The classification of a 3D crystalline topological insulator can be understood in terms of the axion coupling. A scalar quantity that is determined from the ground state wavefunction where is a shorthand notation for the Berry connection matrix where is the cell-periodic part of the ground state Bloch wavefunction. The topological nature of the axion coupling is evident if one considers gauge transformations. In this condensed matter setting a gauge transformation is a unitary transformation between states at the same point Now a gauge transformation will cause ,. Since a gauge choice is arbitrary, this property tells us that is only well defined in an interval of length e.g.. The final ingredient we need to acquire a classification based on the axion coupling comes from observing how crystalline symmetries act on.
The consequence is that if time-reversal or inversion are symmetries of the crystal we need to have and that can only be true if, giving us a classification. Furthermore, we can combine inversion or time-reversal with other symmetries that do not affect to acquire new symmetries that quantize. For example, mirror symmetry can always be expressed as giving rise to crystalline topological insulators, while the first intrinsic magnetic topological insulator MnBiTe has the quantizing symmetry.
So far we have discussed the mathematical properties of the axion coupling. Physically, a non-trivial axion coupling will result in a half-quantized surface anomalous Hall conductivity if the surface states are gapped. To see this, note that in general has two contribution. One comes from the axion coupling, a quantity that is determined from bulk considerations as we have seen, while the other is the Berry phase of the surface states at the Fermi level and therefore depends on the surface. In summary for a given surface termination the perpendicular component of the surface anomalous Hall conductivity to the surface will be The expression for is defined because a surface property can be determined from a bulk property up to a quantum. To see this, consider a block of a material with some initial which we wrap with a 2D quantum anomalous Hall insulator with Chern index. As long as we do this without closing the surface gap, we are able to increase by without altering the bulk, and therefore without altering the axion coupling. One of the most dramatic effects occurs when and time-reversal symmetry is present, i.e. non-magnetic topological insulator. Since is a pseudovector on the surface of the crystal, it must respect the surface symmetries, and is one of them, but resulting in. This forces on every surface resulting in a Dirac cone on every surface and therefore making the boundary of the material conducting. On the other hand, if time-reversal symmetry is absent, other symmetries can quantize and but not force to vanish. The most extreme case is the case of inversion symmetry. Inversion is never a surface symmetry and therefore a non-zero is valid. In the case that a surface is gapped, we have which results in a half-quantized surface AHC. A half quantized surface Hall conductivity and a related treatment is also valid to understand topological insulators in magnetic field giving an effective axion description of the electrodynamics of these materials. This term leads to several interesting predictions including a quantized magnetoelectric effect. Evidence for this effect has recently been given in THz spectroscopy experiments performed at the Johns Hopkins University.
