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

TL;DR

This paper estimates the critical temperature of quantum spin systems on regular trees using a probabilistic random loop representation, proposing a formula likely applicable to large cubic lattices.

Contribution

It introduces a new probabilistic method to estimate critical temperatures for quantum spin models on regular trees, including the XXZ and nematic models.

Findings

Derived a formula for critical temperature on regular trees

Conjectured applicability to large cubic lattices

Validated the approach using random loop representations

Abstract

We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-$\frac12$ 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.