Transfer matrix spectrum for lattice classical O(N) ferromagnetic spin systems at high temperature

Abstract

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 \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s} = (s_1 ,s_2 , \ldots ,s_N ),\alpha _N \equiv \left\langle {\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s} \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s} } \right)^2 } \right\rangle - \frac{{N + 2}}{N}\left\langle {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s} ^2 } \right\rangle ^2 > 0\); 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.