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We study the properties of the surface states in three-dimensionaltopological insulators in the presence of a ferromagnetic exchange field. Wedemonstrate that for layered materials like Bi$_2$Se$_3$ the surface states onthe top surface behave qualitatively different than the surface states at theside surfaces. We show that the group velocity of the surface states can betuned by the direction and strength of the exchange field. If the exchangefield becomes larger than the bulk gap of the material, a phase transition intoa topologically nontrivial semimetallic state occurs. In particular, thematerial becomes a Weyl semimetal, if the exchange field possesses a non-zerocomponent perpendicular to the layers. Associated with the Weyl semimetallicstate we show that Fermi arcs appear at the surface. Under certaincircumstances either one-dimensional or even two-dimensional surface flat bandscan appear. We show that the appearence of these flat bands is related tochiral symmetries of the system and can be understood in terms of topologicalwinding numbers. In contrast to previous systems that have been suggested topossess surface flat bands, the present system has a much larger energy scale,allowing the observation of surface flat bands at room temperature. The flatbands are tunable in the sense that they can be turned on or off by rotation ofthe ferromagnetic exchange field. Our findings are supported by both numericalresults on a finite system as well as approximate analytical results.

Appearance of flat surface bands in three-dimensional topological insulators in a ferromagnetic exchange field