Dynamical symmetry breaking on a lattice

Abstract

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 ${(\overline{\ensuremath{\psi}}\ensuremath{\psi})}^{2}={(\ensuremath{\Sigma}{\ensuremath{\alpha}=1}^{N}{\overline{\ensuremath{\psi}}}_{\ensuremath{\alpha}}{\ensuremath{\psi}}_{\ensuremath{\alpha}})}^{2}$ 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 $\ensuremath{-}\frac{1}{\mathrm{ln}a}$ 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 $N=1$ the model reduces to an antiferromagnetic chain.