The low-lying energy-momentum spectrum for the four-Fermi model on a lattice

We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice four-Fermi or Gross-Neveu model in \(d+1\) space-time dimensions (\(d=1,2,3\)) and with \(N\)-component fermions. Let \(\kappa>0\) be the hopping parameter, \(\lambda>0\) the four-fermion coupling and \(M>0\) denote the fermion mass; and take \(s\times s\) spin matrices, \(s=2,4\). We work in the \(\kappa\ll 1\) regime. Our analysis of the one- and the two-particle spectrum is based on spectral representation for suitable two- and four-fermion correlations. The one-particle energy-momentum spectrum is obtained rigorously and is manifested by \(sN/2\) isolated and identical dispersion curves, and the mass of particles has asymptotic value \(-\ln\kappa\). The existence of two-particle bound states above or below the two-particle band depends on whether Gaussian domination does hold or does not, respectively. Two-particle bound states emerge from solutions to a lattice Bethe-Salpeter equation, in a ladder approximation. Within this approximation, the \(sN(sN/2-1)/4\) identical bound states have \({\cal O}(\kappa^0)\) binding energies at zero system momentum and their masses are all equal, with value \(\approx -2\ln\kappa\). Our results can be validated to the complete model as the Bethe-Salpeter kernel exhibits good decay properties.