We give an overview of the work done during the past ten years on the Casimir interaction in electronic topological materials, our focus being solids which possess surface or bulk electronic band structures with nontrivial topologies, which can be evinced through optical properties that are characterizable in terms of nonzero topological invariants. The examples we review are three-dimensional magnetic topological insulators, two-dimensional Chern insulators, graphene monolayers exhibiting the relativistic quantum Hall effect, and time reversal symmetry-broken Weyl semimetals, which are fascinating systems in the context of Casimir physics, firstly for the reason that they possess electromagnetic properties characterizable by axial vectors (because of time reversal symmetry breaking), and depending on the mutual orientation of a pair of such axial vectors, two systems can experience a repulsive Casimir-Lifshitz force even though they may be dielectrically identical. Secondly, the repulsion thus generated is potentially robust against weak disorder, as such repulsion is associated with a Hall conductivity which is topologically protected in the zero-frequency limit. Finally, the far-field low-temperature behavior of the Casimir force of such systems can provide signatures of topological quantization.
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