# Perspective on Some Recent and Future Developments in Casimir Interactions
## 2 Nontrivial Topology in Materials
The strength and sign of the interaction can be modified using the anisotropy through the off-diagonal elements being suppressed or enhanced with respect to the diagonal terms in the Fresnel reflection matrix. Reference [21] has also found that the van der Waals force oscillates with respect to the angle between the optical axes of two anisotropic TIs, which can lead to van der Waals torque. Interestingly, the torque is weakened when \(\theta_{1}\) and \(\theta_{2}\) have opposite signs. The strength of the Casimir repulsion can also be enhanced in TI multilayered composites [22]. The Casimir-Polder energy between an atom and a TI was found to be generally attractive, although some repulsive regimes were also uncovered due to the interplay between the \(\theta\)-term and the atomic states [23]. It is important to note that the Casimir/van der Waals interaction is closely affected by the optical response of the actual TI materials. The above results have been obtained using generic models for the bulk and surface dielectric response, characterized by adjustable parameters. However, the frequency dependence of the optical properties of TI materials will definitely affect the interaction [1]. For example, the nontrivial topological contribution resulting in a constant conductivity is observed in the low-frequency regime, \(\omega<300\) meV, while transitions between the surface and bulk states are responsible for the higher frequency range. Additionally, temperature, electron density, and other factors can affect the optical properties of TIs [24; 25; 26]. Thus, even though the nontrivial topology is  
Figure 1: Attraction and repulsion at \(T\rightarrow\infty\) and \(\theta_{1}=-\theta_{2}=\theta\). The forbidden region is given in black. (Figure is taken from reference [18]). expected to be present in large distance separations, a more complete model for the optical conductivity \(\sigma(\omega)\) may be necessary to examine the survival of repulsion and the role of distance and other factors in the interaction for material-specific TIs. Chern Insulators: Chern insulators (CIs) are another type of topological materials for which the Casimir/vdW interaction has been studied. CIs are 2D band insulators with a quantized Hall conductivity whose origin comes from the broken time-reversal symmetry [27]. Such materials give rise to the quantum anomalous Hall effect since there is no net magnetic field (external and internal magnetic fields are zero). Several types of lattices have been shown to be CIs, including hexagonal [27], kagome [28], triangular [29], and several others [30]. The signature feature for Chern insulators is the Chern number \(C\), which sums up the signs of the non-zero mass gaps of the Dirac cones and it essentially determines the Hall conductivity at zero frequency \(\sigma_{xy}(\omega=0)=\frac{C\epsilon^{2}}{\hbar}\). One of the first works to consider the Casimir interaction in CIs is reference [31], where it has been shown that at large distances, the zero-temperature quantum mechanical limit of the energy is determined by the zero frequency optical properties and that the anomalous quantum Hall effect plays an important role. In fact, the interaction energy per area in the lowest order of \(\alpha\) is found as  
\[E_{CI}=-\frac{\hbar\alpha\alpha^{2}}{8\pi^{2}d^{3}}C_{1}C_{2}, \tag{4}\]  
where \(C_{1},~{}C_{2}\) are the Chern numbers of the two CIs. Later studies [32; 33; 34] have shown that in the case of planar Chern-Simons materials, forward and backward electromagnetic field scattering must be considered explicitly, which modifies the Casimir energy to  
\[E_{CI}=-\frac{\hbar c}{8\pi^{2}d^{3}}\text{Re}Li_{4}\left(\frac{C_{1}C_{2} \alpha^{2}}{(C_{1}\alpha+i)(C_{2}\alpha+i)}\right), \tag{5}\]  
where the two CIs are with vanishing longitudinal conductivity and \(Li_{4}(z)\) is the polylogarithmic function. Equations (4) and (5) have very similar large distance tails for comparable optical response properties characterized by small Chern numbers (as noted in reference [32]).