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.

Theory

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

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

where ${\mathcal {A}}_{\alpha }$ 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 tranformation will cause $\theta \rightarrow \theta +2\pi n$ , $n\in \mathbb {N}$. Since a gauge choice is arbitary, 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 $\theta$.

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

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 $S=T*\tau _{1/2}$.

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 ($\sigma _{\text{AHC}}^{\text{surf}}=e^{2}/2h$) if the surface states are gapped. To see that note that in general $\sigma _{\text{AHC}}^{\text{surf}}$ 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 we have that

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

The expression for $\sigma _{\text{AHC}}^{\text{surf}}$ is defined ${\text{mod}}\ e^{2}/h$ because a surface propertie ($\sigma _{\text{AHC}}^{\text{surf}}$) 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 $\sigma _{\text{AHC}}^{\text{surf}}$ by $e^{2}/h$ without altering the bulk, and therefore without altering the axion coupling $\theta$.