The chiral condensate, which is constant in vacuum, may become spatially modulated at moderately high densities where in the traditional picture of the QCD phase diagram a first-order chiral phase transition occurs. We review the current status of this idea, which originally dates back to Migdal's pion condensation, but recently received new momentum through studies on the nature of the chiral critical point and by the conjecture of a quarkyonic-matter phase. We discuss how these nonuniform phases emerge in generalized Ginzburg-Landau analyses as well as in specific calculations, both within effective models and in Dyson-Schwinger or large-N c approaches to QCD. Questions about the most favored shape of the modulations and its dimension, and about the effects of nonzero isospin chemical potential, strange quarks, color superconductivity, and external magnetic fields on these inhomogeneous phases will be addressed as well.

We investigate the role of the isoscalar-vector interaction and the dynamics of the Polyakov loop on inhomogeneous phases in the phase diagram of the two-flavor Nambu-Jona-Lasinio model. Thereby we concentrate on inhomogeneous phases with a one-dimensional modulation, explicitly domain-wall solitons and, for comparison, the chiral spiral. While the inclusion of the Polyakov loop merely leads to quantitative changes compared to the original Nambu-Jona-Lasinio model, the inclusion of a repulsive vector-channel interaction has significant qualitative effects: Whereas for homogeneous phases the firstorder phase transition gets weakened and eventually turns into a second-order transition or a crossover, the domain of inhomogeneous phases is less affected. In particular the location of the Lifshitz point in terms of temperature and density is not modified. Consequently, the critical point disappears from the phase diagram and only a Lifshitz point (showing a different critical behavior) remains. In particular, susceptibilities remain finite.

Inhomogeneous chiral-symmetry breaking phases at non-vanishing chemical potential and temperature are studied within a two-flavor quark-meson model in the chiral limit. The analysis is performed beyond the standard mean-field approximation by taking into account the Dirac-sea contributions of the quarks. Compared with the case where the Dirac sea is neglected, we find that the inhomogeneous phase shrinks, but in general does not disappear. It is shown within a Ginzburg-Landau analysis that the Lifshitz point of the inhomogeneous phase coincides with the tricritical point if the ratio between sigma-meson and constituent quark mass in vacuum is chosen to be $m_\sigma/M = 2$, corresponding to the fixed mass ratio in the Nambu--Jona-Lasinio model. In the present model, however, this ratio can be varied, offering the possibility to separate the two points. This is confirmed by our numerical calculations, which demonstrate a strong sensitivity of the size of the inhomogeneous phase on $m_\sigma$. Finally, we uncover a general instability of the model with respect to large wave numbers of the chiral modulations, which calls for further improvements beyond the present approximation.Comment: 21 pages, 12 figures. v2: extended discussions, to be published in PR

Abstract. When the isospin chemical potential exceeds the pion mass, charged pions condense in the zeromomentum state forming a superfluid. Chiral perturbation theory provides a very powerful tool for studying this phase. However, the formalism that is usually employed in this context does not clarify various aspects of the condensation mechanism and makes the identification of the soft modes problematic. We re-examine the pion condensed phase using different approaches within the chiral perturbation theory framework. As a first step, we perform a low-density expansion of the chiral Lagrangian valid close to the onset of the Bose-Einstein condensation. We obtain an effective theory that can be mapped to a Gross-Pitaevskii Lagrangian in which, remarkably, all the coefficients depend on the isospin chemical potential. The lowdensity expansion becomes unreliable deep in the pion condensed phase. For this reason, we develop an alternative field expansion deriving a low-energy Lagrangian analog to that of quantum magnets. By integrating out the "radial" fluctuations we obtain a soft Lagrangian in terms of the Nambu-Goldstone bosons arising from the breaking of the pion number symmetry. Finally, we test the robustness of the second-order transition between the normal and the pion condensed phase when next-to-leading-order chiral corrections are included. We determine the range of parameters for turning the second-order phase transition into a first-order one, finding that the currently accepted values of these corrections are unlikely to change the order of the phase transition.

We use the on-shell effective field theory (OSEFT) for the derivation of the collision terms of the chiral kinetic theory, up to the first subleading order in the energy expansion. We first prove that the OSEFT Lagrangian can also be obtained from a Foldy-Wouthuysen (FW) diagonalization of the QED Lagrangian associated to a very energetic massless fermion. OSEFT is thus the quantum field theory counterpart of the FW diagonalization in relativistic quantum mechanics for massless fermions. It is important to note that in the FW picture the associated fermions are known to interact non-minimally with the electromagnetic fields, acquire magnetic moments, and have a spatial extent of radius one half their Compton wavelength. These facts are essential to give a semi-classical interpretation of the chiral kinetic theory. We find that the leading order collision term in the energy expansion describes particle-particle and particle-antiparticle collisions, mediated by a soft-photon exchange, and the subleading correction reveals the fact that a chiral fermion interacts differently with the two transverse circular polarized photon states which are present in a medium with chiral imbalance.