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Journal of Statistical Physics

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

Ricardo S. Schor, Michael O’Carroll

Published Jun 1, 2000• 34 citations
Publisher
DOI
10.1023/a:1018688722554

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Critical phenomena in ferromagnetic spin systems on lattices

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We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region $(\ensuremath{\beta}\ensuremath{\ll}1).$ Each model is characterized by a single-site a priori spin probability distribution taken to be noneven. We state our results in terms of the parameter $\ensuremath{\alpha}=(〈\overline{s}{}^{4}〉\ensuremath{-}3〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1})/(〈\overline{s}{}^{4}〉\ensuremath{-}〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1}),$ where $\overline{s}=s\ensuremath{-}〈s〉,$ and 〈sk〉〈{s}^{k}〉〈sk〉 denotes the kthk\mathrm{th}kth moment of the single-site distribution. Associated with the model is a lattice quantum field theory which is known to contain a particle of mass $m\ensuremath{\sim}\mathrm{ln}\ensuremath{\beta}.$ Assuming $〈{s}^{3}〉\ensuremath{\ne}0$ we show that for $\ensuremath{\alpha}>0,$ \ensuremath{\beta} small, there exists a bound state with mass below the two-particle threshold 2m.2m.2m. For $\ensuremath{\alpha}<0$ bound states do not exist. These results are obtained using a Bethe-Salpeter (BS) equation in the ladder approximation in conjunction with a representation for the inverse of the two-point function designed to analyze the spectrum below but close to 2m.2m.2m.
We obtain different properties of general d dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (beta<<1). Each model is characterized by a single site a priori spin distribution, taken to be even. We state our results in terms of the parameter alpha=<s(4)>-3<s(2)>(2) where <s(k)> denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory that is known to contain particles. We show that for alpha>0, beta small, there exists a bound state with mass below the two-particle threshold. For alpha<0, bound states do not exist. The existence of the bound state has implications on the decay of correlations, i.e., the four-point function decays at a slower rate than twice that of the two-point function. These results are obtained using a lattice version of the Bethe-Salpeter equation in the ladder approximation. The existence and nonexistence results generalize to N-component models with rotationally invariant a priori spin distributions.