Inhomogeneous superconductors and inhomogeneous superfluids appear in a variety of contexts including quark matter at extreme densities, fermionic systems of cold atoms, type-II cuprates, and organic superconductors. In the present review the focus is on properties of quark matter at high baryonic density, which may exist in the interior of compact stars. The conditions realized in these stellar objects tend to disfavor standard symmetric BCS pairing and may favor an inhomogeneous color superconducting phase. The properties of inhomogeneous color superconductors are discussed in detail and in particular of crystalline color superconductors. The possible astrophysical signatures associated with the presence of crystalline color superconducting phases within the core of compact stars are also reviewed.Comment: Added 3 figures, added section II F, added section with conclusions. Several references added. Improved the quality of the presentation and removed various typos. Almost matches the version accepted for publication of Reviews of Modern Physic

We study the phase diagram of the three flavor Polyakov-Nambu-Jona Lasinio (PNJL) model and in particular the interplay between chiral symmetry restoration and deconfinement crossover. We compute chiral condensates, quark densities and the Polyakov loop at several values of temperature and chemical potential. Moreover we investigate on the role of the Polyakov loop dynamics in the transition from nuclear matter to quark matter.

We study the phase diagram and the pion modes in the electrically neutral two flavor PNJL model. One of the main result of this paper is that in the model with massive quarks, when electrical neutrality is required, pions do not condense in the ground state of the model: the isospin chemical potential µI = −µe/2 is always smaller than the value required for pion condensation to occur. Moreover we investigate on the pions and σ mass spectra. We find that the qualitative behavior of the masses resembles that obtained in the NJL model. We close this paper by studying the intriguing possibility that bound states with the quantum numbers of the pions can be formed above the chiral phase transition.

We compute in the Ginzburg-Landau approximation the gluon Meissner masses for the LarkinOvchinnikov-Fulde-Ferrell (LOFF) phase of QCD with three flavors in the kinematical range where it is energetically favored. We find real Meissner masses and therefore chromomagnetic stability.