Dynamical symmetry breakings on a nontrivial topology.

Abstract

Dynamical symmetry breakings in two-dimensional massless fermion field theory with quartic interactions (the Gross-Neveu model) are investigated in the large-fermion-number ($N$) limit on a space with the ${S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ topology which may correspond to the finite-volume system at finite temperature. Four types and the sum of the spin structures are considered. It is shown that the model has a richer phase structure in all boundary conditions than those in ${R}^{2}$ or ${R}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ space-time. It depends on the effective area and the ratio of the circumferences of the two circles whether or not dynamical symmetry breakings occur. In the sum of the spin structures, the phase diagram is the same as that of the periodic-periodic boundary condition and the critical line equation can be written in modular-invariant form.