The amazing properties of crystalline color superconductors

This paper is a brief journey into the amazing realm of crystalline color superconductors. Starting from a qualitative description of superfluids, superconductors and supersolids, we show how inhomogeneous phases may arise when the system is under stress. These basic concepts are then extended to quark matter, in which a richer variety of phases can be realized. Then, the most interesting properties of the crystalline color superconductors are presented. This brief journey ends with a discussion of crystalline color superconductors in compact stars and related astrophysical observables. We aim at providing a pedagogical introduction for nonexpert in the field to a few interesting properties of crystalline color superconductors, without discussing the methods and the technicalities. Thus, the results are presented without a proof. However, we try to give a qualitatively clear description of the main concepts, using standard quantum field theory and analogies with condensed matter systems.