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ily is due to band inversion between two orbitals with opposite parity, driven by the strong spin-orbit coupling of Bi and Te. Due to such similarity, that family of 3D topological insulators can be described by a 3D version of the HgTe model (see the appendix). First-principle calculations show that the materials have a single Dirac cone on the surface. The spin of the surface state lies in the surface plane and is always perpendicular to the momentum, as shown in figure 4b. 

Known to be excellent thermoelectric materials, Bi$_{2}$Te$_{3}$ and Bi$_{2}$Se$_{3}$ have been investigated independently, particularly at Princeton, where Hasans group observed in ARPES experiments the single Dirac-cone surface state of Bi$_{2}$Se$_{3}$ samples prepared by Robert Cava and coworkers. 7 Furthermore, the group’s spin-resolved mea- surements showed that the electron spin indeed lies in the plane of the surface and is always perpendicular to the momentum, in agreement with theory. However, the ex- periments also observed bulk carriers coexisting with the topological surface states. A pure topological insulator phase without bulk carriers was first observed in Bi$_{2}$Te$_{3}$ by Yulin Chen and Zhi-xun Shen’s group at Stanford in ARPES experiments on material prepared by Ian Fisher and colleagues. 8 As shown in figure 4c, the observed sur- face states indeed disperse linearly, crossing at the point with zero momentum. By mapping all of momentum space, the ARPES experiments show convincingly that the surface states of Bi$_{2}$Te$_{3}$ and Bi$_{2}$Se$_{3}$ consist of a single Dirac cone. Such a state is impossible to construct in a purely 2D system. For example, a 2D graphene sheet has four Dirac cones (see the article by Andrey Geim and Al- lan MacDonald, PHYSICS TODAY, August 2007, page 35). The 2D HgTe quantum well at the crossover point d = d$_{c}$ has two Dirac cones. In a sense, the cones are spatially separated, with one placed on the top surface and the other on the bottom surface, similar to spatial decomposition illustrated in figure 1 for the 1D surfaces of a 2D system. (Particle physicists have been using similar ideas—treating a 3D lattice as the surface of a 4D lattice—in numerical simulation of fermions so as to avoid getting an unwanted doubling of neutrinos.) As we discuss below, a single Dirac cone on the surface directly leads to novel topological properties. 11,12 

## IV. TOPOLOGICAL CLASSIFICATION OF INSULATORS 

Mathematicians group geometric objects into broad topological classes. Objects with different shapes, such as a donut and a coffee cup, can be smoothly deformed into each other and can therefore be grouped into the same topological class. Mathematicians also developed the concept of a topological invariant that uniquely de- fines the topological class. Topological materials in gen- eral, and topological insulators in particular, can be de- fined by physically measurable topological invariants in topological field theories. 

<img src='content_image/399857.jpg'>

We can first divide insulators into two broad classes, according to the presence or absence of TR symmetry. The QH state is a topological insulator state that breaks TR symmetry. David Thouless and coworkers showed that the physically measured integer QH conductance is given by a topological invariant called the first Chern number (see the article by Joseph Avron, Daniel Osad- chy, and Ruedi Seiler, PHYSICS TODAY, August 2003, page 38). For a generally interacting system, the topo- logical properties of the QH state can be described by an effective topological field theory based on the ChernC- Simons theory. 14 Although Duncan Haldane constructed a model of the QH effect without the external magnetic field, that state still breaks TR symmetry. 

For a long time it was widely believed that both TR symmetry breaking and two-dimensionality are neces- sary for an insulator to be topological, but in 2001 the first model of a TR-invariant topological insulator was introduced. 15 That model was originally defined in 4D, but TR-invariant topological insulators in 3D and 2D can be obtained through a simple dimension-reduction procedure. 12 Shuichi Murakami, Naoto Nagaosa, and Zhang, and in parallel MacDonald and colleagues at the 

