We examine four-fermion theory to investigate the restoration of dynamically broken chiral symmetry by varying some parameters of a system: the coupling constant, spacetime topology (finite temperature and size) and curvature, and particle-number density. Four-fermion systems have rich vacuum structures. The ground structure is composed of fermion-antifermion pairs, and chiral symmetry is dynamically broken for large values of coupling constant; therefore, they have some analogy to the low-energy theory of QCD and BCS theory of superconductivity.
We investigated two kinds of four-fermion model, Gross-Neveu model and Nambu and Jona-Lasinio model in the toroidal spacetime. The Gross-Neveu model has also been analyzed in a spherical spacetime at finite-chemical potential, which is related to the fermion-number density. Then, critical values of coupling constant, toroidal dimensions, spacetime curvature, and chemical potential have been derived, at which the dynamically broken chiral symmetry is restored, and the dynamically acquired fermion mass becomes zero, and fermions begin to decouple. Additionly, sine-Gordon theory is investigated on the torus. Critical temperatures have been obtained as a function of finite size, at which the system becomes unstable.