Dynamical symmetry breaking on a lattice

We propose to study dynamical symmetry breaking on a spatial lattice. Experiences with solid-state physics suggest that one should deal with an effective interaction between quarks and antiquarks rather than starting with the fundamental local Lagrangian. As an example, we study the ${\left(\overline{\psi }\psi \right)}^2={\left(\Sigma {\alpha =1\mathrm{}}^N{\overline{\psi }}_{\alpha }{\psi }_{\alpha }\right)}^2$ interaction of massless fermions in two dimensions on a lattice in the approximation of a large number of degrees of freedom, $N$. We show explicitly that the model reduces to one-dimensional superconductivity by following the original methods of Bardeen, Cooper, and Schrieffer. The lattice coupling constant ${g_0}^2$ is found to go as $a$ for large lattice spacing $a$ and as $-\frac{1}{\ln a}$ for small $a$, and to have a finite cut for imaginary values of $a$. The same result may be obtained by a path-integral approach. For $N=1\mathrm{}$ the model reduces to an antiferromagnetic chain.