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We theoretically demonstrate how competition between band inversion andspin-orbit coupling (SOC) results in nontrivial evolution of band topology,taking antiperovskite Ba$_3$SnO as a prototype material. A key observation isthat when the band inversion dominates over SOC, there appear "twin" Diraccones in the band structure. Due to the twin Dirac cones, the band shows highlypeculiar structure in which the upper cone of one of the twin continuouslytransforms to the lower cone of the other. Interestingly, the relative size ofthe band inversion and SOC is controlled in this series of antiperovskiteA$_3$EO by substitution of A (Ca, Sr, Ba) and/or E (Sn, Pb) atoms. Analysis ofan effective model shows that the emergence of twin Dirac cones is general,which makes our argument a promising starting point for finding a singular bandstructure induced by the competing band inversion and SOC.

Evolution of Band Topology by Competing Band Overlap and Spin-Orbit Coupling: Twin Dirac Cones in Ba$_3$SnO as a Prototype