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

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical finite-range scaling in scalar-field theories and Ising models

Per Arne Rikvold, Bryan M. Gorman, M. A. Novotny

Published Mar 1, 1993• 9 citations
Publisher

Abstract

We develop a critical finite-force-range scaling theory for D-dimensional scalar ${\mathrm{\ensuremath{\varphi}}}^{\mathit{n}}$ field theories that is based on a scaling ansatz equivalent to a Ginzburg criterion. To investigate its relationship to other scaling theories we derive equivalent results from renormalization groups and from finite-size crossover scaling for systems with weak long-range forces. By comparing our finite-range scaling relations with finite-size scaling relations for hypercylindrical systems above the upper critical dimension ${\mathit{D}}_{\mathit{c}}$, we arrive at a criterion of critical equivalence that provides an asymptotic mapping between the two kinds of systems. We apply our scaling relations to a ${\mathrm{\ensuremath{\varphi}}}^{4}$ Ginzburg-Landau Hamiltonian, to the one-dimensional Kac model with exponentially decaying interactions, and to the N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} quasi-one-dimensional Ising (Q1DI) model, in which each spin interacts with O(N) others. Near the Gaussian mean-field critical point the Ginzburg-Landau Hamiltonians for all three models become identical, but for the Q1DI model this requires a length rescaling. For the Kac model the resulting scaling relations are those of a D=1 quartic field theory, and for the Q1DI model they are those of a cylindrical Ising system above ${\mathit{D}}_{\mathit{c}}$. Results of specialized numerical scaling techniques applied to transfer-matrix calculations for the Q1DI model with N\ensuremath{\le}1024 strongly support our theoretically obtained scaling relations.

DOI
10.1103/physreve.47.1474

Related papers

Scaling and Finite-Size Scaling above the Upper Critical Dimension

Ralph Kenna B, Bertrand Berche

In the 1960's, four famous scaling relations were developed which relate the six standard critical exponents describing continuous phase transitions in the thermodynamic limit of statistical physics models. They are well understood at a fundamental level through the renormalization group. They have been verified in multitudes of theoretical, computational and experimental studies and are firmly established and profoundly important for our understanding of critical phenomena. One of the scaling relations, hyperscaling, fails above the upper critical dimension. There, critical phenomena are governed by Gaussian fixed points in the renormalization-group formalism. Dangerous irrelevant variables are required to deliver the mean-field and Landau values of the critical exponents, which are deemed valid by the Ginzburg criterion. Also above the upper critical dimension, the standard picture is that, unlike for low-dimensional systems, finite-size scaling is non-universal. Here we report on new developments which indicate that the current paradigm is flawed and incomplete. In particular, the introduction of a new exponent characterising the finite-size correlation length allows one to extend hyperscaling beyond the upper critical dimension. Moreover, finite-size scaling is shown to be universal provided the correct scaling window is chosen. These recent developments also lead to the introduction of a new scaling relation analogous to one introduced by Fisher 50 years ago.

arXiv:2404.09190•WORLD SCIENTIFIC eBooks•Mar 19, 2015•9 citations

Universal finite-size scaling analysis of Ising models with long-range interactions at the upper critical dimensionality: Isotropic case

Daniel Grüneberg, Alfred Hucht

We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.

arXiv:cond-mat/0310252•Physical Review E•Mar 9, 2004•16 citations

Renormalization group for the φ4 theory with long-range interaction and the critical exponent η of the Ising model

Raphael Goll, Peter Kopietz

We calculate the critical exponent η of the D-dimensional Ising model from a simple truncation of the functional renormalization group flow equations for a scalar field theory with long-range interaction. Our approach relies on the smallness of the inverse range of the interaction and on the assumption that the Ginzburg momentum defining the width of the scaling regime in momentum space is larger than the scale where the renormalized interaction crosses over from long range to short range; the numerical value of η can then be estimated by stopping the renormalization group flow at this scale. In three dimensions our result η=0.03651 is in good agreement with recent conformal bootstrap and Monte Carlo calculations. We extend our calculations to fractional dimensions D and obtain the resulting critical exponent η(D) between two and four dimensions. For dimensions 2≤D≤3 our result for η is consistent with previous calculations.

