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:hep-th/0409300 · Physical Review B

Optimization of field-dependent nonperturbative renormalization group flows

Léonie Canet

Published Jan 31, 2005• 33 citations
arXivHTMLPDF

Abstract

We investigate the influence of the momentum cutoff function on the field-dependent nonperturbative renormalization group flows for the three-dimensional Ising model, up to the second order of the derivative expansion. We show that, even when dealing with the full functional dependence of the renormalization functions, the accuracy of the critical exponents can be simply optimized, through the principle of minimal sensitivity, which yields $\ensuremath{\nu}=0.628$ and $\ensuremath{\eta}=0.044$.

DOI
10.1103/physrevb.71.012418

Related papers

Conformal invariance in the nonperturbative renormalization group: A rationale for choosing the regulator

Ivan Balog, Gonzalo De Polsi, Matthieu Tissier +1 more

Field-theoretical calculations performed in an approximation scheme often present a spurious dependence of physical quantities on some unphysical parameters associated with the details of the calculation setup (such as the renormalization scheme or, in perturbation theory, the resummation procedure). In the present article, we propose to reduce this dependence by invoking conformal invariance. Using as a benchmark the three-dimensional Ising model, we show that, within the derivative expansion at order 4, performed in the nonperturbative renormalization group formalism, the identity associated with this symmetry is not exactly satisfied. The calculations which best satisfy this identity are shown to yield critical exponents which coincide to a high accuracy with those obtained by the conformal bootstrap. Additionally, this work gives a strong justification to the success of a widely used criterion for fixing the appropriate renormalization scheme, namely the principle of minimal sensitivity.

arXiv:2004.02521•Physical review. E•Jun 29, 2020•31 citations

Nonperturbative renormalization group approach to the Ising model: A derivative expansion at order ∂4

Léonie Canet, Bertrand Delamotte, D. Mouhanna +1 more

On the example of the three-dimensional Ising model, we show that nonperturbative renormalization group equations allow one to obtain very accurate critical exponents. Implementing the order ${\ensuremath{\partial}}^{4}$ of the derivative expansion leads to $\ensuremath{\nu}=0.632$ and to an anomalous dimension $\ensuremath{\eta}=0.033$ which is significantly improved compared with lower orders calculations.

arXiv:hep-th/0302227•Physical review. B, Condensed matter•Aug 22, 2003•195 citations

Convergence of Nonperturbative Approximations to the Renormalization Group

Ivan Balog, Hugues Chaté, Bertrand Delamotte +2 more

We provide analytical arguments showing that the "nonperturbative" approximation scheme to Wilson's renormalization group known as the derivative expansion has a finite radius of convergence. We also provide guidelines for choosing the regulator function at the heart of the procedure and propose empirical rules for selecting an optimal one, without prior knowledge of the problem at stake. Using the Ising model in three dimensions as a testing ground and the derivative expansion at order six, we find fast convergence of critical exponents to their exact values, irrespective of the well-behaved regulator used, in full agreement with our general arguments. We hope these findings will put an end to disputes regarding this type of nonperturbative methods.

arXiv:1907.01829•Physical Review Letters•Dec 13, 2019•133 citations

Solutions of renormalization-group flow equations with full momentum dependence

Federico Benitez, Jean-Paul Blaizot, Hugues Chaté +3 more

We demonstrate the power of a recently proposed approximation scheme for the nonperturbative renormalization group that gives access to correlation functions over their full momentum range. We solve numerically the leading-order flow equations obtained within this scheme and compute the two-point functions of the O(N) theories at criticality, in two and three dimensions. Excellent results are obtained for both universal and nonuniversal quantities at modest numerical cost.

arXiv:0901.0128•Physical Review E•Sep 9, 2009•92 citations

Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation

Federico Benitez, Jean-Paul Blaizot, Hugues Chaté +3 more

We present the implementation of the Blaizot-Méndez-Wschebor approximation scheme of the nonperturbative renormalization group we present in detail, which allows for the computation of the full-momentum dependence of correlation functions. We discuss its significance and its relation with other schemes, in particular, the derivative expansion. Quantitative results are presented for the test ground of scalar O(N) theories. Besides critical exponents, which are zero-momentum quantities, we compute the two-point function at criticality in the whole momentum range in three dimensions and, in the high-temperature phase, the universal structure factor. In all cases, we find very good agreement with the best existing results.

