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 from field theory

J. C. Le Guillou, Jean Zinn‐Justin

Published May 1, 1980• 1,804 citations
Publisher

Abstract

We present a detailed study of the methods of summation based on Borel transformation and conformal mapping, which we have used to calculate critical exponents of the $n$-vector model through field theoretical methods. In particular we discuss the sensitivity of the results to various changes in the summation procedure.

DOI
10.1103/physrevb.21.3976

Related papers

Accurate critical exponents from field theory

J. C. Le Guillou, Jean Zinn‐Justin

This is the third article in a sequence in which we reexamine the calculation of the critical exponents of the N-vector model from field theory. In the first article we have applied summation methods to the ε-expansion for which new terms have been calculated (up to order ε5). In the second article we have incorporated in the summation method, in the case of Ising-like systems (N = 1), the knowledge of the exact values of exponents in d = 2 dimensions. Since exact values are now also known in the case of polymers (N = 0), we apply here the same method to this case. Moreover, since for the exponent η for N = 1 the results coming from the ε-expansion and perturbation series at fixed dimension are slightly different, we have recalculated the exponents in the latter case using the variant of the summation method developed for the ε-expansion. The results do not change significantly showing both the reliability and the limits of the method.

Journal de physique•Jan 1, 1989•122 citations

Determination of critical exponents and equation of state by field theory method

Jean Zinn‐Justin

Path integrals have played a fundamental role in emphasizing the profound analogies between Quantum Field Theory (QFT), and Classical as well as Quantum Statistical Physics. Ideas coming from Statistical Physics have then led to a deeper understanding of Quantum Field Theory and open the way for a wealth of non-perturbative methods. Conversely QFT methods are become essential for the description of the phase transitions and critical phenomena beyond mean field theory. This is the point we want to illustrate here. We therefore review the methods, based on renormalized phi^4_3 quantum field theory and renormalization group, which have led to an accurate determination of critical exponents of the N-vector model, and more recently of the equation of state of the 3D Ising model. The starting point is the perturbative expansion for RG functions or the effective potential to the order presently available. Perturbation theory is known to be divergent and its divergence has been related to instanton contributions. This has allowed to characterize the large order behaviour of perturbation series, an information that can be used to efficiently "sum" them. Practical summation methods based on Borel transformation and conformal mapping have been developed, leading to the most accurate results available probing field theory in a non perturbative regime. We illustrate the methods with a short discussion of the scaling equation of state of the 3D Ising model. Compared to exponents its determination involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order-dependent mapping.

arXiv:hep-th/9810193•Oct 23, 1998•2 citations

Renormalization group study of the m-vector model between two and four dimensions

Yurij V. Holovatch

In the frames of field theoretical fixed dimension renormalization group scheme critical exponents of m vector model are calculated in the case when space dimension continuously changes from the lower critical dimension to the upper one.

Ferroelectrics•Feb 1, 1997•6 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

Renormalization-Group Functions and Nonuniversal Critical Behaviour

R. Schloms, V. Dohm

The method of Borel resummation is used to obtain an accurate representation of renormalization-group functions of the n -vector model within the minimal renormalization scheme. This permits to relate the nonuniversal critical behaviour of different quantities well away from criticality. The results can be used to estimate the range of applicability of the n -vector model. Application to the λ-transition of 4 He explains the measured effective exponent of the superfluid density along the λ-line.

Europhysics Letters (EPL)•Feb 15, 1987•67 citations

Critical exponents and equation of state from series summation

Jean Zinn‐Justin

Abstract Universal quantities near the phase transition of O(N) symmetric vector models, can be determined, in the framework of the (f2 )2 field theory, and the corresponding renormalization group (RG), in the form of perturbative series. The O(N) symmetric field theories describe, in particular for N = 0, the universal properties of the statistics of long polymers, for N = 1, the liquid–vapour transition, for N = 2, superfluid helium transition, and so on. Universal quantities have been calculated within two different schemes, the Wilson-Fisher ϵ = 4 − d expansion, and perturbative expansion at fixed dimensions 2 and 3 (as suggested by Parisi). In both cases, the series are divergent, and the expansion parameters are not small. In fixed dimensions smaller than 4, the series are proven to be Borel summable. For the ϵ expansion, there are reasons that the property is equally true, but a proof is lacking. With this assumption, in both cases, although the series are divergent, they define unique functions. Since the expansion parameters are not small, summation methods are then required to determine these functions. A specific summation method, based on a parametric Borel transformation and mapping, in which the knowledge of the large order behaviour has been incorporated, has been successfully applied to the series, and has led to a precise evaluation of critical exponents and other universal quantities.

