Resolving Spurious Multifractality in Discrete Systems: A Finite-Size Scaling Protocol for MFDFA in the 2D Ising Model

TL;DR

This paper establishes a finite-size scaling protocol for MFDFA in the 2D Ising model, resolving spurious multifractality issues and accurately identifying genuine multifractal spectra in disordered systems.

Contribution

It introduces a systematic finite-size scaling method for MFDFA, clarifying the distinction between artifacts and true multifractality in discrete lattice models.

Findings

Finite-size effects cause broad spectra in negative moments, which vanish in the thermodynamic limit.

The method recovers the expected monofractal exponent for the pure Ising model.

Disorder induces a genuine broad multifractal spectrum that persists under scaling.

Abstract

Multifractal Detrended Fluctuation Analysis (MFDFA) has emerged as a standard tool for characterizing scale invariance in complex systems, yet its application to discrete spin models is frequently marred by reports of ``spurious multifractality'' that contradict established theory. In this work, we resolve this controversy by establishing a rigorous protocol for the analysis of discrete lattice snapshots. Using the 2D Ising model as a benchmark, we demonstrate that the previously reported broad singularity spectra \cite{Ludescher2011} are finite-size artifacts dominated by lattice discreteness effects in the negative moment regime ($q<0$). By restricting the analysis to positive moments and performing a systematic Finite-Size Scaling (FSS) analysis, we show that the spectral width collapses to zero ($\Delta \alpha \to 0$) in the thermodynamic limit. The method accurately recovers the monofractal exponent of the Ising universality class ($\alpha \approx H \approx 0.875$), consistent with Conformal Field Theory. To validate the discriminatory power of this protocol, we contrast these findings with the Random Bond Ising Model (RBIM), showing that quenched disorder induces a genuine, broad multifractal spectrum ($\Delta \alpha \approx 0.23$) that survives scaling. Furthermore, we propose a theoretical interpretation where the MFDFA polynomial detrending functions as a phenomenological Renormalization Group filter, suppressing analytic background fields (irrelevant operators) to isolate the singular critical behavior. These results define a robust methodology for distinguishing between clean and disorder-dominated criticality in finite systems.