# Hyperscaling breakdown and Ising Spin Glasses: the Binder cumulant

## TL;DR

This paper investigates the breakdown of hyperscaling relations involving the Binder cumulant in Ising models, especially in higher dimensions and spin glasses, revealing different scaling behaviors in various regimes.

## Contribution

It provides a detailed numerical analysis of the Binder cumulant and correlation length scaling in Ising models and spin glasses, highlighting where hyperscaling fails.

## Key findings

- Hyperscaling holds in 3D ferromagnets.
- Hyperscaling breaks down in 5D ferromagnets.
- Binder cumulant becomes size-independent in spin glasses near criticality.

## Abstract

Among the Renormalization Group Theory scaling rules relating critical exponents, there are hyperscaling rules involving the dimension of the system. It is well known that in Ising models hyperscaling breaks down above the upper critical dimension. It was shown by M. Schwartz [Europhys. Lett. {\bf 15}, 777 (1991)] that the standard Josephson hyperscaling rule can also break down in Ising systems with quenched random interactions. A related Renormalization Group Theory hyperscaling rule links the critical exponents for the normalized Binder cumulant and the correlation length in the thermodynamic limit. An appropriate scaling approach for analyzing measurements from criticality to infinite temperature is first outlined. Numerical data on the scaling of the normalized correlation length and the normalized Binder cumulant are shown for the canonical Ising ferromagnet model in dimension three where hyperscaling holds, for the Ising ferromagnet in dimension five (so above the upper critical dimension) where hyperscaling breaks down, and then for Ising spin glass models in dimension three where the quenched interactions are random. For the Ising spin glasses there is a breakdown of the normalized Binder cumulant hyperscaling relation in the thermodynamic limit regime, with a return to size independent Binder cumulant values in the finite-size scaling regime around the critical region.