LONG RANGE ORDER FOR THE QUANTUM HEISENBERG MODEL
Contributed by: Elliott H. Lieb (Princeton Univ.), Jan. 27, 1999.
Abstract The Peierls argument (as modified by Griffiths and by Dobrushin) showed that long range order (LRO) exists in the Ising model for low temperature, T, in dimension D=2 or greater. For many years the analogous statement for the Heisenbeg model, in which the spin symmetry is continuous rather than discrete, was an open problem. It was finally proved for the classical model (for D >2) in . The quantum case was solved in  for T>0, but only for the antiferromagnet and for large enough spin. (Note: classically there is no mathematical difference between the ferro- and antiferromagnets, but there is a huge difference quantum-mechanically. In particular, reflection positivity, which holds for the antiferromagnet, fails for the ferromagnet.) In  LRO was proved for the ground state of the 2D antiferromagnet for spin 1 or greater and in  this was extended to all spins greater than 0 for dimensions D>2. Several open problems seem to be annoyingly difficult, but their solution would probably add some new techniques to the theory.