Hyperscaling violation in Ising Spin Glasses

TL;DR

This paper investigates the breakdown of hyperscaling relations in Ising Spin Glasses across multiple dimensions, revealing violations consistent with theoretical predictions and providing numerical estimates of the hyperscaling violation exponent.

Contribution

It presents numerical evidence of hyperscaling violation in Ising Spin Glasses in dimensions 3, 4, 5, and 7, extending understanding of critical behavior in disordered systems.

Findings

Hyperscaling is violated in dimensions 3 and 4.

Violations are observed above the upper critical dimension D=6.

Estimates of the hyperscaling violation exponent are provided.

Abstract

In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models which are in this class have been intensively studied. Here numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5 and 7. Hyperscaling is clearly violated in dimensions 3 and 4 as well as above the upper critical dimension $D=6$. Estimates are obtained for the "violation of hyperscaling exponent" values in the various models.