Axion coupling edit

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 edit

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]