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The phase diagram of dense quark matter with chiral imbalance is consideredwith the conditions of electric neutrality and $\beta$-equilibrium. It has beenshown recently that chiral imbalance can generate charged pion condensation indense quark matter, so it was interesting to verify that this phenomenon takesplace in realistic physical scenarios such as electrically neutral matter in$\beta$-equilibrium, because a window of pion condensation at dense quarkmatter phase diagram (without chiral imbalance) predicted earlier was closed bythe consideration of these conditions at the physical current quark mass. Inthis paper it has been shown that the charged pion condensation is generated bychiral imbalance in the dense electric neutral quark/baryonic matter in$\beta$-equilibrium, i. e. matter in neutron stars. It has been alsodemonstrated that pion condensation is inevitable phenomenon in dense quarkmatter with chiral imbalance if there is non-zero chiral imbalance in twoforms, chiral and chiral isospin one. It seems that in this case pioncondensation phase can be hardly avoided by any physical constraint on isopinimbalance and that this conclusion can be probably generalized from neutronstar matter to the matter produced in heavy ion collisions or in neutron starmergers. The chiral limit and the physical piont (physical pion mass) has beenconsidered and it was shown that the appearance of pion condensation is notmuch affected by the consideration of non-zero current quark mass.

Electrical neutrality and $\beta$-equilibrium conditions in dense quark matter: generation of charged pion condensation by chiral imbalance