Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0508734

Anomalous Scaling and Fractal Dimensions

Wolfhard Janke E, Adriaan M. J. Schakel

Published Aug 30, 2005• 2 citations
arXivHTMLPDF

Abstract

The relation between critical exponents, characterizing a continuous phase transition, and the fractal structure of physical lines, proliferating at the critical point, is established by considering the two-dimensional O($N$) spin model for which many exact results are available.

DOI
10.48550/arxiv.cond-mat/0508734

Related papers

The critical point

Ross H. McKenzie

Abstract ‘The critical point’ in a phase diagram denotes the conditions under which a continuous transition between two states of matter occurs. Surprisingly, near the critical point diverse systems can have the same dependence of physical properties on parameters such as temperature. The only details that determine the critical exponents are the symmetry of the order parameter and the spatial dimensionality of the system. This independence from chemical and structural details, known as universality, presented a major challenge to theoretical physics for decades. Theory must deal with the large fluctuations in the amount of order that occurs close to the critical point. The successful theory developed by Ken Wilson exploits the fact that different states of a system and different systems can be related to one another by mathematically rescaling the length scales in the system. The mathematics involved can be made tractable by considering the abstract idea of a system with a variable number of spatial dimensions. The idea of rescaling is illustrated with the results of computer simulations of an Ising model and with images of fractals.

•May 25, 2023•0 citations

Scaling, fractal dynamics, and critical exponents in the equilibrium phase transition

Adauto F. Souza, H. Lima, A. L. R. Barbosa +1 more

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.

arXiv:2606.20460•Jun 18, 2026•0 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.

arXiv:2511.20846•The European Physical Journal Special Topics•Apr 28, 2026•1 citation

Phase diagram and critical exponents of the ±J Ising model in finite dimensions by Monte Carlo renormalization group

之康 尾関, Yukiyasu Ozeki, 秀稔 西森 +1 more

•Apr 1, 1987•0 citations

Quantum critical behavior of the transverse-field quantum Ising model on a fractal structure

Hangmo Yi

•Aug 24, 2014•0 citations

Quantum critical behavior of the quantum Ising model on fractal lattices

Hangmo Yi

I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.

arXiv:1408.5595•Physical Review E•Jan 9, 2015•9 citations
The relation between critical exponents, characterizing a continuous phase transition, and the fractal structure of physical lines, proliferating at the critical point, is established by considering the two-dimensional O(NNN) spin model for which many exact results are available.
Abstract ‘The critical point’ in a phase diagram denotes the conditions under which a continuous transition between two states of matter occurs. Surprisingly, near the critical point diverse systems can have the same dependence of physical properties on parameters such as temperature. The only details that determine the critical exponents are the symmetry of the order parameter and the spatial dimensionality of the system. This independence from chemical and structural details, known as universality, presented a major challenge to theoretical physics for decades. Theory must deal with the large fluctuations in the amount of order that occurs close to the critical point. The successful theory developed by Ken Wilson exploits the fact that different states of a system and different systems can be related to one another by mathematically rescaling the length scales in the system. The mathematics involved can be made tractable by considering the abstract idea of a system with a variable number of spatial dimensions. The idea of rescaling is illustrated with the results of computer simulations of an Ising model and with images of fractals.
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.
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.