The Low Lying Energy-Momentum Spectrum for the Lattice Four-Fermi Model

Abstract

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 κ > 0 be the hopping parameter, λ > 0 the four-fermion coupling and M > 0 denote the fermion mass; and take s × s spin matrices, s = 2, 4. We work in the κ ≪ 1 regime. Our analysis of the oneand 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 κ. 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 O(κ 0 ) binding energies at zero system momentum and their masses are all equal, with value ≈ −2 ln κ. Our results can be validated to the complete model as the Bethe-Salpeter kernel exhibits good decay properties.