Anomalous finite size corrections in random field models

TL;DR

This paper introduces a replica-based analytical method to compute anomalous finite size corrections in random field models, demonstrating its effectiveness on mean field Ising models and suggesting broader applicability.

Contribution

The paper presents a novel analytical approach using the replica trick to calculate $O(rac{1}{ oot{2}}N)$ finite size corrections in random field models, extending understanding of finite size effects.

Findings

Derived explicit formulas for finite size corrections in mean field Ising models.

Validated analytical results with numerical simulations.

Suggested the presence of similar effects in finite dimensional models.

Abstract

The presence of a random magnetic field in ferromagnetic systems leads, in the broken phase, to an anomalous $O(\sqrt{1/N})$ convergence of some thermodynamic quantities to their asymptotic limits. Here we show a general method, based on the replica trick, to compute analytically the $O(\sqrt{1/N})$ finite size correction to the average free energy. We apply this method to two mean field Ising models, fully connected and random regular graphs, and compare the results to exact numerical algorithms. We argue that this behaviour is present in finite dimensional models as well.