Short-Range Ising Spin Glass: Multifractal Properties

TL;DR

This study investigates the multifractal nature of the local order parameter in a 3D short-range Ising spin glass model near criticality, revealing universal multifractal spectra and a transition signal above the critical temperature.

Contribution

It provides an exact recursive analysis of the multifractal properties of the Edwards-Anderson order parameter in a 3D spin glass model, highlighting universality and transition behavior.

Findings

Multifractal spectra are required to describe the local order parameter.

Spectra show universality across different coupling distributions.

A dramatic change in the spectra indicates the phase transition.

Abstract

The multifractal properties of the Edwards-Anderson order parameter of the short-range Ising spin glass model on d=3 diamond hierarchical lattices is studied via an exact recursion procedure. The profiles of the local order parameter are calculated and analysed within a range of temperatures close to the critical point with four symmetric distributions of the coupling constants (Gaussian, Bimodal, Uniform and Exponential). Unlike the pure case, the multifractal analysis of these profiles reveals that a large spectrum of the $\alpha $-H\"older exponent is required to describe the singularities of the measure defined by the normalized local order parameter, at and below the critical point. Minor changes in these spectra are observed for distinct initial distributions of coupling constants, suggesting an universal spectra behavior. For temperatures slightly above T_{c}, a dramatic change in the $F(\alpha)$ function is found, signalizing the transition.