Interplay between superconductivity and chiral symmetry breaking in a (2+1)-dimensional model with compactified spatial coordinate

In this paper a (2+1)-dimensional model with four-fermion interactions is investigated in the case when one spatial coordinate is compactified and the space topology takes the form of an infinite cylinder, $R^1\otimes S^1$. It is supposed that the system is embedded into real three-dimensional space and that a magnetic flux $\mathrm{\Phi }$ crosses the transverse section of the cylinder. The model includes four-fermion interactions both in the fermion-antifermion (or chiral) and fermion-fermion (or superconducting) channels. We then study phase transitions in dependence on the chemical potential $\mu$ and the flux $\mathrm{\Phi }$ in the leading order of the large-$N$ expansion technique, where $N$ is the number of fermion fields. It is demonstrated that for arbitrary relations between coupling constants in the chiral and superconducting channels, superconductivity appears in the system at rather high values of $\mu$ (the length $L$ of the circumference $S^1$ is fixed). Moreover, it is shown that at sufficiently small values of $\mu$ the growth of the magnetic flux $\mathrm{\Phi }$ leads to a periodical reentrance of the chiral symmetry breaking or superconducting phase, depending on the values of $\mu$ and coupling constants.