Gapped Surface States in a Strong-Topological-Semimetal

A three-dimensional strong-topological-insulator or -semimetal hosts topological surface states which are often said to be gapless so long as time-reversal symmetry is preserved. This narrative can be mistaken when surface state degeneracies occur away from time-reversal-invariant momenta. The mirror-invariance of the system then becomes essential in protecting the existence of a surface Fermi surface. Here we show that such a case exists in the strong-topological-semimetal Bi${}_4$Se${}_3$. Angle-resolved photoemission spectroscopy and $\mathit{\text{ab}}\mathit{\text{initio}}$ calculations reveal partial gapping of surface bands on the Bi${}_2$Se${}_3$-termination of Bi${}_4$Se${}_3$(111), where an 85 meV gap along $\bar{\mathrm{\Gamma }}\bar{K}$ closes to zero toward the mirror-invariant $\bar{\mathrm{\Gamma }}\bar{M}$ azimuth. The gap opening is attributed to an interband spin-orbit interaction that mixes states of opposite spin-helicity.