Nonlocal interactions and the excitation spectrum in lattice quantum scalar field models

Abstract

We determine the effects of nonlocal, nonlinear interactions on the excitation spectrum of lattice quantum field scalar models. We consider perturbations of a quantized discrete string formally self-adjoint Hamiltonian operator on the lattice d, and with a large mass coefficient for the quadratic term. The low-lying energy–momentum spectrum has an isolated dispersion curve and a two-particle (first) band. We analyse a ladder approximation of the Bethe–Salpeter equation on the lattice, for a weak perturbation of the type ∑d [λ6 : ()6 : +V(()], λ6 > 0, and consider the spectral interval starting at zero and extending to near the three-particle threshold. For space dimension d = 1,2 and V(()) = λ1 : ()4 :, we find that a bound state occurs either below (if λ1 0), but not both. This agrees with recent results where bound states were obtained for the stochastic dynamics generator associated with the relaxation rate to equilibrium in weakly coupled stochastic Ginzburg–Landau models with continuous time and on a spatial lattice d. These results are in contrast, however, with those obtained for V(()) = λ2 : ()3(− Δ)() :. For this case, surprisingly, we show that stable particles exist simultaneously above and below the band, for d = 1,2, regardless of the sign of the coupling λ2. If V(()) = λ3 : ()2(− Δ2)() :, the ladder analysis is inconclusive. If d = 3, 4, ..., no bound states exist in the spectral region we consider.