Duality between chiral symmetry breaking and charged pion condensation at large $N_c$: Consideration of an NJL$_2$ model with baryon-, isospin- and chiral isospin chemical potentials

In this paper we investigate the phase structure of a (1+1)-dimensional schematic quark model with four-quark interaction and in the presence of baryon ($mu_B$), isospin ($mu_I$) and chiral isospin ($mu_{I5}$) chemical potentials. It is established that in the large-$N_c$ limit ($N_c$ is the number of colored quarks) there exists a duality correspondence between the chiral symmetry breaking phase and the charged pion condensation (PC) one. The role and influence of this property on the phase structure of the model are studied. Moreover, it is shown that the chemical potential $mu_{I5}$ promotes the appearance of the charged PC phase with nonzero baryon density.