Mean Field Upper Bound on the Transition Temperature in Multicomponent Ferromagnets 1

Using local Ward identities and a new correlation inequality, we prove that the mean field transition temperature is an upper bound on the true transition temperature in multicomponent Heisenberg-type classical ferromagnets. (49' = 1 i=1 Given a translation-invariant function J(c~-t3) on Z v x Z ~, we consider for A c Z ~ the Hamiltonian D c~,B~A i = 1 cEA and form the partition function with a priori measure the usual one on S ~ The spontaneous magnetization is as usual the right-hand derivative of the pressure limh,0[p(h)-p(O)]/h at h = 0. Here we will prove: Theorem. Let D ~> 2. IfJ(c~-fl) i> 0 and j = ~S(a) < D (1) then the spontaneous magnetization is zero. Remarks 1. j = D is the mean field value. It is probably the best value for which the theorem holds, since using either the methods of Ref. 9 or Ref. 6,