Vertical and reentrant ferromagnetic boundaries below the Nishimori point

Abstract

We prove that below the multicritical (Nishimori) point the ferromagnetic boundary of a biased Ising spin glass, argued by Nishimori (1986) to be exactly vertical, can be vertical and can also be reentrant, so that cooling at fixed disorder destroys order. It is vertical in the Sherrington-Kirkpatrick (SK) model with a Curie-Weiss bias $J_0$, and reentrant near the triple point of fully connected $p$-spin glasses and, at one disorder strength, on a decorated planar $\mathbb{Z}^2$-periodic lattice. For every $T J$. For every integer $p\ge3$, at each bias slightly larger than the triple-point bias, the $p$-spin magnetization is macroscopic at the Nishimori temperature and vanishes at an explicit lower temperature; this is Nishimori's recent replica prediction in two-temperature form. On the lattice ($\mathbb{Z}^2$ with each bond replaced by a fixed series-parallel gadget; maximum degree four), with iid $\pm J$ couplings at $\mathbb{P}(J_e=-1)=9/10000$, the Gibbs state is almost surely unique at high temperature, non-unique in a window containing the Nishimori temperature, and unique again at low temperature. As Dey and Kang observed, the SK boundary is set by the zero-field susceptibility together with an envelope bound on the free-energy gain in a field, which follows from the gauge inequality. We prove that this susceptibility is the curvature at the origin of the zero-field Parisi PDE solution, which Lopatto evaluated for every $β>1$ via his theorem that the support of the Parisi measure accumulates at the origin; we give a short alternative proof of that theorem. The $p$-spin and lattice results and the alternative proof are computer-assisted. The models are proxies for the nearest-neighbour $\pm J$ model on $\mathbb{Z}^2$ and say nothing about $\mathbb{Z}^d$.