$d$-dimensional spherical ferromagnets in random fields: Metastates, continuous symmetry breaking, and spin-glass features
Kalle Koskinen, Christof Külske
TL;DR
This work analyzes $d$-dimensional spherical ferromagnets subject to i.i.d. $d$-vector random fields, comparing models with scaled and non-scaled field strengths. It establishes ferromagnetic order in a high-β, low-disorder regime and shows a rich metastate structure: in non-scaled fields the set of Gibbs-measure cluster points includes a continuum of product states and, for $d\ge2$, a trivial overlap, while in scaled fields the overlaps exhibit replica-symmetry breaking and non-self-averaging, with ultrametricity only in $d=1$. The paper develops a detailed finite-volume analysis via shifted microcanonical measures and exponential tilting, proving uniform convergence, concentration results, and precise limit points, then translates these into AW and NS metastates, as well as overlap distributions. The results link large-volume behavior to random-walk limits, Brownian projections, and sphere-based tilts, illuminating when spin-glass-like features emerge and how they depend on spin dimension and field-scaling. These findings advance understanding of random-field effects in continuous-symmetry spin systems, with implications for metastate descriptions and spin-glass phenomena in high-dimensional disordered ferromagnets.
Abstract
We study the large-volume behavior of the spherical model for $d$-dimensional local spins, in the presence of $d$-dimensional random fields, for $d\geq 2$. We compare two models, one with volume-scaled random fields, and another one with non-scaled random fields, on the level of Aizenman-Wehr metastates, Newman-Stein metastates, as well as overlap distributions. We show that in $d\geq 2$ the metastates are fully supported on a continuity of random product states, with weights which we describe, for both models. For the non-scaled random fields, the set of a.s. cluster points of Gibbs measures contains these product states, but behaves differently in the 'recurrent' spin dimension $d=2$ where it also contains non-trivial mixtures of tilted measures. For the scaled model, moreover the overlap distribution displays spin-glass characteristics, as it is non-self averaging, and shows replica symmetry breaking, although it is ultrametric if and only if $d=1$. For $d\geq 2$ it oscillates chaotically on a set of continuous distributions for large volumes, while the limiting set contains only discrete distributions in $d=1$. Our results are based on concentration estimates, analysis of Gibbs measures in finite but large volumes, and the asymptotics of $d$-dimensional random walks and their spherical projections.