Octupolar edge state in an $e_g$ orbital system on a square lattice

A tight-binding model for $e_g$ orbitals on a square lattice is investigated. We consider only the nearest-neighbor hopping and the model is characterized by two hopping parameters, $t_1$ and $t_2$. There are Dirac points in the electronic band structure and the type of the Dirac points (type-I or type-II) depends on the ratio $t_2/t_1$. For the case of the type-I Dirac points, edge states appear for a lattice with edges perpendicular to the $[11]$ direction. The edge states at a certain momentum along the edges have octupole moments with opposite signs between the right and left edges. Thus, these edge states can be called helical octupolar edge states. This study bridges the research fields of topological phenomena and multipole physics.