Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

Abstract

We consider lattice classical ferromagnetic spin systems at high temperature (β≪1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m≅−lnβ in the energy-momentum spectrum. If α≡〈s ⁴〉−3〈s ²〉²<0, where 〈s ^k〉 is the kth moment of the ssd, and β is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For α>0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1−γ)+O(β), where γ=α(α+2〈s ²〉²)⁻¹. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.