Critical temperature of Heisenberg models on regular trees, via random loops

Abstract

We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-$\frac{1}{2}$ XXZ model and the spin-1 nematic model. Our formula is conjectured to be valid for large-dimensional cubic lattices. Our method of proof uses a probabilistic representation in terms of random loops.

14 pages, to appear in J. Stat. Phys