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The established diversity of electronic topology classes lends theopportunity to pair them into dual topological complexes. Bulk-surfacecorrespondence then ensures the coexistence of a combination of boundary statesthat cannot be realized but only at the various surfaces of such a dualtopological material. We show that the layered compound Bi_2TeI realizes a dualtopological insulator. It exhibits band inversions at two time reversalsymmetry points of the bulk band which classify it as a weak topologicalinsulator with metallic states on its (010) 'side' surfaces. Additional mirrorsymmetry of the crystal structure concurrently classifies it as a topologicalcrystalline insulator. Bi2TeI is therefore predicted to host a pair of Diraccones protected by time reversal symmetry on its 'side' surfaces and threepairs of Dirac cones protected by mirror symmetry on its 'top' and 'bottom'(001) surfaces. We spectroscopically map the top cleaved surface of Bi_2TeI,and crystallographic step edges therein. We show the existence of both twodimensional surface states which are susceptible to mirror symmetry breaking,as well as one dimensional channels that reside along the step edges. Theirmutual coexistence on the step edge where both facets join is facilitated bymomentum and energy segregation. Our observations of a dual topologicalinsulator make way to additional pairing of other dual topology classes withdistinct surface manifestations coexisting at their boundaries.

Coexisting surface states in the weak and crystalline topological insulator Bi_2TeI