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 ($$e^2/2h$$) perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination.

Axion coupling
The $$\mathbb{Z}_2$$ classification of a 3D crystalline topological insulator can be understood in terms of the axion coupling $$\theta$$. A scalar quantity that is determined from the ground state wavefunction


 * $$\theta = -\frac{1}{4\pi}\int_{\rm BZ} d^3k \, \epsilon^{\alpha \beta \gamma} \text{Tr} \Big[ \mathcal{A}_\alpha \partial_\beta \mathcal{A}_\gamma -i\frac{2}{3} \mathcal{A}_\alpha \mathcal{A}_\beta \mathcal{A}_\gamma \Big]$$.

where $$\mathcal{A}_\alpha$$ is a shorthand notation for the Berry connection matrix
 * $$\mathcal{A}_j^{nm}(\mathbf{k}) = \langle u_{n\mathbf{k}} | i\partial_{k_j} | u_{m\mathbf{k}} \rangle$$,

where $$| u_{m\mathbf{k}} \rangle$$ 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 $$\mathbf{k}$$ point
 * $$|\tilde{\psi}_{n\mathbf{k}}\rangle = U_{mn}(\mathbf{k})|\psi_{n\mathbf{k}}\rangle$$.

Now a gauge transformation will cause $$ \theta \rightarrow \theta +2\pi n$$, $$n \in \mathbb{N}$$. Since a gauge choice is arbitrary, this property tells us that $$\theta$$ is only well defined in an interval of length $$2\pi$$ e.g. $$\theta \in [-\pi,\pi]$$.

The final ingredient we need to acquire a $$\mathbb{Z}_2$$ classification based on the axion coupling comes from observing how crystalline symmetries act on $$\theta$$. The consequence is that if time-reversal or inversion are symmetries of the crystal we need to have $$\theta = -\theta $$ and that can only be true if $$\theta = 0$$(trivial),$$\pi$$(non-trivial) (note that $$-\pi$$ and $$\pi$$ are identified) giving us a $$\mathbb{Z}_2$$ classification. Furthermore, we can combine inversion or time-reversal with other symmetries that do not affect $$\theta$$ to acquire new symmetries that quantize $$\theta$$. For example, mirror symmetry can always be expressed as $$m=I*C_2$$ giving rise to crystalline topological insulators, while the first intrinsic magnetic topological insulator MnBi$$_2$$Te$$_4$$ has the quantizing symmetry $$S=T*\tau_{1/2}$$.
 * Fractional lattice translations $$\tau_q$$, n-fold rotations $$C_n$$: $$\theta \rightarrow \theta $$.
 * Time-reversal $$T$$, inversion $$I$$: $$\theta \rightarrow -\theta $$.

Surface anomalous hall conductivity
So far we have discussed the mathematical properties of the axion coupling. Physically, a non-trivial axion coupling ($$\theta = \pi$$) will result in a half-quantized surface anomalous Hall conductivity ($$\sigma^{\text{surf}}_{\text{AHC}}=e^2/2h$$) if the surface states are gapped. To see this, note that in general $$\sigma^{\text{surf}}_{\text{AHC}}$$ has two contribution. One comes from the axion coupling $$\theta $$, a quantity that is determined from bulk considerations as we have seen, while the other is the Berry phase $$\phi $$ 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
 * $$\sigma^{\text{surf}}_{\text{AHC}} = -\frac{e^2}{h}\frac{\theta-\phi}{2\pi} \ \text{mod} \ e^2/h $$.

The expression for $$\sigma^{\text{surf}}_{\text{AHC}}$$ is defined $$\text{mod} \ e^2/h $$ because a surface property ($$\sigma^{\text{surf}}_{\text{AHC}}$$) can be determined from a bulk property ($$\theta$$) up to a quantum. To see this, consider a block of a material with some initial $$\theta$$ which we wrap with a 2D quantum anomalous Hall insulator with Chern index $$C=1$$. As long as we do this without closing the surface gap, we are able to increase $$\sigma^{\text{surf}}_{\text{AHC}}$$ by $$e^2/h$$ without altering the bulk, and therefore without altering the axion coupling $$\theta$$.

One of the most dramatic effects occurs when $$\theta=\pi$$ and time-reversal symmetry is present, i.e. non-magnetic topological insulator. Since $$\boldsymbol{\sigma}^{\text{surf}}_{\text{AHC}}$$ is a pseudovector on the surface of the crystal, it must respect the surface symmetries, and $$T$$ is one of them, but $$T\boldsymbol{\sigma}^{\text{surf}}_{\text{AHC}} =- \boldsymbol{\sigma}^{\text{surf}}_{\text{AHC}}$$ resulting in $$\boldsymbol{\sigma}^{\text{surf}}_{\text{AHC}} = 0$$. This forces $$\phi = \pi$$ 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 $$\theta=\pi$$ and but not force $$\boldsymbol{\sigma}^{\text{surf}}_{\text{AHC}}$$ to vanish. The most extreme case is the case of inversion symmetry (I). Inversion is never a surface symmetry and therefore a non-zero $$\boldsymbol{\sigma}^{\text{surf}}_{\text{AHC}}$$ is valid. In the case that a surface is gapped, we have $$\phi = 0$$ which results in a half-quantized surface AHC $$\sigma^{\text{surf}}_{\text{AHC}} = -\frac{e^2}{2h}$$.

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.