Electromagnetic effects induced by a time-dependent axion field

We studied the dynamics of the so-called $\theta$ term, which exists in topological materials and is related to a hypothetical field predicted by Peccei-Quinn in particle physics, in a magnetic superlattice constructed using a topological insulator and two ferromagnetic insulators, where the ferromagnetic insulators had perpendicular magnetic anisotropies and different magnetic coercive fields. We examined a way to drive the dynamics of the $\theta$ term in the magnetic superlattice through changing the inversion symmetry (from an antiparallel to a parallel magnetic configuration) using an external magnetic field. As a result, we found that unconventional electromagnetic fields, which are magnetic field-induced charge currents and vice versa, are generated by the nonzero dynamics of the $\theta$ term.

Figure 1

(a) Superlattice constructed by [FI1/TI/FI2/spacer]${}_n$. (b) Phase diagram of superlattice, which is categorized by an order parameter $O=M_1{M}_2$ with several tunneling parameters ${\mathrm{\Delta }}_{\pm }={\mathrm{\Delta }}_{\mathrm{T}}\pm {\mathrm{\Delta }}_{\mathrm{M}}$. In the AI phase ($O<{\mathrm{\Delta }}_{-}^2$), the $\theta$ term is given by $\theta ={tan}^{-1}(d_5/d_4)$ within the lowest order of $M_{+}$. (c) Illustration of the magnetic hysteresis loops and magnetic configurations of the superlattice with different magnetic coercive fields ($b_1$ and $b_2$) under an external magnetic field ${\mathit{B}}_{\mathrm{ex}}$ along the layered direction. $M_1$ and $M_2$ correspond to the magnetizations of the FIs. The antiparallel (AP) and parallel (P) magnetic configurations can be generated by ${\mathit{B}}_{\mathrm{ex}}$ though $1\to 2\to 3$ (red arrows) and $3\to 4$, respectively. (d) Time dependence of $M_2\left(t\right)$ by the external magnetic field ${\mathit{B}}_{\mathrm{ex}}$ (inset) during the $3\to 4$ process. (e) Time dependence of the dynamical $\theta$ term with several $|M_1/M_2(t=0)|$ at $M_1=0.01$ eV.

Figure 2

Dynamical $\theta$ term induces magnetic field ${\mathit{B}}_{\theta }$ and charge current ${\mathit{j}}_{\theta }$. (a) Illustration of ${\mathit{B}}_{\theta }$ (red arrows) and ${\mathit{j}}_{\theta }$ (orange arrows) in the presence of the dynamical $\theta$ term during the $3(t\le 0)\to 4(t\gg t_{\mathrm{f}})$ process. ${\mathit{B}}_{\mathrm{ex}}$ is an external magnetic field and ${\mathit{M}}_{+}=[M_1+M_2\left(t\right)]\widehat{\mathit{z}}$ corresponds to the total magnetization of the FIs. (b) Time dependence of $B_{\theta }$ and (c) time dependence of $j_{\theta }$. We use the parameters ${\mathrm{\Delta }}_{+}=0.1$ eV, $M_1=10$ meV, ${\tau }_{\mathrm{f}}=1$ ns, $\epsilon =8.8\times {10}^{-12}$ F/m, and $B_{\mathrm{ex}}=0.1$ T.

Figure 3

Time dependence of magnetic field $B_{\theta }\equiv B_{\theta }(j_e=0)+B_{\theta }(j_e\ne 0)$ due to ${\partial }_t\delta \theta$ in presence of external magnetic field ${\mathit{B}}_{\mathrm{ex}}$ and charge current ${\mathit{j}}_e=j_e\widehat{\mathit{z}}$ along layered direction in (a) $j_e>0$ and (b) $j_e<0$, with several $j_e$ at $|M_1/M_2(t=0)|=1$. The parameters are the same as those used in Fig. 2.

Figure 4

Magnetic field ${\mathit{B}}_{\theta }=B_{\theta ,z}\widehat{\mathit{z}}+B_{\theta ,x}\widehat{\mathit{x}}$ induced by dynamics of $\theta$ under charge current along $x$ direction (${\mathit{j}}_e=j_e\widehat{\mathit{x}}$). (a) Time dependence of $B_{\theta ,z}$ with several $|M_1/M_2|$ [see also Fig. 2]. (b) Time dependence of $B_{\theta ,x}$ at $j_e=\pm {10}^7$ A/m${}^2$ with $|M_1/M_2(t=0)|=1,2$. The parameters are the same as those used in Fig. 2. Inset: Illustration of the magnetic fields due to the dynamics of $\theta$.