Hot and dense quark matter with isospin and chiral imbalances is investigated in the framework of the (3+1)-dimensional Nambu-Jona-Lasinio model (NJL) in the large-Nc limit (Nc is the number of quark colors). Its phase structure is considered in terms of barion -µB, isospin -µI and chiral isospin -µI5 chemical potentials. It is shown in the paper that (i) in the chiral limit there is a duality between chiral symmetry breaking (CSB) and charged pion condensation (PC) phenomena. (ii) At the physical point, i.e. at nonzero bare quark mass m0, and temperature this duality relation is only approximate, although rather accurate. (iii) We have shown that the chiral isospin chemical potential µI5 in dense quark matter generates charged pion condensation both at zero and nonzero m0, and at µI5 = 0 this phase might be observed up to temperatures as high as 100 MeV. (iv) Pseudo-critical temperature of the chiral crossover transition rises in the NJL model with increasing µI5. (v) It has been found an agreement between particular sections of the phase diagram in the framework of NJL model and corresponding ones in lattice QCD simulations. Two different plots from different lattice simulations that are completely independent and are not connected at the first sight are in reality dual to each other, it means that lattice QCD simulations support the hypothesis that in real quark matter there exists the (approximate) duality between CSB and charged PC. Moreover, we can reverse the logic and we can predict the increase of pseudo-critical temperature with chiral chemical potential, the much debated effect recently, just by the duality notion, hence bolster confidence in this result (lattice QCD showed this feature for unphysically large pion mass) and put it on the considerably more solid ground.

In this paper the phase structure of the dense quark matter has been investigated in the presence of baryon µB, isospin µI , chiral µ5 and chiral isospin µI5 chemical potentials in the framework of Nambu-Jona-Lasinio model. It has been shown that in the large-Nc limit (Nc is the number of quark colours) there exist three duality correspondences in the model. The first duality is between the chiral symmetry breaking and the charged pion condensation phenomena. And there are two new dualities that hold only for chiral symmetry breaking and charged pion condensation phenomena separately. These dualities show that chiral symmetry breaking phenomenon does not feel the difference between chiral µ5 and chiral isospin µI5 chemical potentials and charged pion condensation phenomenon does not feel the difference between isospin µI and chiral µ5 chemical potentials. It was shown that µ5 can generate charged pion condensation, but this generation occurs at not so large baryon densities. In the case of both chiral imbalances (chiral µ5 and chiral isospin µI5 chemical potentials) the phase portrait is rather rich, and charged pion condensation in dense quark matter is shown to take up a large part of the phase diagram. Charged pion condensation in dense quark matter happens even in the case of zero isospin imbalance and requires only chiral imbalances, this fact can be demonstrated with use of one of the new dualities and this is only one example when these dualities are of the great use in exploring the phase diagram.

In this short review we tried to give an outline of investigations of charged pion condensation (PC) in dense baryonic (quark) matter in the framework of effective Nambu-Jona-Lasinio (NJL) type models. The possibility of charged PC phase in dense quark matter with isospin asymmetry is investigated. First, it is demonstrated that this phase can be realized in the framework of massless NJL model. But the existence of this phase is enormously fragile to the values of current quark mass and we show that charged PC phase is forbidden in electrically neutral dense quark matter with β-equilibrium when current quark masses are close to their physical value of 5.5 MeV. Nevertheless, then it is shown that in real physical systems there could be conditions promoting the appearance of charged PC phenomenon in dense quark matter, namely, it was shown that if one includes into consideration the fact that system can have finite size, then a dense charged PC phase can be realized there. It was also demonstrated that the possibility of inhomogeneous pion condensate might allow this phase to appear. And more recently it was revealed that there is another interesting factor that can induce a charged PC phase in dense quark matter even without isospin imbalance. It is a chiral imbalance of the system (non-zero difference between densities of left-and right-handed quarks). This results can be interesting in heavy ion collision experiments, where it is expected to get high baryon densities. It is of interest also in the context of the neutron stars, where quark matter might be realized in the core and very high baryon and isospin densities are attained.

In this paper the phase structure of dense quark matter has been investigated at zero temperature in the presence of baryon, isospin and chiral isospin chemical potentials in the framework of massless (3+1)-dimensional Nambu-Jona-Lasinio model with two quark flavors. It has been shown that in the large-Nc limit (Nc is the number of colors of quarks) there exists a duality correspondence between the chiral symmetry breaking phase and the charged pion condensation one. The key conclusion of our studies is the fact that chiral isospin chemical potential generates charged pion condensation in dense quark matter with isotopic asymmetry.