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

## TL;DR

This paper theoretically explores how the interplay between band inversion and spin-orbit coupling leads to unique band topologies, exemplified by twin Dirac cones in Ba$_3$SnO, with implications for designing novel electronic materials.

## Contribution

It introduces the concept of twin Dirac cones caused by competing band inversion and SOC, providing a general framework applicable to various antiperovskite materials.

## Key findings

- Twin Dirac cones appear when band inversion dominates over SOC.
- The band structure exhibits a continuous transformation between twin Dirac cones.
- The emergence of twin Dirac cones is a general phenomenon in related materials.

## Abstract

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