Magnetic topological insulators are three dimensional magnetic materials with a non-trivial topological index protected by a symmetry other than time-reversal.^{[1][2][3][4][5]} 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 (${\displaystyle e^{2}/2h}$) perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination.^[6]

Theory

Axion coupling

The ${\displaystyle \mathbb {Z} _{2))$ classification of a 3D crystalline topological insulator can be understood in terms of the axion coupling ${\displaystyle \theta }$. A scalar quantity that is determined from the ground state wavefunction^[7]

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

where ${\displaystyle {\mathcal {A))_{\alpha ))$ is a shorthand notation for the Berry connection matrix

${\displaystyle {\mathcal {A))_{j}^{nm}(\mathbf {k} )=\langle u_{n\mathbf {k} }|i\partial _{k_{j))|u_{m\mathbf {k} }\rangle }$,

where ${\displaystyle |u_{m\mathbf {k} }\rangle }$ 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 ${\displaystyle \mathbf {k} }$ point

${\displaystyle |{\tilde {\psi ))_{n\mathbf {k} }\rangle =U_{mn}(\mathbf {k} )|\psi _{n\mathbf {k} }\rangle }$.

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

The final ingredient we need to acquire a ${\displaystyle \mathbb {Z} _{2))$ classification based on the axion coupling comes from observing how crystalline symmetries act on ${\displaystyle \theta }$.

• Fractional lattice translations ${\displaystyle \tau _{q))$, n-fold rotations ${\displaystyle C_{n))$: ${\displaystyle \theta \rightarrow \theta }$.
• Time-reversal ${\displaystyle T}$, inversion ${\displaystyle I}$: ${\displaystyle \theta \rightarrow -\theta }$.

The consequence is that if time-reversal or inversion are symmetries of the crystal we need to have ${\displaystyle \theta =-\theta }$ and that can only be true if ${\displaystyle \theta =0}$(trivial),${\displaystyle \pi }$(non-trivial) (note that ${\displaystyle -\pi }$ and ${\displaystyle \pi }$ are identified) giving us a ${\displaystyle \mathbb {Z} _{2))$ classification. Furthermore, we can combine inversion or time-reversal with other symmetries that do not affect ${\displaystyle \theta }$ to acquire new symmetries that quantize ${\displaystyle \theta }$. For example, mirror symmetry can always be expressed as ${\displaystyle m=I*C_{2))$ giving rise to crystalline topological insulators,^[8] while the first intrinsic magnetic topological insulator MnBi${\displaystyle _{2))$Te${\displaystyle _{4))$^[9][10] has the quantizing symmetry ${\displaystyle S=T*\tau _{1/2))$.

Surface anomalous hall conductivity

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

${\displaystyle \sigma _{\text{AHC))^{\text{surf))=-{\frac {e^{2)){h)){\frac {\theta -\phi }{2\pi ))\ {\text{mod))\ e^{2}/h}$.

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

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

A half quantized surface Hall conductivity and a related treatment is also valid to understand topological insulators in magnetic field ^[11] giving an effective axion description of the electrodynamics of these materials.^[12] This term leads to several interesting predictions including a quantized magnetoelectric effect.^[13] Evidence for this effect has recently been given in THz spectroscopy experiments performed at the Johns Hopkins University.^[14]

Experimental realizations