When we think of topology, we normally think of objects that cannot be simply transformed into each other, such as a rubber band and a Möbius strip (top). The metallic surface of a topological insulator is different from an ordinary surface because its metallic nature is protected by certain symmetry invariants. In this sense, it cannot be simply transformed into the surface of a normal insulator. The sketches (bottom) show the electronic structure (energy versus momentum) for a “trivial” insulator (left) and a strong topological insulator (right), such as Bi1-xSbx. In both cases, there are allowed electron states (black lines) introduced by the surface that lie in the bulk band gap (the bulk valence and conduction bands are indicated by the green and blue lines, respectively). In the trivial case, even a small perturbation (say, changing the chemistry of the surface) can open a gap in the surface states, but in the nontrivial case, the conducting surface states are protected. Note that in the topological insulator, the surface states are linear in momentum and meet at an odd number of points in k-space.