arXiv:1804.04150•Physical review. E•Aug 28, 2018•6 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

Finite-range-scaling analysis of metastability in an Ising model with long-range interactions

Bryan M. Gorman, Per Arne Rikvold, M. A. Novotny

We apply both a scalar field theory and a recently developed transfer-matrix method to study the stationary properties of metastability in a two-state model with weak, long-range interactions: the N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} quasi-one-dimensional Ising model. Using the field theory, we find the analytic continuation f\ifmmode \tilde{}\else \~{}\fi{} of the free energy across the first-order transition, assuming that the system escapes the metastable state by the nucleation of noninteracting droplets. We find that corrections to the field dependence are substantial, and, by solving the Euler-Lagrange equation for the model numerically, we have verified the form of the free-energy cost of nucleation, including the first correction. In the transfer-matrix method, we associate with the subdominant eigenvectors of the transfer matrix a complex-valued ``constrained'' free-energy density ${\mathit{f}}_{\mathrm{\ensuremath{\alpha}}}$ computed directly from the matrix. For the eigenvector with an associated magnetization most strongly opposed to the applied magnetic field, ${\mathit{f}}_{\mathrm{\ensuremath{\alpha}}}$ exhibits finite-range scaling behavior in agreement with f\ifmmode \tilde{}\else \~{}\fi{} over a wide range of temperatures and fields, extending nearly to the classical spinodal. Some implications of these results for numerical studies of metastability are discussed.

arXiv:cond-mat/9402007•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 1, 1994•11 citations

Critical properties of the three-dimensional equivalent-neighbor model and crossover scaling in finite systems

Erik Luijten

Accurate numerical results are presented for the three-dimensional equivalent-neighbor model on a cubic lattice, for 12 different interaction ranges (coordination number between 18 and 250). These results allow the determination of the range dependences of the critical temperature and various critical amplitudes, which are compared to renormalization-group predictions. In addition, the analysis yields an estimate for the interaction range at which the leading corrections to scaling vanish for the spin-1 / 2 model, and confirms earlier conclusions that the leading Wegner correction must be negative for the three-dimensional (nearest-neighbor) Ising model. By complementing these results with Monte Carlo data for systems with coordination numbers as large as 52 514, the full finite-size crossover curves between classical and Ising-like behavior are obtained as a function of a generalized Ginzburg parameter. Also, the crossover function for the effective magnetic exponent is determined.