arXiv:1110.2665•Physical Review E•Feb 28, 2012•105 citations

Optimizing Wilson’s momentum shell renormalization group

A. Tröster

Previous attempts to accurately compute critical exponents from Wilson's momentum shell renormalization prescription suffered from the difficulties posed by the presence of an infinite number of irrelevant couplings. Taking the example of the one-dimensional long-ranged Ising model, we calculate the momentum shell renormalization flow in the plane spanned by the coupling constants $({u}_{0},{r}_{0})$ for different values of the momentum shell thickness parameter $b$ by simulation using our recently developed Fourier Monte Carlo algorithm. We report strong anomalies in the $b$ dependence of the fixed-point couplings and the resulting exponents ${y}_{\ensuremath{\tau}}$ and $\ensuremath{\omega}$ in the vicinity of a shell parameter ${b}^{\ensuremath{\ast}}<1$ characterizing a thin but finite momentum shell. Evaluation of the exponents for this value of $b$ yields a dramatic improvement of their numerical accuracy, indicating a strong damping of the influence of irrelevant couplings for $b={b}^{\ensuremath{\ast}}$.

Physical Review B•Mar 31, 2010•8 citations
We investigate the influence of the momentum cutoff function on the field-dependent nonperturbative renormalization group flows for the three-dimensional Ising model, up to the second order of the derivative expansion. We show that, even when dealing with the full functional dependence of the renormalization functions, the accuracy of the critical exponents can be simply optimized, through the principle of minimal sensitivity, which yields \ensuremathν=0.628\ensuremath{\nu}=0.628\ensuremathν=0.628 and \ensuremathη=0.044\ensuremath{\eta}=0.044\ensuremathη=0.044.
Field-theoretical calculations performed in an approximation scheme often present a spurious dependence of physical quantities on some unphysical parameters associated with the details of the calculation setup (such as the renormalization scheme or, in perturbation theory, the resummation procedure). In the present article, we propose to reduce this dependence by invoking conformal invariance. Using as a benchmark the three-dimensional Ising model, we show that, within the derivative expansion at order 4, performed in the nonperturbative renormalization group formalism, the identity associated with this symmetry is not exactly satisfied. The calculations which best satisfy this identity are shown to yield critical exponents which coincide to a high accuracy with those obtained by the conformal bootstrap. Additionally, this work gives a strong justification to the success of a widely used criterion for fixing the appropriate renormalization scheme, namely the principle of minimal sensitivity.
We provide analytical arguments showing that the "nonperturbative" approximation scheme to Wilson's renormalization group known as the derivative expansion has a finite radius of convergence. We also provide guidelines for choosing the regulator function at the heart of the procedure and propose empirical rules for selecting an optimal one, without prior knowledge of the problem at stake. Using the Ising model in three dimensions as a testing ground and the derivative expansion at order six, we find fast convergence of critical exponents to their exact values, irrespective of the well-behaved regulator used, in full agreement with our general arguments. We hope these findings will put an end to disputes regarding this type of nonperturbative methods.
We present the implementation of the Blaizot-Méndez-Wschebor approximation scheme of the nonperturbative renormalization group we present in detail, which allows for the computation of the full-momentum dependence of correlation functions. We discuss its significance and its relation with other schemes, in particular, the derivative expansion. Quantitative results are presented for the test ground of scalar O(N) theories. Besides critical exponents, which are zero-momentum quantities, we compute the two-point function at criticality in the whole momentum range in three dimensions and, in the high-temperature phase, the universal structure factor. In all cases, we find very good agreement with the best existing results.
Previous attempts to accurately compute critical exponents from Wilson's momentum shell renormalization prescription suffered from the difficulties posed by the presence of an infinite number of irrelevant couplings. Taking the example of the one-dimensional long-ranged Ising model, we calculate the momentum shell renormalization flow in the plane spanned by the coupling constants (u0,r0)({u}_{0},{r}_{0})(u0​,r0​) for different values of the momentum shell thickness parameter bbb by simulation using our recently developed Fourier Monte Carlo algorithm. We report strong anomalies in the bbb dependence of the fixed-point couplings and the resulting exponents ${y}_{\ensuremath{\tau}}$ and $\ensuremath{\omega}$ in the vicinity of a shell parameter ${b}^{\ensuremath{\ast}}<1$ characterizing a thin but finite momentum shell. Evaluation of the exponents for this value of bbb yields a dramatic improvement of their numerical accuracy, indicating a strong damping of the influence of irrelevant couplings for $b={b}^{\ensuremath{\ast}}$.