Theory of the spin-orbit coupling and topological flat band in the polyhedral π-conjugated molecules

The research on the spin-orbit coupling (SOC) has been one of the main themes in materials science since it induces the topological aspects in matter. Recently, the topological flat band has been focused on because of its new type of topological nature. In the present study, we focus on the triangular lattice of the organic molecules with C₃ symmetry. The band structure has been analyzed based on the fragment-molecular-orbital (fMO) picture, where the flat band appears and touches the dispersive band at the Γ point [1]. The dispersive and flat bands are constructed by the π-orbital, which is extended in the two-dimensional plane, and thus this system is an ideal two-dimensional system. Considering the Haldane model on this lattice in which the next-nearest spin-dependent hopping is introduced, it has been clarified that the infinitesimal spin dependent hopping induces a gap at the Γ point, and the topological flat band is realized with a nonzero Chern number [1]. In the present work, we derive the effective fMO model of the π-polyhedral molecule, including the effect of the intrinsic SOC. By treating the SOC term in the second-order perturbation theory, we relate the SOC to the next-nearest-neighbor hopping term in the polyhedral π-system. The structure of the effective Hamiltonian is different from that of graphene. We evaluate the order of magnitude of the spin-dependent hopping parameters and discuss their relevance to the material.

[1] Y. Shuku, R. Suizu, S. Nakano, M. Tsuchiizu, and K. Awaga, Phys Rev. B 107, 155123 (2023).