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arXiv:2605.15441

Fractal signatures of strong universality in the disordered Ising model

H. Lima, Kaue H. Abbehausen, Ismael S. S. Carrasco, Jairo R. L. de Almeida, Fernando A. Oliveira

Published May 14, 2026• 0 citations
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Abstract

Disordered systems are very rich laboratories for exploring complex systems. In particular, disordered magnetic systems have been extremely important in the last five decades for understanding a wide range of phenomena. In this work, we use the Edwards-Anderson Hamiltonian to obtain the thermodynamic properties of disordered magnetic systems. In this way, we conduct a systematic investigation of magnetization, correlation functions, order parameter, and fractal dimensions, in function of disorder. In this context, the autocorrelation function for order--parameter fluctuations, introduced by Fisher ( Journal of Mathematical Physics 5, 944322 (1964)), provides an important mathematical framework for understanding the second-order phase transition at equilibrium. However, his analysis is restricted to a Euclidean space of dimension $d$, and an exponent $η$ is introduced to correct the spatial behavior of the correlation function at $T=T_c$. In recent work, Lima et al ( Phys. Rev. E 110, L062107 (2024)) demonstrated that at $T_c$ a fractal analysis is necessary for a complete description of the correlation function. We use Monte Carlo simulations to validate analytical results and to show how disorder alters critical exponents , giving rise to different universality classes. On the other hand, there is a subgroup of critical exponents and fractal dimensions that are invariant with disorder. This subgroup heralds a strong universality class.

