Predictive power of MCT: Numerics and Finite size scaling for a mean field spin glass

TL;DR

This paper critically tests Mode Coupling Theory predictions in a mean field spin glass model, revealing partial validation and highlighting the impact of finite size effects and sample fluctuations on the theory's applicability.

Contribution

It provides a detailed numerical analysis of MCT in a fully connected spin glass model, introducing a modified finite size scaling approach for disordered systems.

Findings

Some MCT predictions are verified, others are violated.

Strong pre-asymptotic effects influence the results.

Standard FSS fails due to sample fluctuations, leading to a new modified FSS.

Abstract

The aim of this paper is to test numerically the predictions of the Mode Coupling Theory (MCT) of the glass transition and study its finite size scaling properties in a model with an exact MCT transition, which we choose to be the fully connected Random Orthogonal Model. Surprisingly, some predictions are verified while others seem clearly violated, with inconsistent values of some MCT exponents. We show that this is due to strong pre-asymptotic effects that disappear only in a surprisingly narrow region around the critical point. Our study of Finite Size Scaling (FSS) show that standard theory valid for pure systems fails because of strong sample to sample fluctuations. We propose a modified form of FSS that accounts well for our results. {\it En passant,} we also give new theoretical insights about FSS in disordered systems above their upper critical dimension. Our conclusion is that the quantitative predictions of MCT are exceedingly difficult to test even for models for which MCT is exact. Our results highlight that some predictions are more robust than others. This could provide useful guidance when dealing with experimental data.