Magnetic warping in topological insulators

We analyze the electronic structure of topological surface states in family magnetic insulators MnBi$_{2n}$Te$_{3n+1}$. show that, at natural-cleavage surfaces, Dirac cone warping changes its symmetry from hexagonal to trigonal ordering temperature. In particular, an energy splitting develops between same band index but opposite momenta upon formation long-range order. As a consequence, measurements such splittings constitute simple protocol detect via structure, alternative detection gap. Interestingly, while latter signals nonzero flux, predicted here is, addition, sensitive direction flux.