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I'm currently reading a paper about an antiferromagnetic axion insulator candidate and am having trouble understanding exactly what is meant by the words axion insulator in this case. To my knowledge TR-symmetric TIs are characterized by a magnetoelectric effect of topological origin, where applying an external electric field will induce a magnetic field in response. It can be described by adding the so-called axion action $$ S_\theta = \int dtd^3r \frac{\theta e^2}{4\pi^2 \hbar c}\textbf{E}\cdot\textbf{B} $$ to the standard Maxwell action. Here, $\theta$ is the value of the topological invariant and for both a TI and axion insulator $\theta = \pi$, although in axion insulators we don't have TRS but rather an effective TRS $S = T\times \tau_{1/2}$ where $T$ is TR and $\tau_{1/2}$ is a translation by half a unit cell.

So to me this just seems like saying that an axion insulator is an antiferromagnetic topological insulator but I feel like I'm missing something?

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