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. B, Condensed matter

Critical exponents in three dimensions from a scalarfield theory with Gaussian propagator

Claude Bervillier, Jean-Michel Drouffe, Claude Godrèche

Published Sep 1, 1979• 1 citation
Publisher

Abstract

Critical exponents are derived from a ${\ensuremath{\phi}}^{4}$ field theory with a Gaussian propagator. They arise from the study of the singular behavior of the correlation functions in two coupling constants. The corresponding double series are resummed through a Pad\'e variational procedure. A check of universality is provided since our numerical estimates agree with the already known values.

DOI
10.1103/physrevb.20.2097

Related papers

Critical exponents from a regularized field theoretical model

Claude Bervillier, Claude Godrèche

We use a regularized (ϕ2)2 field theory for the determination of the critical exponents γ and ν in three dimensions. The regularization is done by a Gaussian propagator. This enables any order in g for any correlation function to be easily calculated but introduces a second expansion parameter. In spite of the difficulties encountered when dealing with double series, we obtain values for the critical exponents in good agreement with those found with the renormalized theory. This and the discussion we give of the link between the renormalized theory and our model illustrate the concept of universality in critical phenomena.

Journal de physique•Jan 1, 1982•2 citations

Critical behavior of φ3 field theories in three dimensions

Jeffrey S. Reeve, Anthony John Guttmann, B W Keck

The methods of renormalized perturbation theory are used to find the critical exponents of the percolation and Yang-Lee edge singularity problems. Both of the above critical phenomena are described by ${\ensuremath{\varphi}}^{3}$ field theories, and the Wilson functions are calculated explicitly in three dimensions to order three loops and summed using the Pad\'e-Borel method. The resulting critical exponent values are consistent with direct series expansion results.

Physical review. B, Condensed matter•Oct 1, 1982•10 citations

Universal combination of critical amplitudes from field theory

Claude Bervillier, Claude Godrèche

Near a continuous phase transition, there exist universal combinations of critical amplitudes. We present an estimation of such a universal quantity, in the framework of the ${\ensuremath{\varphi}}^{4}$ field theory in three dimensions, using information recently available in the literature, and already used to determine precise values for critical exponents. This leads to an accuracy only obtained, hitherto, in the framework of field theory, for critical exponents. The universal quantity calculated, often denoted by $R_{\ensuremath{\xi}}^{}{}_{}{}^{+}$, combines the amplitude of the correlation length and that of the singular part of the specific heat in the vicinity of the critical point.

Physical review. B, Condensed matter•Jun 1, 1980•108 citations

Ising exponents from the functional renormalization group

Daniel F. Litim, Dario Zappalà

We study the 3d Ising universality class using the functional renormalization group. With the help of background fields and a derivative expansion up to fourth order we compute the leading index, the subleading symmetric and antisymmetric corrections to scaling, the anomalous dimension, the scaling solution, and the eigenperturbations at criticality. We also study the cross correlations of scaling exponents, their dependence on dimensionality, and the numerical convergence of the derivative expansion. Collecting all available data from functional renormalization group studies to date, we estimate that systematic errors are in good agreement with findings from Monte Carlo simulations, $ϵ$-expansion techniques, and resummed perturbation theory.

arXiv:1009.1948•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 7, 2011•125 citations

Critical thermodynamics of the two-dimensional systems in five-loop renormalization-group approximation

E. V. Orlov, Aleksandr Ivanovich Sokolov

The RG functions of the 2D $n$-vector $ϕ^4$ model are calculated in the five-loop approximation. Perturbative series for the $β$ function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case $n = 1$ as well as in the others ($n = 0$, $n = -1$, $n = 2, 3,...32$) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the $β$ function that can not be found perturbatively. The evaluation of the critical exponents for $n = 1$, $n = 0$ and $n = -1$ in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.

