically changes, leading to the emergence of the time-reversal breaking phase with complex gap functions. It is shown that this intriguing pairing state is characterized by the finite momentum-space distributions of the orbital and spin angular momentum with three-fold rotational symmetry on the Fermi-surface pockets around K and K ′ points. The present mechanism for the time-reversal breaking superconductivity can ubiquitously be applied to spin-orbit-coupled metals in noncentrosymmetric crystal structures.

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