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:1206.5062 · Physical Review Letters

Generalized Scaling Theory for Critical Phenomena Including Essential Singularities and Infinite Dimensionality

Tomoaki Nogawa, Takehisa Hasegawa, Koji Nemoto

Published Jun 21, 2012• 14 citations
arXivHTMLPDF

Abstract

We propose a generic scaling theory for critical phenomena that includes power-law and essential singularities in finite and infinite dimensional systems. In addition, we clarify its validity by analyzing the Potts model in a simple hierarchical network, where a saddle-node bifurcation of the renormalization-group fixed point governs the essential singularity.

DOI
10.1103/physrevlett.108.255703

Related papers

Singularities and Scaling Functions at the Potts-Model Multicritical Point

Michael Nauenberg, Douglas J. Scalapino

Differential renormalization equation for the $q$-state Potts model are proposed, and the critical behavior of the model near $q={q}_{c}$ discussed. The equations give rise to critical and tricritical fixed points which merge at $q={q}_{c}$ when a dilution field becomes marginal, to an essential singularity in the latent heat as a function of $q={q}_{c}$, in accordance with the exact result of Baxter, and, for $q={q}_{c}$, to a logarithm correction to the power-law behavior of the free energy as a function of $T\ensuremath{-}{T}_{c}$.

Physical Review Letters•Mar 31, 1980•170 citations

Renormalization group and scaling

Hidetoshi Nishimori, Gerardo Ortíz

Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.

•Dec 2, 2010•0 citations

Finite-size scaling in the canonical ensemble

Youjin Deng, Henk W. J. Blöte

We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent $y_1$ that depends on the temperature exponent $y_t$ and the dimensionality $d$ as $y_1=-|2y_t-d|$. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes.

arXiv:cond-mat/0508348•Aug 15, 2005•1 citation

Normal Form for Renormalization Groups

Archishman Raju, Colin B. Clement, Lorien X. Hayden +4 more

The results of the renormalization group are commonly advertised as the existence of power-law singularities near critical points.The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis.We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables.We recover and explain the existing literature and predict the nonlinear generalization for the universal homogeneous scaling functions.We show that this procedure leads to a better handling of the singularity even in classic cases and elaborate our framework using several examples.

arXiv:1706.00137•Physical Review X•Apr 23, 2019•35 citations

Power-law singularities and critical exponents in n-vector models

J. Kaupužs

Power-law singularities and critical exponents in n-vector models are considered from different theoretical points of view. It includes a theoretical approach called the GFD (grouping of Feynman diagrams) theory, as well as the perturbative renormalization group (RG) treatment. A non-perturbative proof concerning corrections to scaling in the two-point correlation function of the phi^4 model is provided, showing that predictions of the GFD theory rather than those of the perturbative RG theory can be correct. Critical exponents determined from highly accurate experimental data very close to the lambda-transition point in liquid helium, as well as the Goldstone mode singularities in n-vector spin models, evaluated from Monte Carlo simulation results, are discussed with an aim to test the theoretical predictions. Our analysis shows that in both cases the data can be well interpreted within the GFD theory.

arXiv:1103.0888•Mar 4, 2011•0 citations

Entropy as a Universal Scaling Law for Critical Phenomena

Revista, Zen, PHYSICS, 10

Critical phenomena, characterized by abrupt changes in macroscopic properties near phase transition points, exhibit universal scaling behavior independent of microscopic details. This paper explores the hypothesis that entropy serves as a fundamental universal scaling law governing these phenomena. We review the theoretical foundations of critical phenomena, including Landau theory and the renormalization group, highlighting the role of critical exponents and scaling functions. We then propose a framework where the configurational entropy, entanglement entropy, and other entropic measures exhibit universal scaling, linking them directly to the underlying critical exponents. Our methodology involves analyzing how entropy behaves under coarse-graining transformations near critical points, identifying a potential entropic fixed point in the renormalization group flow. We present theoretical arguments suggesting that the information content, as quantified by various entropy definitions, reflects the diverging correlation length and the emergent scale invariance at criticality. The discussion extends to how different entropic forms, such as Rényi and Tsallis entropies, might provide a more generalized framework for understanding critical behavior in complex and non-extensive systems. By demonstrating entropy's role as a universal scaling law, this work offers a unifying perspective on critical phenomena, deepening our understanding of phase transitions and the emergence of macroscopic order from microscopic fluctuations.

•Nov 28, 2025•0 citations
We propose a generic scaling theory for critical phenomena that includes power-law and essential singularities in finite and infinite dimensional systems. In addition, we clarify its validity by analyzing the Potts model in a simple hierarchical network, where a saddle-node bifurcation of the renormalization-group fixed point governs the essential singularity.
Differential renormalization equation for the qqq-state Potts model are proposed, and the critical behavior of the model near q=qcq={q}_{c}q=qc​ discussed. The equations give rise to critical and tricritical fixed points which merge at q=qcq={q}_{c}q=qc​ when a dilution field becomes marginal, to an essential singularity in the latent heat as a function of q=qcq={q}_{c}q=qc​, in accordance with the exact result of Baxter, and, for q=qcq={q}_{c}q=qc​, to a logarithm correction to the power-law behavior of the free energy as a function of $T\ensuremath{-}{T}_{c}$.
Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.
We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent y1y_1y1​ that depends on the temperature exponent yty_tyt​ and the dimensionality ddd as y1=−∣2yt−d∣y_1=-|2y_t-d|y1​=−∣2yt​−d∣. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes.
The results of the renormalization group are commonly advertised as the existence of power-law singularities near critical points.The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis.We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables.We recover and explain the existing literature and predict the nonlinear generalization for the universal homogeneous scaling functions.We show that this procedure leads to a better handling of the singularity even in classic cases and elaborate our framework using several examples.
Power-law singularities and critical exponents in n-vector models are considered from different theoretical points of view. It includes a theoretical approach called the GFD (grouping of Feynman diagrams) theory, as well as the perturbative renormalization group (RG) treatment. A non-perturbative proof concerning corrections to scaling in the two-point correlation function of the phi^4 model is provided, showing that predictions of the GFD theory rather than those of the perturbative RG theory can be correct. Critical exponents determined from highly accurate experimental data very close to the lambda-transition point in liquid helium, as well as the Goldstone mode singularities in n-vector spin models, evaluated from Monte Carlo simulation results, are discussed with an aim to test the theoretical predictions. Our analysis shows that in both cases the data can be well interpreted within the GFD theory.
Critical phenomena, characterized by abrupt changes in macroscopic properties near phase transition points, exhibit universal scaling behavior independent of microscopic details. This paper explores the hypothesis that entropy serves as a fundamental universal scaling law governing these phenomena. We review the theoretical foundations of critical phenomena, including Landau theory and the renormalization group, highlighting the role of critical exponents and scaling functions. We then propose a framework where the configurational entropy, entanglement entropy, and other entropic measures exhibit universal scaling, linking them directly to the underlying critical exponents. Our methodology involves analyzing how entropy behaves under coarse-graining transformations near critical points, identifying a potential entropic fixed point in the renormalization group flow. We present theoretical arguments suggesting that the information content, as quantified by various entropy definitions, reflects the diverging correlation length and the emergent scale invariance at criticality. The discussion extends to how different entropic forms, such as Rényi and Tsallis entropies, might provide a more generalized framework for understanding critical behavior in complex and non-extensive systems. By demonstrating entropy's role as a universal scaling law, this work offers a unifying perspective on critical phenomena, deepening our understanding of phase transitions and the emergence of macroscopic order from microscopic fluctuations.