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-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.1991 Mathematics Subject Classification. 60K35, 82B20, 82B26, 82B31.

On a class of orthogonal-invariant quantum spin systems on the complete graph

“…where we used π λ implies bn λ,ρ = 0 which implies b n λ,ρ = 0, since b n λ,ρ ≤ bn λ,ρ . We also have, from (11) αn…”

“…This model was introduced by Ueltschi [33], who showed, for certain values of the parameters, equivalence with an interchange process with "reversals". For these parameters, the model and interchange process have been studied on Z d [10], [14], trees [11], [21], graphs of bounded degree [25], and the complete graph [9], [8], the latter of which computes many observables. Our methods allow us to deal with all values of the parameters, not just those for which the probabilistic representation holds.…”

Phase transition for loop representations of quantum spin systems on trees

“…In a very recent preprint [5], they extend these results to the case where θ = 1. The justification for studying a tree is that for very high space dimensions, the difference between a d-regular tree and Z d in terms of percolation questions should be small.…”

Sharp phase transition for random loop models on trees

“…As it is the case in the random stirring model, the most interesting (but also apparently the most challenging) graph to study these models on is Z d . Mathematical results exist for the complete graph [9,13], the 2-dimensional Hamming graph [1], Galton-Watson trees [8] and the d-ary tree [12], again in the regime of high degrees. Unfortunately, for θ = 1, the weighted measures involve intricate correlations and the techniques of our paper do not directly apply.…”