In a recent article we have shown that, by applying sophisticated summation methods to Wilson-Fisher's ε-expansion, it is possible from the presently known terms of the series to obtain accurate values of critical exponents for the 0 ( n ) symmetric n-vector model : these values are consistent with the best estimates obtained from three-dimensional Renormalization Group calculations and, in the case of Ising-like systems, with the exactly known two-dimensional values of the Ising model. The controversial conjecture has been recently formulated that some fractal lattices could interpolate regular lattices in non-integer dimensions. Numerical calculations have been done for the Ising model. To allow for direct comparison with Renormalization Group values, we present here estimates for exponents in non-integer dimensions d(1In a recent article we have shown that, by applying sophisticated summation methods to Wilson-Fisher's ε-expansion, it is possible from the presently known terms of the series to obtain accurate values of critical exponents for the 0 ( n ) symmetric n-vector model : these values are consistent with the best estimates obtained from three-dimensional Renormalization Group calculations and, in the case of Ising-like systems, with the exactly known two-dimensional values of the Ising model. The controversial conjecture has been recently formulated that some fractal lattices could interpolate regular lattices in non-integer dimensions. Numerical calculations have been done for the Ising model. To allow for direct comparison with Renormalization Group values, we present here estimates for exponents in non-integer dimensions d(1

Journal de physiqueJan 1, 1987189 citations

An Estimate of the Dependence of the Ising Critical Exponents on the Lattice Dimensionality

P. R. Silva

Abstract Upper and lower bounds are estimated of the two‐point correlation function exponent, η, of the Ising model, as a function of the lattice dimensionality, d. The average value of these two bounds together with the correlation length critical exponent v(d) evaluated by Thompson, lead to a set of critical indexes which satisfy the hyperscaling relation. This set of exponents also shows an adequate agreement with the results obtained by Le‐Guillou and Zinn‐Justin for the Ising model in non‐integer dimensions, in a large range of the parameter d.Abstract Upper and lower bounds are estimated of the two‐point correlation function exponent, η, of the Ising model, as a function of the lattice dimensionality, d. The average value of these two bounds together with the correlation length critical exponent v(d) evaluated by Thompson, lead to a set of critical indexes which satisfy the hyperscaling relation. This set of exponents also shows an adequate agreement with the results obtained by Le‐Guillou and Zinn‐Justin for the Ising model in non‐integer dimensions, in a large range of the parameter d.

physica status solidi (b)Sep 1, 19910 citations

A new fractal mean-field analysis in phase transition

Ismael S. S. Carrasco, H. Lima, Fernando Albuquerque Oliveira

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 $\eta$ 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 $\eta$ 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.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 $$\eta$$ 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 $$\eta$$ 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.

arXiv:2511.20846The European Physical Journal Special TopicsApr 28, 20261 citation

\eta

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

\eta

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.