cannot be removed—it is one of the topological signatures of a QSH insulator. Less than one year after the theoretical prediction, a team at the University of W¨ u rzburg led by Laurens Molenkamp observed the QSH effect in HgTe quantum wells grown by molecular-beam epitaxy. 3 The edge states provide a direct way to experimentally distinguish the QSH insulator from the trivial insulator. The two edge states of the QSH insulator act as two conducting 1D channels, which each contribute one quantum of conduc- tance, e $^{2}$/h . That perfect transmission is possible be- cause of the principle of antireflection explained earlier. In contrast, a trivial insulator phase is “really” insulat- ing, with vanishing conductance. Such a sharp conduc- tance difference between thin and thick quantum wells was observed experimentally, as shown in figure 3c. From figure 3b we see that the 2D topological insula- tor has a pair of 1D edge states crossing at momentum k = 0. Near the crossing point, the dispersion of the states is linear. That’s exactly the dispersion one gets in quantum field theory from the Dirac equation for a massless relativistic fermion in 1D, and thus that equa- tion can be used to describe the QSH edge state. Such a picture can be simply generalized to a 3D topological insulator, for which the surface state consists of a single 2D massless Dirac fermion and the dispersion forms a so-called Dirac cone, as illustrated in figure 4. Similar to the 2D case, the crossing point—the tip of the cone—is located at a TR-invariant point, such as at k = 0, and the degeneracy is protected by TR symmetry. Liang Fu and Kane predicted 4 that the alloy Bi$_{1}$$_{−}$$_{x}$Sb$_{x}$ would be a 3D topological insulator in a special range of x , and with angle-resolved photoemission spectroscopy (ARPES) Zahid Hasan and coworkers at Princeton Uni- versity observed the topological surface states in that system. 5 However, the surface states and the underlying mechanism turn out to be extremely complex. In collab- oration with Zhong Fangs group at the Chinese Academy of Sciences, the two of us predicted that Bi$_{2}$Te$_{3}$, Bi$_{2}$Se$_{3}$, and Sb$_{2}$Te$_{3}$, all with the layered structure in figure 4a, are 3D topological insulators, whereas a related material, Sb$_{2}$Se$_{3}$, is not. 6 As in HgTe, the nontrivial topology of the Bi$_{2}$Te$_{3}$ fam- III. FROM TWO TO THREE DIMENSIONS FIG. 2: (a) On a lens with antireflection coating, light waves reflected by the top (blue line) and the bottom (red line) surfaces interfere destructively, which leads to suppressed re- flection. (b) A quantum spin Hall edge state can be scattered in two directions by a nonmagnetic impurity. Going clockwise along the blue curve, the spin rotates by π ; counterclockwise along the red curve, by − π . A quantum mechanical phase factor of − 1 associated with that difference of 2 π leads to de- structive interference of the two paths—the backscattering of electrons is suppressed in a way similar to that of photons off the antireflection coating. FI G. 3: Mercury telluride quantum wells are two-dimensional top ological insulators. (a) The behavior of a mercury tell uride-cadmium telluride quantum well depends on the thi c kness d of the HgTe layer. Here the blue curve shows the potential-energy well experienced by electrons in the con- duc tion band; the red curve is the barrier for holes in the val e nce band. Electrons and holes are trapped laterally by tho se potentials but are free in the other two dimensions. For qua ntum wells thinner than a critical thickness d c ≃ 6 . 5nm, the energy of the lowest energy conduction subband, labeled E1, is higher than that of the highest-energy valence band, lab eled H1. But for d > d c , those electron and hole bands are inv erted. (b) The energy spectra of the quantum wells. The thi n quantum well has an insulating energy gap, but inside the gap in the thick quantum well are edge states, shown by red and blue lines. (c) Experimentally measured resistance of thin and thick quantum wells, plotted against the voltage applied to a gate electrode to change the chemical potential. 3 The thin quantum well has a nearly infinite resistance within the gap, whereas the thick quantum well has a quantized resis- tance plateau at R = h/ 2 e $^{2}$, due to the perfectly conducting edge states. Moreover, the resistance plateau is the same for samples with different widths, from 0.5 µ m (red) to 1.0 µ m (blue), proof that only the edges are conducting. b a 3