Transfer Matrix Spectrum for Lattice Classical O(N) Ferromagnetic Spin Systems at High Temperature

We obtain new properties of d-dimensional lattice ferromagnetic classical N-component vector spin systems in the high temperature region. Each model is characterized by a single site a priori single spin probability distribution (sspd) which we take to be rotationally invariant. Associated with the model is a discrete imaginary time lattice quantum field theory which is known to contain particles of mass m. Letting 〈⋅〉 denote the sspd expectation we show that there exists two bound states below but near the two-particle threshold 2m if, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s} = (s_1 ,s_2 , \ldots ,s_N ),\alpha _N \equiv \left\langle {\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s} \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s} } \right)^2 } \right\rangle - \frac{{N + 2}}{N}\left\langle {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s} ^2 } \right\rangle ^2 > 0$$ \end{document}; if αN < 0 there are no bound states. These results are obtained using a lattice version of the Bethe–Salpeter equation in a ladder approximation.