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:2307.01075 · Physical review. E

Finding critical points and correlation length exponents using finite size scaling of Gini index

Soumyaditya Das, Soumyajyoti Biswas, Anirban Chakraborti, Bikas K. Chakrabarti

Published Feb 20, 2024• 2 citations
arXivHTMLPDF

Abstract

The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.g., temperature T for ferromagnetic to paramagnetic transition, or lattice occupation probability p in percolation), N is the system size, d is the spatial dimension and ν is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice, and the fiber bundle model of fracture.

DOI
10.1103/physreve.109.024121

Related papers

Critical Behaviors and Finite-Size Scaling of Principal Fluctuation Modes in Complex Systems*

Xiao-Teng Li, Xiaosong Chen

Abstract Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N × N correlation matrix C can be obtained. The principal fluctuation modes are defined by the eigenvectors of C. Near the critical point of a complex system, we anticipate that the principal fluctuation modes have the critical behaviors similar to that of the susceptibity. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes have been studied. The eigenvalues of the first 9 principal fluctuation modes have been invesitigated. Our Monte Carlo data demonstrate that these eigenvalues of the system with size L and the reduced temperature t follow a finite-size scaling form λ n (L, t) = L γ/ν f n (tL 1/ν ), where γ is critical exponent of susceptibility and ν is the critical exponent of the correlation length. Using eigenvalues λ 1 , λ 2 and λ 6 , we get the finite-size scaling form of the second moment correlation length ξ ( L , t ) = L ξ ˜ ( t L 1 / ν ) . It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic.

Communications in Theoretical Physics•Sep 1, 2016•18 citations

Conjecture on the lower bound of the length-scale critical exponent ν at continuous phase transitions

Andrea Pelissetto C, Ettore Vicari

A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent $ν$, which should hold for the large class of continuous transitions associated with $d$-dimensional Landau-Ginzburg-Wilson (LGW) $Φ^4$ theories with a multicomponent scalar field $φ$ and a unique $φ\cdot φ$ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If $Δ_φ=(d-2+η)/2$ is the dimension of the order-parameter field $φ$, and $Δ_\varepsilon=d-1/ν$ is the RG dimension of the energy operator $\varepsilon$, which can be identified with $[φ\cdot φ]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality $Δ_\varepsilon \ge 2 Δ_φ$, which implies $ν\ge (2-η)^{-1}$ and $γ= (2-η)ν\ge 1$. These inequalities are supported by general arguments for ferromagnetic lattice models, by $ε$-expansion results for generic LGW $Φ^4$ theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW $Φ^4$ theories. In particular, since unitarity requires $η\ge 0$, the above inequality implies $ν\ge 1/2$ for unitary theories. This lower bound is more restrictive than $ν> 1/d$, derived by noting that $ν=1/d$ characterizes the singular finite-size behavior at first-order transitions.

arXiv:2510.17637•Physical review. B./Physical review. B•Jun 9, 2026•1 citation

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

Finite- Size Scaling of Correlation Function

Xin Zhang, Gaoke Hu, Yongwen Zhang +2 more

We propose the finite-size scaling of correlation function in a finite system near its critical point. At a distance ${\bf r}$ in the finite system with size $L$, the correlation function can be written as the product of $|{\bf r}|^{-(d-2+η)}$ and its finite-size scaling function of variables ${\bf r}/L$ and $tL^{1/ν}$, where $t=(T-T_c)/T_c$. The directional dependence of correlation function is nonnegligible only when $|{\bf r}|$ becomes compariable with $L$. This finite-size scaling of correlation function has been confirmed by correlation functions of the Ising model and the bond percolation in two-diemnional lattices, which are calculated by Monte Carlo simulation. We can use the finite-size scaling of correlation function to determine the critical point and the critical exponent $η$.

arXiv:1805.08607•May 22, 2018•1 citation

Universal Critical Phase Diagram Using Gini Index

Soumyaditya Das, Soumyajyoti Biswas

The critical phase surface of a system, in general, can depend on one or more parameters. We show that by calculating the Gini index ($g$) of any suitably defined response function of a system, the critical phase surface can always be reduced to that of a single parameter, starting from $g=0$ and terminating at $g=g_f$, where $g_f$ is a universal number for a chosen response function in a given universality class. We demonstrate the construction with analytical and numerical calculations of mean field transverse field Ising model and site diluted Ising model on the Bethe lattice, respectively. Both models have two parameter critical phase surfaces -- transverse field and temperature for the first case and site dilution and temperature in the second case. Both can be reduced to single parameter transition points in terms of the Gini index. We have additionally demonstrated the validity of the method for a mean field two parameter opinion dynamics model that includes a tri-critical point. The method is generally applicable for any multi-parameter critical transition.

arXiv:2409.01453•Journal of the Physical Society of Japan•Sep 19, 2025•0 citations

Large-deviation tails of critical order-parameter distributions

Jinhong Zhu, Yihao Xu, Abbas Ali Saberi +1 more

Large-deviation tails of critical probability distributions provide a sensitive probe of universality beyond standard finite-size scaling. We study these tails for critical percolation and Fortuin--Kasteleyn Ising models on two-dimensional lattices, three-dimensional lattices, and complete graphs. We consider two rescaled order parameters: the magnetization-like variable $x_m=|M|/\langle |M|\rangle$, including a signed cluster-mass analogue for percolation, and the largest-cluster variable $x_C=C_1/\langle C_1\rangle$. For $x_m$, we test the expected stretched-exponential large-deviation tail and show that the same form applies to the percolation analogue. For $x_C$, guided by the exact complete-graph result and scaling arguments, we propose universal scaling forms for both tails of the cumulative distribution and test them by extensive Monte Carlo simulations. In the complete-graph FK-Ising model, the left tail is governed by rare configurations with percolation-like scaling rather than by the typical Ising scaling. Our results show that the tails of order-parameter distributions reveal universal features of critical fluctuations that are not captured by averaged observables alone.

