Spherical Model in a Random Field

TL;DR

This paper studies the spherical model in a random field, revealing how boundary fields influence self-averaging properties and analyzing phase transitions and correlations at criticality.

Contribution

It demonstrates the conditional self-averaging of magnetization, the dominance of boundary fields in the ferromagnetic phase, and the phase transition of the effective field at critical temperature.

Findings

Magnetization is not self-averaging on the critical line.

Weak boundary fields restore self-averaging of thermodynamic observables.

Effective field exhibits long-range correlations at criticality.

Abstract

We investigate the properties of the Gibbs states and thermodynamic observables of the spherical model in a random field. We show that on the low-temperature critical line the magnetization of the model is not a self-averaging observable, but it self-averages conditionally. We also show that an arbitrarily weak homogeneous boundary field dominates over fluctuations of the random field once the model transits into a ferromagnetic phase. As a result, a homogeneous boundary field restores the conventional self-averaging of thermodynamic observables, like the magnetization and the susceptibility. We also investigate the effective field created at the sites of the lattice by the random field, and show that at the critical temperature of the spherical model the effective field undergoes a transition into a phase with long-range correlations $\sim r^{4-d}$.