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

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 ($Δα\to 0$) in the thermodynamic limit. The method accurately recovers the monofractal exponent of the Ising universality class ($α\approx H \approx 0.875$), 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 ($Δα\approx 0.23$) 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.

Figures (5)

• Figure 1: Schematic of the MFDFA detrending interpreted as an RG filter. The raw integrated profile $Y(x)$ (top, blue) is a superposition of singular critical fluctuations and a smooth analytic background. The polynomial trend $P_\nu(x)$ (red dashed) captures the macroscopic background field driven by irrelevant operators. The MFDFA detrending operation acts as a projection $\hat{\mathcal{P}}_\nu$, subtracting this background to isolate the singular scaling part (bottom, green). This filtering grants access to the intrinsic critical exponents even in finite systems where correction-to-scaling terms are significant.
• Figure 2: Evolution of the Hurst exponent $H$ ($q=2$) vs Temperature for $L=256$. The intersection with the theoretical value $H=0.875$ (blue dotted line) occurs precisely at the critical temperature $T_c$ (red dashed line).
• Figure 3: Evolution of the singularity spectra $f(\alpha)$ for $q>0$. Darker curves correspond to smaller system sizes ($L=32$), while lighter curves represent larger systems ($L=256$). A clear narrowing effect is observed as $L$ increases.
• Figure 4: Finite-Size Scaling (FSS) of the singularity exponent $\langle \alpha \rangle$. The convergence towards the theoretical value $\alpha = 0.875$ (green line) is evident as the system size increases ($1/L \to 0$).
• Figure 5: Sensitivity of MFDFA to disorder. While the pure Ising model converges to a vertical line (green dashed line, $\Delta \alpha \to 0$), the Random Bond Ising Model (RBIM, red circles) McCoy1968 displays a broad multifractal spectrum ($\Delta \alpha \approx 0.23$), revealing the complex fluctuation structure induced by quenched impurities.