Inhomogeneous phases and chiral symmetry breaking

A significant fraction of the current efforts in the QCD research community is focused on characterizing the phases of strong-interaction matter that occur at finite densities and temperatures. So far, most of the experimental probes have been limited to the relatively narrow window of the QCD phase diagram characterized by high temperatures and low chemical potentials, explored in high-energy ion collision experiments at RHIC and LHC. More recently, some new insight on the finite chemical potential region has been obtained by the energy-beam scan program at RHIC, aimed at possibly determining the existence of a critical point in the QCD phase transition. On the theoretical side, studies of strong interactions are also limited by the reliability of available methods. While the zero density, finite temperature region or the zero temperature, superdense region can be investigated with the help of well-established approaches like lattice and weakly coupled QCD respectively, the study of the intermediate densities and temperatures region has to rely on effective models and nonperturbative methods, some of which are still being developed. In the past few years, a growing number of compelling arguments, backed up by model calculations, pointed out that the intermediate-density region of the QCD phase diagram may be characterized by the formation of inhomogeneous condensates which spontaneously break some of the spatial symmetries of the theory. In the following we provide a brief recapitulation of these arguments and describe some recent results in a 3+1-dimensional QCD-inspired NJL model with quark-hole condensation in the form of a plane wave in the scalar and tensor channels. This model exhibits particular features in close analogy to its 1+1-dimensional counterpart, most notably an asymmetric spectral density and the arising of an anomalous contribution to the free energy.