Charge density functional plus U calculations are carried out to examine the validity of molecular J eff = 1/2 and 3/2 states in lacunar spinel GaM4Se8 (M = Nb, Mo, Ta, and W). With LDA (spin-unpolarized local density approximation)+U , which has recently been suggested as a more desirable choice than LSDA (local spin density approximation)+U , we examine the band structure in comparison with the previous prediction based on the spin-polarized version of functional and with the prototypical J eff = 1/2 material Sr2IrO4. It is found that the previously suggested J eff = 1/2 and 3/2 band characters remain valid still in LDA+U calculations while the use of charge-only density causes some minor differences. Our result provides further support for the novel molecular J eff state in this series of materials, which can hopefully motivate the future exploration toward its verification and the further search for new functionalities.

Topological insulators are characterized by $${\Bbb Z}_2$$ Z 2 indices, and their metallic surface states are protected by time-reversal symmetry. In topological crystalline insulators, on the other hand, crystal symmetry plays a key role in protecting surface states. Therefore, perturbation breaking time-reversal or crystal symmetry induces a phase transition to a conventional band insulator. In dual topological insulators (DTIs), in which time-reversal and crystal symmetries coexist, the topological character is more robust against perturbation. Here, we propose that NaZnBi is a new DTI with $${\Bbb Z}_2$$ Z 2 invariants $$(\nu _0;\nu _1\nu _2\nu _3) = (1;000)$$ ( ν 0 ; ν 1 ν 2 ν 3 ) = ( 1 ; 000 ) and odd mirror Chern numbers ±1. We find that the characteristic Dirac cone is preserved even if either time-reversal or mirror symmetry is broken, verifying the robustness of the DTI phase. Given that gapless surface states can be experimentally observable even under perturbations that break any lattice symmetry or time-reversal, NaZnBi can be a good candidate material for future device applications.

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