References
- Berry Phases in Electronic Structure Theory, David Vanderbilt
- Essin, Moore, Vanderbilt, “Magnetoelectric Polarizability and Axion Electrodynamics in Crystalline Insulators”, PRL 102, 146805 (2009)
- Li et al. “Dynamical axion field in topological magnetic insulators”, Nature physics, 6 (2010)
To recap, the linear ME effect is defined by a matrix,
This coupling includes a “Chern-Simons” contribution which connects to topological insulators and contributes to the isotropic or “axion” ME response. The tensor has 9 independent components and can be decomposed as
\alpha_{ij} = \tilde{\alpha}_{ij} + \frac{\theta e^2}{2\pi h}\delta_{ij} $$where first term is traceless and second term is the _pseudoscalar_ part of the coupling. The $\theta$ term is called the _axion angle_. $\theta$ changes sign under TR or inversion. # Chern Simons Coupling Electrodynamics is invariant under $\theta \rightarrow \theta + 2\pi$. Can alternatively be described in terms of surface Hall conductivity $\sigma_H = \theta e^2/2\pi h$ which is determined by bulk properties only modulo the quantum $e^2/h$ corresponding to adsorbing a surface layer of non-zero Chern number $C$. When time-reversal ($\mathcal{T}$) is present, TKNN invariants vanish, but other invariants arise such as $\mathbb{Z}_2$ invariant in 2D, distinguishing ordinary from quantum spin Hall states. In 3D, a similar invariant can be computed from 2D invariant on certain planes or from Fu-Kane parity criterion, classifying strong and weak topological insulators. - T maps $\theta \rightarrow -\theta$ and the ambiguity mod $2\pi$ allows $\theta = 0,\pi$, with the latter corresponding to the strong [[Topological Insulators|topological insulators]] - If T extends to surfaces, these become metallic by virtue of topologically protected edge modes. If T-breaking perturbation, then $\sigma_H = e^2/2h$ mod $e^2/h$ at surface of STI. Here focus of magnetoelectric coupling resulting from _orbital_ (frozen lattice) magnetization and polarization, referred to as the _orbital magnetoelectric polarizability_ (OMP # Computational Methodologies The calculation of $\theta_{CS}$ is somewhat problematic. Most methods require a two-step procedure in which a smooth gauge is first constructed on a given $\mathbf{k}$-mesh, and then the Berry connections $\mathbf{A}(\mathbf{k})$ and its derivatives are evaluated using finite differences. The Chern-Simons angle is then calculated by numerical integration on the mesh. The problem of finding a smooth gauge is the same as finding well-localized Wannier functions. However, there are situations where it is necessary to break symmetry in the gauge to construct a smooth gauge (Soluyanov and Vanderbilt 20??). For strong and weak TIs, TR must be broken in the gauge. As a result, if one uses this method for computing $\theta_{CS}$ for a strong TI, the TR breaking in the gauge propagates into $\theta_{CS}$ such that it is only approximately equal to $\pi$. The Chern-Simons angle does converge in the limit of a fine $\mathbf{k}$-point mesh, but the convergence is rather slow (Coh and Vanderbilt, 2013). # Axion Electrodynamics # Dynamical Axions When _axion field_ $\theta(\mathbf{r}, t)$ is _constant_, it plays _no role_ in electrodynamics. However can have profound consequences at _surfaces and interfaces_, where _gradients_ in $\theta(\mathbf{r})$ appear. #### Kavli IPMU and ISSP - [Story](https://www.ipmu.jp/en/story/7788) titled "Instability in magnetic materials with dynamical axion field" - Joint research effort in Tokyo between elementary particle physics and condensed matter physics - Axion is hypothetical elementary particle postulated to resolve the strong CP problem in quantum chromodynamics - To detect axion, experimental proposals such as applying a strong magnetic field to convert the axion to a photon in a crystal of germanium - Similar axions excitations exist in topological insulators - Magnetic fluctuations plays a role in the dynamical axion field