Ising Spin Glasses and Renormalization Group Theory: the Binder cumulant

TL;DR

This paper investigates the scaling behavior of the Binder cumulant and correlation length in Ising models and spin glasses, revealing violations of expected hyperscaling rules in spin glasses through numerical analysis.

Contribution

It provides the first numerical evidence that the Josephson hyperscaling rule does not hold for Ising spin glasses in three and four dimensions.

Findings

The Renormalization Group theory rule linking critical exponents is obeyed in Ising ferromagnets.

The rule is violated in Ising spin glasses in 3D and 4D.

Standard hyperscaling relations break down in spin glasses.

Abstract

Numerical data on scaling of the normalized Binder cumulant and the normalized correlation length are shown for the Thermodynamic limit regime, first for canonical Ising ferromagnet models and then for a range of Ising spin glass models. A fundamental Renormalization Group Theory rule linking the critical exponents for the two observables is well obeyed in the Ising models, but not for the Ising spin glasses in dimensions three and four. We conclude that there is a violation of a standard Josephson hyperscaling rule in Ising spin glasses.