DOI
10.48550/arxiv.2605.15441

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Disordered systems are very rich laboratories for exploring complex systems. In particular, disordered magnetic systems have been extremely important in the last five decades for understanding a wide range of phenomena. In this work, we use the Edwards-Anderson Hamiltonian to obtain the thermodynamic properties of disordered magnetic systems. In this way, we conduct a systematic investigation of magnetization, correlation functions, order parameter, and fractal dimensions, in function of disorder. In this context, the autocorrelation function for order--parameter fluctuations, introduced by Fisher ( Journal of Mathematical Physics 5, 944322 (1964)), provides an important mathematical framework for understanding the second-order phase transition at equilibrium. However, his analysis is restricted to a Euclidean space of dimension ddd, and an exponent ηηη is introduced to correct the spatial behavior of the correlation function at T=TcT=T_cT=Tc​. In recent work, Lima et al ( Phys. Rev. E 110, L062107 (2024)) demonstrated that at TcT_cTc​ a fractal analysis is necessary for a complete description of the correlation function. We use Monte Carlo simulations to validate analytical results and to show how disorder alters critical exponents , giving rise to different universality classes. On the other hand, there is a subgroup of critical exponents and fractal dimensions that are invariant with disorder. This subgroup heralds a strong universality class.
Statistical methods are essential for understanding thermodynamic systems with many degrees of freedom. For systems in equilibrium, a very useful method is that of correlation functions, which establish a correlation between a field phi(x), which depends on the spatial position x, and the same field evaluated at another position, phi(x0). Fisher [Journal of Mathematical Physics 5, 944322 (1964)] introduced the autocorrelation function for fluctuations of the order parameter, which has been an important mathematical tool for understanding second-order phase transitions in equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent eta is introduced to correct the spatial behavior of the correlation function at T = Tc. In a recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that a modern fractional differential analysis is necessary for a complete description of the correlation function at Tc. In this study, we highlight the deep connection among scaling behavior, critical exponents, and fractal geometry. Our results provide a unified geometric interpretation of critical exponents and fractal dimensions, broadly applicable to thermodynamic phase transitions. However, the approach does not apply to topological phase transitions, which lack local order parameters and the associated scale-invariant fractal geometry. We verify its predictions for several cornerstone thermodynamic models: the Ising, Potts, XY, and Heisenberg systems.
Abstract Understanding phase transitions requires not only identifying order parameters but also characterizing how their correlations behave across scales. By quantifying how fluctuations at distinct spatial or temporal points are related, correlation functions reveal the structural organization of complex systems. In this work, we reexamine the theoretical foundations of these correlations in systems undergoing second-order phase transitions, with emphasis on the Ising model extended to non-integer spatial dimensions. We revisit the hypotesis that, at criticality, the equilibrium dynamics become effectively confined to the fractal edge of spin clusters and redo the analysis using fractional calculus. Within this framework, the fractal dimension that governs the correlations in that subspace is directly related to Fisher’s exponent η\etaη η , which quantifies the singular behavior of the correlation function near criticality. Importantly, this correlation fractal dimension is distinct from the fractal dimension associated with the order parameter. The fractional approach allows us to directly compute the correlation fractal dimension and to establish an explicit geometrical relation connecting the two fractal dimensions. Moreover, the formulation naturally extends to non-integer spatial dimensions, remaining valid below the upper critical dimension and yielding the correct value of Fisher’s exponent η\etaη η for a continuous spatial dimension d . Within this framework, we also provide empirical functions describing how the main critical exponents vary continuously as a function of the space dimension.
Phase transitions and their critical scaling properties, especially in systems with disorder, are important both for our theoretical understanding of our environment, but also for their practical use in applications and materials in our everyday life. This thesis presents results from finite size scaling analysis of critical phenomena in systems with disorder, using high-precision Monte Carlo simulations and state of the art numerical methods. Specifically, theoretical models suitable for simulations in the presence of uncorrelated or correlated disorder are studied. Uncorrelated strong disorder, as present in the two dimensional gauge glass model to study the vortex glass phase of high temperature superconductors in an applied magnetic field is shown to lack a finite temperature phase transition. Further, results from dynamic quantities, such as resistance and autocorrelation functions, indicate the existence of two distinct diverging correlation times, one associated with local relaxation and one associated with vortex phase slips. Correlated disorder is studied both in the superfluid transition of helium-4 and in the anisotropic critical scaling of a transverse Meissner-like transition in an experimental setup of a high temperature superconductor. For the superfluid helium transition, it is shown that the presence of fractally correlated disorder presumably alters the universality class of the pure model. Also, a comparison with experimental data suggests that the critical scaling theory describing the heat capacity of helium-4 may need to be modified in the presence of the disorder. In the case of superconductors, analyzing experimental data from resistance measurements in a system with columnar defects together with an anisotropy in the applied magnetic field, reveals a fully anisotropic scaling regime. Finally, a data analysis is presented from simulations of a charged particle gas system in three dimensions, where the normal Coulomb interaction between charges is changed into a logarithmic interaction. Previous work indicates the possibility of a transition similar to the Kosterlitz-Thouless transition in certain two dimensional systems. On the contrary, our simulations seem to favor a system whose critical scaling behavior is consistent with a transition occurring only at zero critical temperature. Overall, disorder in the model systems studied leads to important modifications of the critical scaling properties of pure systems, and thereby also to possible changes of the corresponding universality classes. This results in interesting predictions with experimentally relevant consequences.
The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.
Some physical systems, although dierent by their microscopic nature, in the vicinity of critical points can have similar thermodynamic and structural properties governed by universal power laws.Such systems are grouped into the so-called universality classes.The latter are determined by such global characteristics as the type of interaction, space dimensionality, symmetry, and the number of components of the order parameter.Therefore, it is interesting to investigate how the mutual inuence of these characteristics can aect the critical features of the system.The goal of our research is to analyze the changes in the critical behavior of a many-particle magnetic system under the inuence of two competing factors: long-range interaction and weak structural disorder.Using the example of an n-vector model in a d-dimensional space, we will investigate ferromagnetic ordering in a structurally disordered magnet with the long-range interaction decaying with the distance x as J(x) ∼ x -d-σ , with the control parameter σ.The analysis is carried out using the eld-theoretical renormalization group methods, which makes it possible to identify the universality classes of the system under consideration, their change with global parameters, and to determine the universal characteristics of critical behavior (critical exponents and marginal dimensions).It will be shown that there exists such a region of parameters (d, n, σ), where the long-range interaction and structural disorder lead to a synergistic eect and the appearance of a new, random long-range universality class.We obtain the renormalization group functions in the three-loop approximation, and calculate the correlation length critical exponent ν(ϵ ′ , n) as a perturbation theory series in ϵ ′ = 2σ -d.Quantitative estimates are obtained using asymptotic series resummation methods.
We report extensive Monte Carlo simulations of disordered Ising systems in the ferromagnetic region with concentrations of magnetic sites between p=1.0 and 0.5. The magnetization, the susceptibility, and the caloric properties have been studied in the critical region. The critical exponents \ensuremath{\beta} and \ensuremath{\gamma} as well as the universal amplitude A of the magnetization and the ratio C+{\mathit{C}}_{+}C+​/${\mathit{C}}_{\mathrm{\ensuremath{-}}}$ of the susceptibility amplitudes have been determined with high precision. In addition to the cusplike specific heat, we have also measured the magnetization-energy correlation function \ensuremath{\Gamma}, which is a divergent thermal quantity. The corresponding critical exponent \ensuremath{\zeta}, which is related to the other exponents by \ensuremath{\zeta}=1-\ensuremath{\beta}=(\ensuremath{\alpha}+\ensuremath{\gamma})/2, has been determined. We have found that all quantities show power-law behavior within the temperature range of our simulation. All critical exponents change continuously with dilution. Even in the range of weak dilution (p\ensuremath{\ge}0.8), the effective critical exponents are concentration dependent and are clearly different from their pure system values. In the strongly diluted regime the critical exponents gradually reach new asymptotic behavior at p=0.5--0.6 with values of \ensuremath{\beta}=0.335\ifmmode\pm\else\textpm\fi{}0.01 and \ensuremath{\gamma}=1.49\ifmmode\pm\else\textpm\fi{}0.02. The exponent \ensuremath{\alpha} of the specific heat becomes -0.17\ifmmode\pm\else\textpm\fi{}0.04, which corresponds to a cusplike singularity. We conclude that disorder profoundly changes the critical behavior for weakly as well as strongly disordered spin systems.