We investigate the possibility of spatially homogeneous and inhomogeneous chiral fermion-antifermion condensation and superconducting fermion-fermion pairing in the (1+1)-dimensional model by Chodos {it et al.} [ Phys. Rev. D 61, 045011 (2000)] gener
alized to continuous chiral invariance. The consideration is performed at nonzero values of temperature $T$, electric charge chemical potential $mu$ and chiral charge chemical potential $mu_5$. It is shown that at $G_1<G_2$, where $G_1$ and $G_2$ are the coupling constants in the fermion-antifermion and fermion-fermion channels, the $(mu,mu_5)$-phase structure of the model is in a one-to-one correspondence with the phase structure at $G_1>G_2$ (called duality correspondence). Under the duality transformation the (inhomogeneous) chiral symmetry breaking (CSB) phase is mapped into the (inhomogeneous) superconducting (SC) phase and vice versa. If $G_1=G_2$, then the phase structure of the model is self-dual. Nevertheless, the degeneracy between the CSB and SC phases is possible in this case only when there is a spatial inhomogeneity of condensates.
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 $0<G_2<G_1$.
The properties of two-flavored massless Nambu-Jona-Lasinio model in (1+1)-dimensional $R^1times S^1$ spacetime with compactified space coordinate are investigated in the presence of isospin and quark number chemical potentials $mu_I$, $mu$. The consi
deration is performed in the large $N_c$ limit, where $N_c$ is the number of colored quarks. It is shown that at $L=infty$ ($L$ is the length of the circumference $S^1$) the pion condensation (PC) phase with {it zero quark number density} is realized at arbitrary nonzero $mu_I$ and for rather small values of $mu$. However, at arbitrary finite values of $L$ the phase portrait of the model contains the PC phase with {it nonzero quark number density} (in the case of periodic boundary conditions for quark fields). Hence, finite sizes of the system can serve as a factor promoting the appearance of the PC phase in quark matter with nonzero baryon densities. In contrast, the phase with chiral symmetry breaking may exist only at rather large values of $L$.