arXiv:2606.19030•Jun 17, 2026•0 citations
The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.g., temperature T for ferromagnetic to paramagnetic transition, or lattice occupation probability p in percolation), N is the system size, d is the spatial dimension and ν is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice, and the fiber bundle model of fracture.
Abstract Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N × N correlation matrix C can be obtained. The principal fluctuation modes are defined by the eigenvectors of C. Near the critical point of a complex system, we anticipate that the principal fluctuation modes have the critical behaviors similar to that of the susceptibity. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes have been studied. The eigenvalues of the first 9 principal fluctuation modes have been invesitigated. Our Monte Carlo data demonstrate that these eigenvalues of the system with size L and the reduced temperature t follow a finite-size scaling form λ n (L, t) = L γ/ν f n (tL 1/ν ), where γ is critical exponent of susceptibility and ν is the critical exponent of the correlation length. Using eigenvalues λ 1 , λ 2 and λ 6 , we get the finite-size scaling form of the second moment correlation length ξ ( L , t ) = L ξ ˜ ( t L 1 / ν ) . It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic.
A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent ννν, which should hold for the large class of continuous transitions associated with ddd-dimensional Landau-Ginzburg-Wilson (LGW) Φ4Φ^4Φ4 theories with a multicomponent scalar field φφφ and a unique φ⋅φφ\cdot φφ⋅φ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If Δφ=(d−2+η)/2Δ_φ=(d-2+η)/2Δφ​=(d−2+η)/2 is the dimension of the order-parameter field φφφ, and Δε=d−1/νΔ_\varepsilon=d-1/νΔε​=d−1/ν is the RG dimension of the energy operator ε\varepsilonε, which can be identified with [φ⋅φ][φ\cdot φ][φ⋅φ] (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality Δε≥2ΔφΔ_\varepsilon \ge 2 Δ_φΔε​≥2Δφ​, which implies ν≥(2−η)−1ν\ge (2-η)^{-1}ν≥(2−η)−1 and γ=(2−η)ν≥1γ= (2-η)ν\ge 1γ=(2−η)ν≥1. These inequalities are supported by general arguments for ferromagnetic lattice models, by εεε-expansion results for generic LGW Φ4Φ^4Φ4 theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW Φ4Φ^4Φ4 theories. In particular, since unitarity requires η≥0η\ge 0η≥0, the above inequality implies ν≥1/2ν\ge 1/2ν≥1/2 for unitary theories. This lower bound is more restrictive than ν>1/dν> 1/dν>1/d, derived by noting that ν=1/dν=1/dν=1/d characterizes the singular finite-size behavior at first-order transitions.
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.
We propose the finite-size scaling of correlation function in a finite system near its critical point. At a distance r{\bf r}r in the finite system with size LLL, the correlation function can be written as the product of ∣r∣−(d−2+η)|{\bf r}|^{-(d-2+η)}∣r∣−(d−2+η) and its finite-size scaling function of variables r/L{\bf r}/Lr/L and tL1/νtL^{1/ν}tL1/ν, where t=(T−Tc)/Tct=(T-T_c)/T_ct=(T−Tc​)/Tc​. The directional dependence of correlation function is nonnegligible only when ∣r∣|{\bf r}|∣r∣ becomes compariable with LLL. This finite-size scaling of correlation function has been confirmed by correlation functions of the Ising model and the bond percolation in two-diemnional lattices, which are calculated by Monte Carlo simulation. We can use the finite-size scaling of correlation function to determine the critical point and the critical exponent ηηη.
The critical phase surface of a system, in general, can depend on one or more parameters. We show that by calculating the Gini index (ggg) of any suitably defined response function of a system, the critical phase surface can always be reduced to that of a single parameter, starting from g=0g=0g=0 and terminating at g=gfg=g_fg=gf​, where gfg_fgf​ is a universal number for a chosen response function in a given universality class. We demonstrate the construction with analytical and numerical calculations of mean field transverse field Ising model and site diluted Ising model on the Bethe lattice, respectively. Both models have two parameter critical phase surfaces -- transverse field and temperature for the first case and site dilution and temperature in the second case. Both can be reduced to single parameter transition points in terms of the Gini index. We have additionally demonstrated the validity of the method for a mean field two parameter opinion dynamics model that includes a tri-critical point. The method is generally applicable for any multi-parameter critical transition.
Large-deviation tails of critical probability distributions provide a sensitive probe of universality beyond standard finite-size scaling. We study these tails for critical percolation and Fortuin--Kasteleyn Ising models on two-dimensional lattices, three-dimensional lattices, and complete graphs. We consider two rescaled order parameters: the magnetization-like variable xm=∣M∣/⟨∣M∣⟩x_m=|M|/\langle |M|\ranglexm​=∣M∣/⟨∣M∣⟩, including a signed cluster-mass analogue for percolation, and the largest-cluster variable xC=C1/⟨C1⟩x_C=C_1/\langle C_1\ranglexC​=C1​/⟨C1​⟩. For xmx_mxm​, we test the expected stretched-exponential large-deviation tail and show that the same form applies to the percolation analogue. For xCx_CxC​, guided by the exact complete-graph result and scaling arguments, we propose universal scaling forms for both tails of the cumulative distribution and test them by extensive Monte Carlo simulations. In the complete-graph FK-Ising model, the left tail is governed by rare configurations with percolation-like scaling rather than by the typical Ising scaling. Our results show that the tails of order-parameter distributions reveal universal features of critical fluctuations that are not captured by averaged observables alone.