The unoccupied part of the band structure of topological insulators Bi 2 Te x Se 3−x (x = 0,2,3) is studied by angle-resolved two-photon photoemission and density functional theory. For all surfaces linearly dispersing surface states are found at the center of the surface Brillouin zone at energies about 1.3 eV above the Fermi level. Theoretical analysis shows that this feature appears in a spin-orbit-interaction-induced and inverted local energy gap. This inversion is insensitive to variation of electronic and structural parameters in Bi 2 Se 3 and Bi 2 Te 2 Se. In Bi 2 Te 3 small structural variations can change the character of the local energy gap, depending on whether or not an unoccupied Dirac state exists. Circular dichroism measurements confirm the expected spin texture. From these findings we assign the observed state to an unoccupied topological surface state.

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We use the N -terminal scheme for studying the edge-state transport in two-dimensional topological insulators. We find the universal nonlocal response in the ballistic transport approach. This macroscopic exhibition of the topological order offers different areas for applications.

Energy bounds of knotted and linked vortex states in a charged two-component system are considered. It is shown that a set of local minima of free energy contains new classes of universality. When the mutual linking number of vector order parameter vortex lines is less than the Hopf invariant, these states have lower-lying energies.