Electronic states of pseudospin-1 fermions in $\alpha-\mathcal{T}_3$ lattice ribbons in a magnetic field

The electronic states on a finite width $\alpha-\mathcal{T}_3$ ribbon in a magnetic field are studied in the framework of low-energy effective theory. Both zigzag and armchair types of boundary conditions are analyzed. The analytical solutions are compared with the results of numerical tight-binding calculations. It is found that the flat band of zero energy survives for all types of boundary conditions. The analytical estimates for the spectral gap in a weak magnetic field are discussed. For zigzag type boundary conditions the approximate expressions for the edge and bulk electron states in the strong magnetic field are found.