arXiv:cond-mat/9811332•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 1, 1999•26 citations
We develop a critical finite-force-range scaling theory for D-dimensional scalar \ensuremathφn{\mathrm{\ensuremath{\varphi}}}^{\mathit{n}}\ensuremathφn field theories that is based on a scaling ansatz equivalent to a Ginzburg criterion. To investigate its relationship to other scaling theories we derive equivalent results from renormalization groups and from finite-size crossover scaling for systems with weak long-range forces. By comparing our finite-range scaling relations with finite-size scaling relations for hypercylindrical systems above the upper critical dimension Dc{\mathit{D}}_{\mathit{c}}Dc​, we arrive at a criterion of critical equivalence that provides an asymptotic mapping between the two kinds of systems. We apply our scaling relations to a \ensuremathφ4{\mathrm{\ensuremath{\varphi}}}^{4}\ensuremathφ4 Ginzburg-Landau Hamiltonian, to the one-dimensional Kac model with exponentially decaying interactions, and to the N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} quasi-one-dimensional Ising (Q1DI) model, in which each spin interacts with O(N) others. Near the Gaussian mean-field critical point the Ginzburg-Landau Hamiltonians for all three models become identical, but for the Q1DI model this requires a length rescaling. For the Kac model the resulting scaling relations are those of a D=1 quartic field theory, and for the Q1DI model they are those of a cylindrical Ising system above Dc{\mathit{D}}_{\mathit{c}}Dc​. Results of specialized numerical scaling techniques applied to transfer-matrix calculations for the Q1DI model with N\ensuremath{\le}1024 strongly support our theoretically obtained scaling relations.
In the 1960's, four famous scaling relations were developed which relate the six standard critical exponents describing continuous phase transitions in the thermodynamic limit of statistical physics models. They are well understood at a fundamental level through the renormalization group. They have been verified in multitudes of theoretical, computational and experimental studies and are firmly established and profoundly important for our understanding of critical phenomena. One of the scaling relations, hyperscaling, fails above the upper critical dimension. There, critical phenomena are governed by Gaussian fixed points in the renormalization-group formalism. Dangerous irrelevant variables are required to deliver the mean-field and Landau values of the critical exponents, which are deemed valid by the Ginzburg criterion. Also above the upper critical dimension, the standard picture is that, unlike for low-dimensional systems, finite-size scaling is non-universal. Here we report on new developments which indicate that the current paradigm is flawed and incomplete. In particular, the introduction of a new exponent characterising the finite-size correlation length allows one to extend hyperscaling beyond the upper critical dimension. Moreover, finite-size scaling is shown to be universal provided the correct scaling window is chosen. These recent developments also lead to the introduction of a new scaling relation analogous to one introduced by Fisher 50 years ago.
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.
We calculate the critical exponent η of the D-dimensional Ising model from a simple truncation of the functional renormalization group flow equations for a scalar field theory with long-range interaction. Our approach relies on the smallness of the inverse range of the interaction and on the assumption that the Ginzburg momentum defining the width of the scaling regime in momentum space is larger than the scale where the renormalized interaction crosses over from long range to short range; the numerical value of η can then be estimated by stopping the renormalization group flow at this scale. In three dimensions our result η=0.03651 is in good agreement with recent conformal bootstrap and Monte Carlo calculations. We extend our calculations to fractional dimensions D and obtain the resulting critical exponent η(D) between two and four dimensions. For dimensions 2≤D≤3 our result for η is consistent with previous calculations.
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.
We apply both a scalar field theory and a recently developed transfer-matrix method to study the stationary properties of metastability in a two-state model with weak, long-range interactions: the N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} quasi-one-dimensional Ising model. Using the field theory, we find the analytic continuation f\ifmmode \tilde{}\else \~{}\fi{} of the free energy across the first-order transition, assuming that the system escapes the metastable state by the nucleation of noninteracting droplets. We find that corrections to the field dependence are substantial, and, by solving the Euler-Lagrange equation for the model numerically, we have verified the form of the free-energy cost of nucleation, including the first correction. In the transfer-matrix method, we associate with the subdominant eigenvectors of the transfer matrix a complex-valued ``constrained'' free-energy density ${\mathit{f}}_{\mathrm{\ensuremath{\alpha}}}$ computed directly from the matrix. For the eigenvector with an associated magnetization most strongly opposed to the applied magnetic field, ${\mathit{f}}_{\mathrm{\ensuremath{\alpha}}}$ exhibits finite-range scaling behavior in agreement with f\ifmmode \tilde{}\else \~{}\fi{} over a wide range of temperatures and fields, extending nearly to the classical spinodal. Some implications of these results for numerical studies of metastability are discussed.
Accurate numerical results are presented for the three-dimensional equivalent-neighbor model on a cubic lattice, for 12 different interaction ranges (coordination number between 18 and 250). These results allow the determination of the range dependences of the critical temperature and various critical amplitudes, which are compared to renormalization-group predictions. In addition, the analysis yields an estimate for the interaction range at which the leading corrections to scaling vanish for the spin-1 / 2 model, and confirms earlier conclusions that the leading Wegner correction must be negative for the three-dimensional (nearest-neighbor) Ising model. By complementing these results with Monte Carlo data for systems with coordination numbers as large as 52 514, the full finite-size crossover curves between classical and Ising-like behavior are obtained as a function of a generalized Ginzburg parameter. Also, the crossover function for the effective magnetic exponent is determined.