Figure 1

(a) Superlattice potential (grey lines). The hopping between $F=9/2$ levels is laser-assisted via an intermediate $F=7/2$ state. The coupling, detuned by $d+\delta$, is induced by an off-resonant Raman transition with Rabi frequency $\Omega$. (b) Scheme of the four states of the $F=9/2$ manifold [(red) vertices], connected by laser-induced hoppings [(green) boxes]. (c) Time-evolution of the populations of the neighboring hyperfine levels. The solid (dashed) line is used for site $i$ ($i+1$); the red (black) line is used for $m_F=9/2$ ($m_F=7/2$). A clear spin-preserving Rabi oscillation between neighboring sites is shown. (d) The same as before for a spin-flipping hopping. Notice the need for a superlattice staggering (10–20 kHz) in order to avoid on- site spin-flipping.Reuse & Permissions

Figure 2

(a) In-gap zero-energy modes [dashed (red) lines] for $\mathit{q}=(k_x,k_y)=\mathbf{0}$, $m/4\le m_x\le 3m/4$, and $m_y=m/2$, $m_z=m/4$, for $N={40}^3$ sites and open boundaries at $z=0$, $L$. (b) Boundary massless Dirac fermion at $z=0$, $\mathit{q}=\mathbf{0}$, and $m_x=m_y=m_z=m/2$. (c) Scheme for a fractional magnetic capacitor consisting of an axion well: $\theta (\mathit{r},t)=-\pi$ if $z\in [z_l,z_r]$, and $\theta (\mathit{r},t)=0$ elsewhere, pierced by a magnetic field $\mathit{B}=B\widehat{\mathit{z}}$. This is designed by tuning $m_x=m_y=m_z/2=m/4$ globally, whereas $\tilde{m}\gg m$ is only applied to $z_l<z<z_r$. (d) Accumulated charge on the “plates” of the capacitor, for $N={30}^3$ sites, $m_x=m_y=m_z/2=m/4$, $\tilde{m}=10m$ (leading to $\theta =-\pi$ for $12<z<18$), and flux $\varphi /{\varphi }_0=2\pi /15$.Reuse & Permissions

Figure 3

(a) Axion index as a function of the masses $m_y/m$, $m_x/m$, and setting $m_z=m/2$. In the ${\mathcal{U}}_a$ invariant regime, only fixed values of the axion $\theta =\{0,\pi \}$ are allowed. (b) Perturbations to the axion term $\delta \theta$ in the ${\mathcal{U}}_a$-breaking regime. (d) Total axion term $\theta$ in the ${\mathcal{U}}_a$-breaking regime.Reuse & Permissions