arXiv subjects

Henrique A. de Lima

Publications and source records attributed to Henrique A. de Lima.

3 recordsLinked to original sources

arXiv2026

A new Fractal Mean-Field analysis in phase transition

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 hypothesis 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 $η$, 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 $η$ 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.

cond-mat.stat-mech↗

arXiv2026

Multifractal and Ergodic Properties of Conductance Fluctuations under Strong Disorder

Understanding the stochastic properties of conductance fluctuations in disordered mesoscopic systems is fundamental to quantum transport. In this work, we investigate the multifractal and ergodic properties of the fictitious time series of conductance in two-dimensional tight-binding models under varying Anderson disorder. Using standard multifractal analysis, we show that conductance fluctuations exhibit a transition from non-ergodic to ergodic behavior as the disorder strength increases, as evidenced by the decay of the conductance correlation function. Remarkably, multifractality persists in both regimes; however, it becomes insensitive to shuffling in the strong-disorder (ergodic) regime, suggesting that distributional effects dominate temporal organization. On the contrary, in the weakly disordered (non-ergodic) regime, long-range correlations play a significant role. These findings are robust against changes in lead geometry (asymmetric vs. symmetric). Our results provide new insights into the interplay between ergodicity, multifractality, and rare events in disordered quantum transport.

cond-mat.mes-hall↗

arXiv2025

Scaling, Fractal Dynamics and Critical Exponents: Application in a non-integer dimensional Ising model

Moving beyond simple associations, researchers need tools to quantify how variables influence each other in space and time. Correlation functions provide a mathematical framework for characterizing these essential dependencies, revealing insights into causality, structure, and hidden patterns within complex systems. In physical systems with many degrees of freedom, such as gases, liquids, and solids, a statistical analysis of these correlations is essential. For a field $Ψ(\vec{x},t)$ that depends on spatial position $\vec{x}$ and time $t$, it is often necessary to understand the correlation with itself at another position and time $Ψ(\vec{x}_0,t_0)$. This specific function is called the autocorrelation function. In this context, the autocorrelation function for order--parameter fluctuations, introduced by Fisher, 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 demonstrated that at $T_c$ a fractal analysis is necessary for a complete description of the correlation function. In this study, we investigate the fundamental physics and mathematics underlying phase transitions. In particular, we show that the application of modern fractional differentials allows us to write down an equation for the correlation function that recovers the correct exponents below the upper critical dimension. We obtain the exact expression for the Fisher exponent $η$. Furthermore, we examine the Rushbrooke scaling relation, which has been questioned in certain magnetic systems, and, drawing on results from the Ising model, we confirm that both our relations and the Rushbrooke scaling law hold even when $d$ is not an integer.

cond-mat.stat-mech↗