Suppression of superconductivity by inhomogeneous chiral condensation in the NJL$_2$ model

We investigate the possibility of spatially inhomogeneous chiral and Cooper, or superconducting, pairing in the (1+1)-dimensional model by Chodos et al [ Phys. Rev. D61, 045011 (2000)] generalized to continuous chiral invariance. The consideration is performed at nonzero temperature $T$ and quark number chemical potential $mu$. It is shown in the framework of the Fulde--Ferrel inhomogeneity ansatz for chiral and Cooper condensates that if $G_1>G_2$, where $G_1$ and $G_2$ are the coupling constants in the quark-antiquark and diquark channels, then in the $(mu,T)$-phase diagram the superconducting phase is suppressed by spatially inhomogeneous chiral spiral phase with broken chiral symmetry. In contrast, in the above mentioned original Chodos et al model, where only the opportunity for homogeneous condensates were taken into account, the superconducting phase is realized at sufficiently high values of $mu$ at arbitrary values of $G_2>0$, including the interval $0<G_2<G_1$.