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Borophene is a monolayer materials made of boron. A perfect planar boropehenecalled $\beta_{12}$ borophene has Dirac cones and they are well reproduced by atight-binding model according to recent experimental and first-principlescalculation results. We explicitly derive a Dirac theory for them. Dirac conesare gapless when the inversion symmetry exists, while they are gapped when itis broken. In addition, three-band touching points emerge together withpseudospin triplet fermions when all transfer energy is equal and all on-siteenergy is equal. The three-band touching is slightly resolved otherwise. Weconstruct effective three-band theories for triplet fermions. We also study theedge states of borophene nanoribbons, which show various behaviors depending onthe way of edge terminations.

Triplet Fermions and Dirac Fermions in Borophene