arXiv:hep-th/0003140•Mar 16, 2000•1 citation

Critical exponents of O(N) models in fractional dimensions

Alessandro Codello, Nicolò Defenu, Giulio D’Odorico

We compute critical exponents of $O(N)$ models in fractional dimensions between $d=2$ and 4, and for continuous values of the number of field components $N$, in this way completing the RG classification of universality classes for these models. These curves represent nonperturbative approximation to the exact results, they respect all the qualitative features expected from such quantities conciliating previously known perturbative results in three dimensions with exact results in two dimensions and giving a strong indication of what could be the exact behavior of such curves. We also report critical exponents for some multicritical universality classes in the cases $N\ensuremath{\ge}2$ and $N=0$. Finally, in the large-$N$ limit our critical exponents correctly approach those of the spherical model, allowing us to set $N\ensuremath{\sim}100$ as the threshold for the quantitative validity of leading order large-$N$ estimates.

arXiv:1410.3308•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•May 5, 2015•70 citations
Critical exponents are derived from a \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 field theory with a Gaussian propagator. They arise from the study of the singular behavior of the correlation functions in two coupling constants. The corresponding double series are resummed through a Pad'e variational procedure. A check of universality is provided since our numerical estimates agree with the already known values.
We use a regularized (ϕ2)2 field theory for the determination of the critical exponents γ and ν in three dimensions. The regularization is done by a Gaussian propagator. This enables any order in g for any correlation function to be easily calculated but introduces a second expansion parameter. In spite of the difficulties encountered when dealing with double series, we obtain values for the critical exponents in good agreement with those found with the renormalized theory. This and the discussion we give of the link between the renormalized theory and our model illustrate the concept of universality in critical phenomena.
Near a continuous phase transition, there exist universal combinations of critical amplitudes. We present an estimation of such a universal quantity, in the framework of the ${\ensuremath{\varphi}}^{4}$ field theory in three dimensions, using information recently available in the literature, and already used to determine precise values for critical exponents. This leads to an accuracy only obtained, hitherto, in the framework of field theory, for critical exponents. The universal quantity calculated, often denoted by $R_{\ensuremath{\xi}}^{}{}_{}{}^{+}$, combines the amplitude of the correlation length and that of the singular part of the specific heat in the vicinity of the critical point.
We study the 3d Ising universality class using the functional renormalization group. With the help of background fields and a derivative expansion up to fourth order we compute the leading index, the subleading symmetric and antisymmetric corrections to scaling, the anomalous dimension, the scaling solution, and the eigenperturbations at criticality. We also study the cross correlations of scaling exponents, their dependence on dimensionality, and the numerical convergence of the derivative expansion. Collecting all available data from functional renormalization group studies to date, we estimate that systematic errors are in good agreement with findings from Monte Carlo simulations, ϵϵϵ-expansion techniques, and resummed perturbation theory.
The RG functions of the 2D nnn-vector ϕ4ϕ^4ϕ4 model are calculated in the five-loop approximation. Perturbative series for the βββ function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case n=1n = 1n=1 as well as in the others (n=0n = 0n=0, n=−1n = -1n=−1, n=2,3,...32n = 2, 3,...32n=2,3,...32) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the βββ function that can not be found perturbatively. The evaluation of the critical exponents for n=1n = 1n=1, n=0n = 0n=0 and n=−1n = -1n=−1 in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.
We compute critical exponents of O(N)O(N)O(N) models in fractional dimensions between d=2d=2d=2 and 4, and for continuous values of the number of field components NNN, in this way completing the RG classification of universality classes for these models. These curves represent nonperturbative approximation to the exact results, they respect all the qualitative features expected from such quantities conciliating previously known perturbative results in three dimensions with exact results in two dimensions and giving a strong indication of what could be the exact behavior of such curves. We also report critical exponents for some multicritical universality classes in the cases $N\ensuremath{\ge}2$ and N=0N=0N=0. Finally, in the large-NNN limit our critical exponents correctly approach those of the spherical model, allowing us to set $N\ensuremath{\sim}100$ as the threshold for the quantitative validity of leading order large-NNN estimates.