•Apr 15, 2021•0 citations
We present a detailed study of the methods of summation based on Borel transformation and conformal mapping, which we have used to calculate critical exponents of the nnn-vector model through field theoretical methods. In particular we discuss the sensitivity of the results to various changes in the summation procedure.
This is the third article in a sequence in which we reexamine the calculation of the critical exponents of the N-vector model from field theory. In the first article we have applied summation methods to the ε-expansion for which new terms have been calculated (up to order ε5). In the second article we have incorporated in the summation method, in the case of Ising-like systems (N = 1), the knowledge of the exact values of exponents in d = 2 dimensions. Since exact values are now also known in the case of polymers (N = 0), we apply here the same method to this case. Moreover, since for the exponent η for N = 1 the results coming from the ε-expansion and perturbation series at fixed dimension are slightly different, we have recalculated the exponents in the latter case using the variant of the summation method developed for the ε-expansion. The results do not change significantly showing both the reliability and the limits of the method.
Path integrals have played a fundamental role in emphasizing the profound analogies between Quantum Field Theory (QFT), and Classical as well as Quantum Statistical Physics. Ideas coming from Statistical Physics have then led to a deeper understanding of Quantum Field Theory and open the way for a wealth of non-perturbative methods. Conversely QFT methods are become essential for the description of the phase transitions and critical phenomena beyond mean field theory. This is the point we want to illustrate here. We therefore review the methods, based on renormalized phi^4_3 quantum field theory and renormalization group, which have led to an accurate determination of critical exponents of the N-vector model, and more recently of the equation of state of the 3D Ising model. The starting point is the perturbative expansion for RG functions or the effective potential to the order presently available. Perturbation theory is known to be divergent and its divergence has been related to instanton contributions. This has allowed to characterize the large order behaviour of perturbation series, an information that can be used to efficiently "sum" them. Practical summation methods based on Borel transformation and conformal mapping have been developed, leading to the most accurate results available probing field theory in a non perturbative regime. We illustrate the methods with a short discussion of the scaling equation of state of the 3D Ising model. Compared to exponents its determination involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order-dependent mapping.
The method of Borel resummation is used to obtain an accurate representation of renormalization-group functions of the n -vector model within the minimal renormalization scheme. This permits to relate the nonuniversal critical behaviour of different quantities well away from criticality. The results can be used to estimate the range of applicability of the n -vector model. Application to the λ-transition of 4 He explains the measured effective exponent of the superfluid density along the λ-line.
Abstract Universal quantities near the phase transition of O(N) symmetric vector models, can be determined, in the framework of the (f2 )2 field theory, and the corresponding renormalization group (RG), in the form of perturbative series. The O(N) symmetric field theories describe, in particular for N = 0, the universal properties of the statistics of long polymers, for N = 1, the liquid–vapour transition, for N = 2, superfluid helium transition, and so on. Universal quantities have been calculated within two different schemes, the Wilson-Fisher ϵ = 4 − d expansion, and perturbative expansion at fixed dimensions 2 and 3 (as suggested by Parisi). In both cases, the series are divergent, and the expansion parameters are not small. In fixed dimensions smaller than 4, the series are proven to be Borel summable. For the ϵ expansion, there are reasons that the property is equally true, but a proof is lacking. With this assumption, in both cases, although the series are divergent, they define unique functions. Since the expansion parameters are not small, summation methods are then required to determine these functions. A specific summation method, based on a parametric Borel transformation and mapping, in which the knowledge of the large order behaviour has been incorporated, has been successfully applied to the series, and has led to a precise evaluation of critical exponents and other universal quantities.