Vacuum polarization and spontaneous symmetry breaking in a two-dimensional massless fermion field theory with quartic interactions (the Gross-Neveu model) are studied in a cylindrical ( R ' x S ' ) space-time topology. In the case of untwisted fermion fields the symmetry-breaking behavior is similar to the model in Minkowski space-time except that the location of the effective potential minima depends upon the size of the space. In the case of a twisted fermion field there exists a critical size of the space such that the symmetry breaking occurs only for space larger than the critical one.
The Gross-Neveu model is analyzed by the Gaussian approximation in the functional Schrodinger picture. It is shown that in the large-N limit the Gaussian approximation exactly reproduces the Gross-Neveu results, but for finite N it contains more information than the large-N approximation. There are two nontrivial phases of the theory depending upon the sign of the infinitesimal bare coupling constant. Dynamical symmetry breaking occurs in one of the phases. We also apply our analysis to the chiral Gross-Neveu model.
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