Critical behavior in a class ofO(N)-invariant field theories in two dimensions

Abstract

Critical behavior in a class of two-dimensional field theories which exhibit dynamical symmetry breaking at zero temperature is analyzed in the $\frac{1}{N}$ approximation. We show that, in the case of an $\mathrm{O}(N)$-invariant theory of massless, $N$-component, Fermi fields, a phase transition takes place in the limit as $N$ goes to infinity. The critical temperature, above which the model becomes symmetric, is given in terms of the induced fermion mass at zero temperature, ${m}_{f}^{0}$, as ${m}_{f}^{0}{\ensuremath{\beta}}_{c}=1.764$. The equivalence between the critical parameters of the theory and those predicted by the BCS theory of superconductivity is established. We show that the BCS gap equation follows from the stability conditions imposed on the effective potential. The phase transition is discussed in a thermodynamical analog of the model. The analysis of the symmetry behavior of the theory is carried out by functional methods.