# Perspective on Some Recent and Future Developments in Casimir Interactions
## 2 Nontrivial Topology in Materials  
Topological Insulators: The discovery of 3D Topological Insulators (TIs) has been inspired by the search for topological invariants in condensed matter systems, and in particular by the prediction of the quantum spin Hall effect [6]. Experiments have shown that TI phases exist in Bi\({}_{x}\)Sb\({}_{1-x}\), layered Hg(Cd)Te structures, half-Heuslers, and others [15]. TIs differ from ordinary materials as their bulk is typically gapped or insulating, while the surface states are gapless and they are a consequence of the unique band structure of the material. In fact, because of the insensitivity of the topology to perturbations, the surface states are considered to be rather stable. A distinguishing feature for the electromagnetism of TIs is the presence of coupling between electric and magnetic fields in the electromagnetic action \(S\). Specifically, in addition to the usual component, the total action contains a topologically nontrivial term that is proportional to the fine structure constant \(\alpha\) and the axion field \(\theta(\mathbf{r},t)\). In the case of a system whose bulk is insulating under a perturbation that breaks the time reversal symmetry on the surface, the axion field \(\theta=(2n+1)\pi\) becomes quantized. The integer \(n\) depends on the type of perturbation, which usually is a magnetic layer on the surface of the TI [16]. Furthermore, the direction of polarization of the layer can result in \(\theta\) having a positive or negative sign. Trivial materials are characterized by \(\theta=0\), while TIs exhibit topological magnetoelectric susceptibility \(\theta\) that can be positive or negative, inducing mixing between the electric and magnetic fields at the surface. The presence of \(\theta\) modifies the constitutive equations for the electric and magnetic fields in TIs accordingly, thus that in frequency space  
\[\mathbf{D}=\epsilon\mathbf{E}+\frac{\alpha\theta}{\pi}\mathbf{B}, \tag{1}\]  
\[\mathbf{H}=\mu^{-1}\mathbf{B}-\frac{\alpha\theta}{\pi}\mathbf{E} \tag{2}\]  
As a result, the electromagnetic boundary conditions now yield off-diagonal components of Fresnel coefficients, i.e., the two polarizations are coupled. This situation challenges the calculation of the Casimir interaction, especially in determining analytical expressions in various limiting cases. It has been realized that the \(\theta\) term plays an important role in achieving repulsion in the Casimir interaction between semi-infinite TIs [17; 18; 19] at certain distance ranges. In fact, at large separations (corresponding to the high-temperature limit), the force per unit area between two isotropic TIs with the same bulk dielectric function \(\epsilon(\omega)\), taken as a Drude-Lorentz model, has been found to have the form  
\[F_{TI}=\frac{k_{B}T}{\pi d^{3}}f[\epsilon(0),\theta_{1},\theta_{2}] \tag{3}\]  
where \(\theta_{1},\theta_{2}\) are the magnetoelectric topological susceptibilities for the surfaces of the materials. The function \(f[\epsilon(0),\theta_{1},\theta_{2}]\) is complicated [18], and its numerical evaluation shows that repulsion is possible when \(\theta_{1}\) and \(\theta_{2}\) have opposite signs. Figure 1 shows the regimes of attraction and repulsion in terms of \(\epsilon(0)\) and \(\theta=\theta_{1}=-\theta_{2}\), in the limit of high temperature. More generally, the (unstable) equilibrium distance \(d_{eq}\) separating the attractive and repulsive regions was found to have the characteristic behavior \(d_{eq}\sim|\theta_{1}\theta_{2}|/T\) and in the case of zero temperature \(d_{eq}\sim|\theta_{1}\theta_{2}|\). The Casimir interaction between TI slabs of finite thickness has also been studied [20]. It has been shown that the repulsive regime depends on the slab thickness, and for slabs thinner than the inter-slab separation, the interaction is attractive. Additionally, the anisotropy of the TIs has been found to be important [18; 19; 20].