Magnetic topological insulator
Magnetic topological insulators are three dimensional magnetic materials with a non-trivial topological index protected by a symmetry other than time-reversal. In contrast with a non-magnetic topological insulator, a magnetic topological insulator can have naturally gapped surface states as long as the quantizing symmetry is broken at the surface. These gapped surfaces exhibit a topologically protected half-quantized surface anomalous Hall conductivity (formula_1) perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination. Theory. Axion coupling. The formula_2 classification of a 3D crystalline topological insulator  can be understood in terms of the axion coupling formula_3. A scalar quantity that is determined from the ground state wavefunction where formula_5 is a shorthand notation for the Berry connection matrix where formula_7 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 formula_8 point Now a gauge transformation will cause formula_10 , formula_11. Since a gauge choice is arbitrary, this property tells us that formula_3 is only  well defined in an interval of length formula_13 e.g. formula_14. The final ingredient we need to acquire a formula_2 classification based on the axion coupling comes from observing how crystalline symmetries act on formula_3. The consequence is that if time-reversal or inversion are symmetries of the crystal we need to have formula_23 and that can only be true if formula_24(trivial),formula_25(non-trivial) (note that formula_26 and formula_25 are identified) giving us a formula_2 classification. Furthermore, we can combine inversion or time-reversal with other symmetries that do not affect formula_3 to acquire new symmetries that quantize formula_3. For example, mirror  symmetry can always be expressed as formula_31 giving rise to crystalline topological insulators, while the first intrinsic magnetic topological insulator MnBiformula_32Teformula_33 has the quantizing symmetry formula_34. Surface anomalous hall conductivity. So far we have discussed the mathematical properties of the axion coupling. Physically, a non-trivial axion coupling (formula_35) will result in a half-quantized surface anomalous Hall conductivity (formula_36) if the surface states are gapped. To see this, note that in general formula_37 has two contribution. One comes from the axion coupling formula_38, a quantity that is determined from bulk considerations as we have seen, while the other is  the Berry phase formula_39 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 formula_37 is defined formula_42 because a surface property (formula_37) can be determined from a bulk property (formula_3) up to a quantum. To see this, consider a block of a material with some initial formula_3 which we wrap with a 2D quantum anomalous Hall insulator with Chern index formula_46. As long as we do this without closing the surface  gap, we are able to increase formula_37 by formula_48 without altering the bulk, and therefore without altering the axion coupling formula_3. One of the most dramatic effects occurs when formula_50 and time-reversal symmetry is present, i.e. non-magnetic topological insulator. Since formula_51 is a pseudovector on the surface of the crystal, it must respect the surface symmetries, and formula_20 is one of them, but formula_53 resulting in formula_54. This forces formula_55 on "every surface" resulting in a Dirac cone (or more generally an odd number of Dirac cones) 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 formula_50 and but not force formula_51 to vanish. The most extreme case is the case of inversion symmetry (I). Inversion is never a surface symmetry and therefore a non-zero formula_51 is valid. In the case that a surface is gapped, we have formula_59 which results in a half-quantized surface AHC formula_60. 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.