Pemantle–Steif phase-transition conjecture for Heisenberg models

Let $\Gamma$ be a tree with branching number $\operatorname{br}(\Gamma)$, let $d\geq 1$, and let $\rho^d(J)$ denote the first normalized spherical harmonic coefficient associated with the $d$-dimensional Heisenberg interaction at parameter $J$.

Pemantle–Steif conjecture. If

$$\operatorname{br}(\Gamma)\rho^d(J)<1,$$

then the $d$-dimensional Heisenberg model on $\Gamma$ with parameter $J$ does not exhibit a phase transition.

The paper proves the corresponding criterion for robust phase transition and notes that the conjecture would establish coincidence of phase transition and robust phase transition for Heisenberg models. It also proves the claim when the branching number is $1$.