# Perspective on Some Recent and Future Developments in Casimir Interactions
## 2 Nontrivial Topology in Materials
It is interesting to note that the \(d^{-3}\) scaling law is the same as for typical metal, however, the interaction magnitude is quite small due to the proportionality to \(\alpha^{2}\). These results further indicate that when the Chern numbers are small, the interactions between different pairs of CIs essentially differ only by the product of their Chern numbers (or Hall conductivities equivalently). Given that \(C_{1}C_{2}\) can be positive or negative, in that case, Casimir repulsion or attraction can be obtained. In general, however, the sign of the interaction is determined by the real part of the polylogarithmic function in Equation (5). References [32; 33; 34] have further examined this problem showing that the behavior shown in Equations (4) and (5) may be overtaken by making the longitudinal conductivity of the interacting materials more prominent or by considering shorter separations, for which the interaction is similar to the one for two regular dielectrics. In the case of dielectric substrates covered with Chern-Simons layers, however, a repulsive Casimir regime is observed at shorter separations, while at larger distances, the interaction resembles one of ordinary dielectrics [35; 36]. The Casimir-Polder energy between an atom and a Chern-Simons surface may be repulsive or attractive depending on the sign of the Chern number [37]. The recent discoveries of graphene-related materials, such as silicene, germanene, and stanene [38; 39; 40], have offered a platform to further study Casimir phenomena in 2D materials with tunable topological properties. The finite buckling between the two sublattices and strong spin-orbit interaction in silicene, germanene, and stanene allow accessing several phases specific for CIs using external electric fields and laser irradiation [41]. Reference [42] reports a rich phase diagram of the Casimir energy between such 2D materials arranged in a Fabry-Perot cavity under externally applied fields (Figure 2a). It is interesting to note that the interaction is precisely of the form as in Equation (4) when the external electric and laser fields drive both materials in the anomalous quantum Hall effect regime. Indeed, in this case, the quantum mechanical energy is quantized and repulsive (denoted by the purple color in Figure 2b) when the product of Chern numbers is negative. Nevertheless, through external field modulations, one can access various interaction regimes characterized by widely different magnitudes and scaling laws (as depicted by the different colors in Figure 2b). Increasing temperature leads to smearing the boundaries between the different phases and for \(k_{B}T/\lambda_{SOC}\gtrsim 10^{-2}\) (\(\lambda_{SOC}\) is the spin-orbit coupling strength) the interaction is described by the classical result \(E=-\zeta(3)k_{B}T/15\pi d^{2}\). Here, it is important to compare these results with the special case of graphene/graphene systems, which has been extensively studied in recent years [43; 44; 45]. The zero-temperature limit of the Casimir energy has been found to be  
\[E_{g}=-\frac{\hbar c\alpha}{32\pi d^{3}}=-\frac{e^{2}}{32\pi d^{3}} \tag{6}\]  
This shows that, after taking into account the fine structure constant relation, the energy does not depend on any fundamental constants except the electronic charge, and the distance dependence is the same as the one for perfect metals. The result in Equation (6) characterizes the large distance separation, and it has been obtained based on the constant optical graphene conductivity \(\sigma_{0}\), specific to the low-frequency regime. Equation (6) applies in the limit of \(\frac{2\pi\sigma_{0}}{c}\ll 1\) appropriate for graphene. Other studies have shown that at smaller separations, the spatial dispersion and higher frequency dispersion in the optical response can lead to deviations from this asymptotic limit [46; 47; 48; 49].