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arXiv:2603.04609 · Physical review. E

Finite-size scaling to resolve spurious multifractality in the two-dimensional Ising model

S. Jaroszewicz, Nahuel Daniel Mendez, Maria Pia Beccar-Varela, Maria Cristina Mariani

Published Jun 29, 2026• 0 citations
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Abstract

Multifractal detrended fluctuation analysis 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 two-dimensional Ising model as a benchmark, we demonstrate that the previously reported broad singularity spectra 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→0) in the thermodynamic limit. The method accurately recovers the monofractal exponent of the Ising universality class (α≈H≈0.875), consistent with conformal field theory. Applying this FSS protocol to the random bond Ising model, we reveal that the broad multifractal dispersion often attributed to quenched disorder also flattens systematically with increasing system size. These results demonstrate that apparent multifractality in finite discrete lattices is predominantly driven by preasymptotic discreteness artifacts, underscoring the universal necessity of FSS to avoid false positives in statistical mechanics.

DOI
10.1103/23vj-5v8c

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Multifractal detrended fluctuation analysis 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 two-dimensional Ising model as a benchmark, we demonstrate that the previously reported broad singularity spectra 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→0) in the thermodynamic limit. The method accurately recovers the monofractal exponent of the Ising universality class (α≈H≈0.875), consistent with conformal field theory. Applying this FSS protocol to the random bond Ising model, we reveal that the broad multifractal dispersion often attributed to quenched disorder also flattens systematically with increasing system size. These results demonstrate that apparent multifractality in finite discrete lattices is predominantly driven by preasymptotic discreteness artifacts, underscoring the universal necessity of FSS to avoid false positives in statistical mechanics.
Multifractal properties of the two-dimensional Ising model are studied by means of the $f(\ensuremath{\alpha})$ spectra of singularities of probability measures supported by energy spectra. These measures are determined through calculations of energy degeneracies for finite-size systems up to $12\ifmmode\times\else\texttimes\fi{}12$ spins in cases of square and triangular lattices and up to $22\ifmmode\times\else\texttimes\fi{}22$ spins in the case of the hexagonal lattice. The calculations are performed in an exact manner using the transfer-matrix method. It is argued that, in the thermodynamic limit, the scaling exponent ${\ensuremath{\alpha}}_{\mathrm{max}}$ associated with the most probable energy of the system takes at the critical temperature a minimum value and, consequently, it is argued that a given system reveals a phase transition at a finite temperature if ${\ensuremath{\alpha}}_{\mathrm{max}}$ possesses a minimum at the finite temperature. It is also shown that, in the thermodynamic limit, the spectrum $f(\ensuremath{\alpha})$ shrinks when the critical temperature is approached.
The recent progress in the study of finite-size scaling (FSS) properties of the Ising model is briefly reviewed. We calculate the universal FSS functions for the Binder parameter g and the magnetization distribution function p(m) for the Ising model on L1×L2 two-dimensional lattices with tilted boundary conditions. We show that the FSS functions are universal for fixed sets of the aspect ratio L1/L2 and the tilt parameter. We also study the percolating properties of the Ising model, giving attention to the effects of the aspect ratio of finite systems. We elucidate the origin of the complex structure of p(m) for the system with large aspect ratio by the multiple-percolating-cluster argument.
We briefly review the Ising model with uncorrelated, quenched random-site or random-bond disorder, which has been controversial in both two and four dimensions. In these dimensions, the leading exponent alpha, which characterizes the specific-heat critical behaviour, vanishes and no Harris prediction for the consequences of quenched disorder can be made. In the two-dimensional case, the controversy is between the strong universality hypothesis which maintains that the leading critical exponents are the same as in the pure case and the weak universality hypothesis, which favours dilution-dependent leading critical exponents. Here the random-site version of the model is subject to a finite-size scaling analysis, paying special attention to the implications for multiplicative logarithmic corrections. The analysis is fully supportive of the scaling relations for logarithmic corrections and of the strong scaling hypothesis in the 2D case. In the four-dimensional case unusual corrections to scaling characterize the model, and the precise nature of these corrections has been debated. Progress made in determining the correct 4D scenario is outlined.
The equilibrium ensemble approach to disordered systems is used to investigate the critical behavior of the two-dimensional Ising model in the presence of quenched random site dilution. The numerical transfer matrix technique in semi-infinite strips of finite width, together with phenomenological renormalization and conformal invariance, is particularly suited to putting the equilibrium ensemble approach to work. A method by which to extract with great precision the critical temperature of the model is proposed and applied. A more systematic finite-size scaling analysis than in previous numerical studies has been performed. A parallel investigation, along the lines of the two main scenarios currently under discussion, namely, the logarithmic corrections scenario (with critical exponents fixed in the Ising universality class) versus the weak universality scenario (critical exponents varying with the degree of disorder), is carried out. In interpreting our data, maximum care is constantly taken to be open to both possibilities. A critical discussion shows that an unambiguous discrimination between the two scenarios is still not possible on the basis of the available finite-size data.
We present a high-precision finite-size scaling (FSS) analysis of the two-dimensional Ising model using single-spin Metropolis Monte Carlo simulations on lattice sizes L=12L = 12L=12–128128128. Critical behaviour is characterised through multiple independent observables, including susceptibility peaks, Binder cumulants, and scaling collapse, yielding a consistent estimate of the critical temperature Tc=2.2690(8)T_c = 2.2690(8)Tc=2.2690(8) with excellent agreement to established benchmarks. Temperature-resolved integrated autocorrelation times τint(T) au_{mathrm{int}}(T)τint(T) are computed to account for statistical correlations, demonstrating classical critical slowing down with scaling τint∝Lz au_{mathrm{int}} propto L^zτint∝Lz, where z≈2.1z approx 2.1z≈2.1. All uncertainties are corrected using correlation-aware error propagation. Building on these results, we introduce a Scale–Coherence Junction (SCJ) interpretation, defining a thermodynamic coherence energy scale: E_coh = k_B * T_c This provides a phenomenological bridge between critical phenomena and coherence thresholds, suggesting that maximal system-wide correlations at criticality correspond to enhanced sensitivity to environmental interactions.
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-size-scaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlation-length amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10(-11) or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.