Magnetically robust topological edge states and flat bands

We study thin strips of three dimensional topological insulators in the presence of a spin-splitting Zeeman field. We show that under certain conditions the topological edge states at the sides of a strip remain robust against a time-reversal symmetry breaking Zeeman field. For a particle-hole symmetric system with Zeeman field perpendicular to the strip we strictly proof that the dispersion and the spin-orbital structure of the edge states remains unchanged. When the Zeeman field lies parallel to the strip, the Dirac spectrum becomes flat, but remains intact. Above a critical value of the Zeeman field a topological flat band appears at the edge. We present numerical calculations for a lattice model of Bi$_2$Se$_3$. These calculations show that even though particle-hole symmetry is not strictly fulfilled in this system, these special features are still present. The flat band is tunable by the Zeeman field and can be realistically achieved in Bi$_2$Se$_3$-ferromagnet heterosystems at room temperature.