Borophene is a monolayer materials made of boron. A perfect planar boropehene called ${\beta }_{12}$ borophene has Dirac cones and they are well reproduced by a tight-binding model according to recent experimental and first-principles calculation results. We explicitly derive a Dirac theory for ${\beta }_{12}$ borophene. Dirac cones are gapless when the inversion symmetry exists, while they are gapped when it is broken. In addition, three-band touching points emerge together with pseudospin triplet fermions when all transfer energy is equal and all onsite energy is equal. The three-band touching is slightly resolved otherwise. We construct effective three-band theories for triplet fermions. We also study the edge states of borophene nanoribbons, which show various behaviors depending on the way of edge terminations.

• Received 30 March 2017
• Revised 24 May 2017

DOI:https://doi.org/10.1103/PhysRevB.96.035425