# Perspective on Some Recent and Future Developments in Casimir Interactions
## 2 Nontrivial Topology in Materials
The interplay between temperature, various models of graphene optical response, and other factors (such as doping, for example) have been studied in the Casimir and Casimir-Polder interactions involving single-layered and multilayered graphene systems [50; 51; 52; 53]. Interesting proposals of fluctuation-induced forces from real charges competing with the Casimir force, Faraday screening effects in vdW interactions, and quantum friction involving graphene systems have also been put forward [54; 55; 56]. Weyl Semimetals: Very recently, researchers started investigating Casimir interactions in other 3D topological materials, such as Dirac and Weyl semimetals, in order to probe novel axion electrodynamics features. There are many materials characterized as Dirac semimetals with degenerate cones of linear dispersion in 3D, which makes these materials similar to 2D graphene [7]. When either inversion or time-reversal symmetries are broken, the cones are not degenerate anymore, and the Dirac semimetal becomes a Weyl semimetal [8]. The nodes of the nondegenerate cones are sources of magnetic monopoles in reciprocal space and Weyl semimetals have topologically nontrivial surface states termed as Fermi arcs. Additionally, Weyl semimetals with broken time-reversal symmetry can  
Figure 2: (**a**) Fabry–Perot cavity of two graphene-like materials under an external electric field \(E_{z}\) and laser light \(\Lambda\); (**b**) Casimir force phase diagram of two identical dissipationless graphene-like materials with \(\frac{d\lambda_{SOC}}{\hbar c}=1\). (Figure taken from reference [42]). exhibit an anomalous Hall effect, which was found to be important for the unique Casimir interaction in 2D Chern insulators. Weyl semimetals can also host an axion electromagnetic field, whose origin does not depend on the details of the electronic structure, thus it is topological in nature, but it is not quantized. The resultant mixing between the electric and magnetic field changes the constitutive electrodynamic equations and the boundary conditions, which are necessary for calculating the Casimir interaction in Weyl semimetals. It turns out that the optical response of Weyl semimetals, even for the simplest effective models, is anisotropic with many characteristic dependences [57], which coupled with the complicated Fresnel reflection matrices leads to difficulties in obtaining analytical expressions for the Casimir interaction [58; 59]. Nevertheless, it was found that at short separations, the 3D anomalous Hall effect leads to the following Casimir interaction energy between Weyl semimetals,  
\[E=\frac{\hbar\alpha a^{2}}{96\pi^{4}d}b_{1}b_{2} \tag{7}\]  
where \(b_{1},b_{2}\) are the Dirac cone separations of the interacting materials. Since the product \(b_{1}b_{2}\) can be positive or negative as the cone separation can be in positive or negative momentum space domains, the Casimir interaction can be attractive or repulsive [58; 59]. Note that the \(b\)-parameters, which essentially determine the axion term in the electrodynamics, are continuous. Thus, the interaction energy is not quantized, which contrasts to the case of Chern or 3D topological insulators, where repulsive quantized force is possible. Even though this is an interesting result, it appears that the Casimir energy is dominated by the bulk diagonal components of the optical response leading to metallic-like behavior. In effect, the topological signatures are secondary in the Casimir energy, which is essentially similar to the one for 3D Dirac semimetals [58]. To enhance the repulsive nature and tunability of the Casimir force between Weyl semimetals, it was suggested that the gap in-between is filled with chiral media [60]. Photons with different chiralities propagate with different velocities leading to time-reversal symmetry breaking [61], which can lead to conditions of a repulsive force.