University of Texas in Austin, developed the theory of the intrinsic spin Hall effect in doped semiconductors and identified spin-orbit coupling as the crucial ingredi- ent; later, Murakami, Nagaosa, and Zhang extended the theory to TR-invariant insulators. Kane and Mele first introduced the topological band theory of TR-invariant QSH insulators in 2D and showed that they fall into two distinct topological classes, generally referred to as the Z$_{2}$ classification. 9 That beautiful topological band the- ory was soon generalized to three dimensions. 11 The two of us and our colleagues have developed a unifying topo- logical field theory that defines the general concept of a topological insulator in terms of a physically measurable topological field theory. 12 We now have two precise definitions of TR-invariant topological insulators, one in terms of noninteracting topological band theory11 and one in terms of topological field theory. 12 If we approximate an insulator with nonin- teracting electrons filling a certain number of bands, the topological band theory can evaluate an explicit topolog- ical invariant that can give only binary values of 0 or 1: a Z$_{2}$ classification that defines trivial and nontrivial insula- tors. For materials with inversion symmetry, a powerful algorithm developed by Fu and Kane 4 can be easily in- tegrated into electronic structure calculations to numer- ically evaluate the topological band invariant. However, since all insulators in nature are necessarily interacting, it is important to have a general definition of topologi- cal insulators that is valid for interacting systems and is experimentally measurable. Both problems were solved with the topological field theory, 12 which can be generally defined for all insulators, with or without interactions. In the noninteracting case, both definitions agree. Sur- prisingly, the topological field theory can be explained in terms of elementary concepts in undergraduate-level electromagnetism. Inside an insulator, the electric field E and the mag- netic field B are both well defined. In a Lagrangian- based field theory, the insulator’s electromagnetic re- sponse can be described by the effective action S$_{0}$ = 1 / 8 π ∫ d $^{3}$xdt ( ϵE 2 − 1 /µB $^{2}$), with ϵ the electric permit- tivity and µ the magnetic permeability, from which Maxwell’s equations can be derived. The integrand de- pends on geometry, though, so it is not topological. To see that dependence, one can write the action in terms of F$_{µν}$ , the 4D electromagnetic field tensor: S$_{0}$ = 1 / 16 π ∫ d $^{3}$xdtF$_{µν}$ F $^{µν}$. The implied summation over the repeated indices µ and ν depends on the metric tensor— that is, on geometry. (Indeed, it is that dependence that leads to the gravitational lensing of light.) There is, how- ever, another possible term in the action of the electro- magnetic field: The field of topological insulators is growing rapidly, and many remarkable experiments have been carried out. In nonlocal transport measurements in a series of HgTe devices, the W¨ urzburg group confirmed that transport current is carried by the QSH edge states. The topolog- Such classification is valid for a periodic system. For a real solid with a finite boundary, a topological insulator is insulating only in the bulk; it has an odd number of gapless Dirac cones on the surface that describe conduct- ing surface states. If we uniformly cover the surface with a thin ferromagnetic film, an insulating gap also opens up on the boundary; the TR symmetry is preserved in the bulk but broken on the surface. The last identity of the equation above for S$_{θ}$ shows that the bulk topo- logical term is in fact a total derivative, expressible as a surface term, given by the expression in parenthesis. That surface term is the same Chern-Simons term that describes the topological field theory of the QH state. In the QH field theory, the term’s coefficient specifies the value of the Hall conductance. 14 Here the coefficient of θ = π translates into a Hall conductance of 1 / 2 e $^{2}$/h , half the conductance of the first QH plateau. That value is uniquely associated with the single Dirac cone on the surface of topological insulators. Any random disorder can change a system’s Hall conductance only by an in- teger multiple of e $^{2}$/h , thus the half-QH conductance of 1 / 2 e $^{2}$/h can never be reduced to zero by disorder—the surface states are topologically robust. If we integrate out all the microscopic fermionic de- grees of freedom to obtain the effective action S$_{θ}$ , all nonmagnetic insulators in the universe would fall into two distinct topological classes, described by effective topological field theories with θ = 0 or with θ = π . Unlike ϵ and µ , the physically measurable θ parame- ter is universally quantized, with the two possible values defining the topologically trivial and nontrivial insula- tors, respectively—the Z$_{2}$ classification again. Since the E field is invariant under TR, whereas the B field changes sign, S$_{θ}$ naively breaks TR symmetry. For a periodic system, however, there are two values of θ , namely θ = 0 or θ = π , that preserve the TR symmetry. 12 One can easily understand that conclusion by an analogy. If we have a 1D ring with a magnetic flux inside, a general value of the flux Φ would break the TR symmetry. However, for two special values of the flux, Φ = 0 or Φ = hc/ 2 e , an electron’s wavefunction changes its phase by 0 or π when the electron circles the ring ei- ther clockwise or counterclockwise, and TR symmetry is maintained. θ is a parameter, and ϵ$_{µνρτ}$ is the fully asymmetric 4D Levi-Civita tensor. Unlike the Maxwell action, S$_{θ}$ is a topological term—it depends only on the topology of the underlying space, not on the geometry. Written using the field tensor, the term is independent of the metric. 5 V. OUTLOOK where α = e $^{2}$/ ℏ c ≃ 1 / 137 is the fine-structure constant, S$_{θ}$ = θα 4 π 2 ∫ d $^{3}$xdt E · B ≡ θα 32 π 2 ∫ d $^{3}$xdtϵ$_{µνρτ}$ F $^{µν}$F ρτ = θ 2 π α 4 π ∫ d $^{3}$xdt∂ $^{µ}$( ϵ$_{µνρτ}$ A $^{ν}$∂$^{ρ}$A$^{τ}$) (1)