https://doi.org/10.1140/epjs/s11734-026-02327-6
Regular Article
A new fractal mean-field analysis in phase transition
1
International Center of Physics, Institute of Physics, University of Brasilia, 70910-900, Brasilia, Federal District, Brazil
2
Departamento de Física, Universidade Federal de Viçcosa, 36570-900, Viçosa, MG, Brazil
a
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Received:
24
October
2025
Accepted:
15
April
2026
Published online:
28
April
2026
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 
, 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 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.
Henrique A. de Lima and Fernando A. Oliveira contributed equally to this work.
© The Author(s) 2026
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