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:1009.1948 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Ising exponents from the functional renormalization group

Daniel F. Litim, Dario Zappalà

Published Apr 7, 2011• 125 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.83.085009

Related papers

Subleading critical exponents from the renormalisation group

Daniel F. Litim, Lautaro Vergara

We study exact renormalisation group equations for the 3d Ising universality class. At the Wilson–Fisher fixed point, symmetric and antisymmetric correction-to-scaling exponents are computed with high accuracy for an optimised cutoff to leading order in the derivative expansion. Further results are derived for other cutoffs including smooth, sharp and background field cutoffs. An estimate for higher order corrections is given as well. We establish that the leading antisymmetric corrections to scaling are strongly subleading compared to the leading symmetric ones.

arXiv:hep-th/0310101•Physics Letters B•Dec 30, 2003•43 citations

Benchmarking the Ising universality class in $3 \le d < 4$ dimensions

Claudio Bonanno, Andrea Cappelli A, Mikhail Vladimirovich Kompaniets +2 more

The Ising critical exponents \eta η , \nu ν and \omega ω are determined up to one-per-thousand relative error in the whole range of dimensions 3 \le d < 4 3 ≤ d < 4 , using numerical conformal-bootstrap techniques. A detailed comparison is made with results by the resummed epsilon expansion in varying dimension, the analytic bootstrap, Monte Carlo and non-perturbative renormalization-group methods, finding very good overall agreement. Precise conformal field theory data of scaling dimensions and structure constants are obtained as functions of dimension, improving on earlier findings, and providing benchmarks in 3 \le d < 4 3 ≤ d < 4 .

arXiv:2210.03051•SciPost Physics•May 30, 2023•9 citations

Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group

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

We compute the critical exponents ν, η and ω of O(N) models for various values of N by implementing the derivative expansion of the nonperturbative renormalization group up to next-to-next-to-leading order [usually denoted O(∂^{4})]. We analyze the behavior of this approximation scheme at successive orders and observe an apparent convergence with a small parameter, typically between 1/9 and 1/4, compatible with previous studies in the Ising case. This allows us to give well-grounded error bars. We obtain a determination of critical exponents with a precision which is similar or better than those obtained by most field-theoretical techniques. We also reach a better precision than Monte Carlo simulations in some physically relevant situations. In the O(2) case, where there is a long-standing controversy between Monte Carlo estimates and experiments for the specific heat exponent α, our results are compatible with those of Monte Carlo but clearly exclude experimental values.

arXiv:2001.07525•Physical review. E•Apr 14, 2020•112 citations

A new functional renormalization group syudy of universality classes with O(N) symmetry

Beretta Tassano, Piero Sebastian

The functional renormalization group (FRG) is a powerful tool that has facilitated the investigation of various strongly interacting theories, encompassing systems characterized by both bosonic and fermionic variables. Specifically, this project focuses on the analysis of theories with O(N) symmetry with bosonic scalar fields. These theories are of particular interest due to their broad applicability in real physical systems. For example, Z2 symmetry (the O(1) model) describes the well-known Ising universality class, which in turn models the liquid-gas transition. On the other hand, the O(2) model belongs to the universality class of the XY model, used to describe the transition from fluid to superfluid in 4 He, while the O(3) model, known as the Heisenberg model, describes the ferromagnetic transition in isotropic materials. Finally the O(0) model is related to the self-avoiding walk (SAW). To characterize these theories, various methods have been implemented within the framework of the renormalization group. Perturbation theory has been applied since the early days of the subject, specifically in the form of the ε-expansion, which has reached high loop order in recent years. Alternatively, a non-perturbative approach can be chosen, which is known as the non-perturbative renormalization group (NPRG). Within the NPRG, there are several approximation schemes, with the derivative expansion being particularly noteworthy. The ultimate goal of these efforts is to calculate the critical exponents, which define the given universality class. The standard non-perturbative method involves solving an equation that depends on a regulator or cut-off, which represents one of the main challenges of this approach. Although the regulators are designed so that the theory does not depend on them at the scales of interest, the results for the critical exponents can vary depending on the regulator used once approximations are made. In this project, we propose an approach that eliminates the need for an explicit mass cut-off significantly simplifying the calculation of critical exponents while mitigating the regulator dependence of the results. In particular, we show that it is possible to use dimensional regularization (DR) beyond the ε-expansion in the context of RG calculations of critical properties. Based on this we propose a new functional RG scheme called Functional Dimensional Regularization (FDR) and apply it to the O(N) model in three dimension, finding excellent agreement with state-of-the-art computations

•Jan 1, 2025•0 citations

Rethinking Dimensional Regularization in Critical Phenomena

P. Beretta, Alessandro Codello

We show that it is possible to use dimensional regularization (DR) beyond the usual $\varepsilon$-expansion in the context of renormalization group (RG) calculations in Critical Phenomena. Based on this fact, we propose a new functional RG scheme - Functional Dimensional Regularization (FDR) - and apply it to a scalar theory in three dimensions. We compute the critical exponents of the Ising universality class directly in $d=3$ under various typical approximations. The method that emerges combines the agility typical of DR with the generality proper of functional RG. Moreover, at a given order of approximation, FDR seems to provide faster convergence and better estimates than other functional RGs.

arXiv:2604.25103•Apr 28, 2026•0 citations

Conformal perturbation of off-critical correlators in the 3D Ising universality class

Michele Caselle, Gianluca Costagliola, Nicodemo Magnoli

Thanks to the impressive progress of conformal bootstrap methods we have now very precise estimates of both scaling dimensions and operator product expansion coefficients for several 3D universality classes. We show how to use this information to obtain similarly precise estimates for off-critical correlators using conformal perturbation. We discuss in particular the $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$, $⟨\ensuremath{\epsilon}(r)\ensuremath{\epsilon}(0)⟩$ and $⟨\ensuremath{\sigma}(r)\ensuremath{\epsilon}(0)⟩$ two-point functions in the high and low temperature regimes of the 3D Ising model and evaluate the leading and next to leading terms in the $s={\mathrm{tr}}^{{\mathrm{\ensuremath{\Delta}}}_{t}}$ expansion, where $t$ is the reduced temperature. Our results for $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$ agree both with Monte Carlo simulations and with a set of experimental estimates of the critical scattering function.

arXiv:1605.05133•Physical review. D/Physical review. D.•Jul 11, 2016•12 citations
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.
We study exact renormalisation group equations for the 3d Ising universality class. At the Wilson–Fisher fixed point, symmetric and antisymmetric correction-to-scaling exponents are computed with high accuracy for an optimised cutoff to leading order in the derivative expansion. Further results are derived for other cutoffs including smooth, sharp and background field cutoffs. An estimate for higher order corrections is given as well. We establish that the leading antisymmetric corrections to scaling are strongly subleading compared to the leading symmetric ones.
Benchmarking the Ising universality class in 3≤d<43 \le d < 43≤d<4 dimensions
The Ising critical exponents \eta η , \nu ν and \omega ω are determined up to one-per-thousand relative error in the whole range of dimensions 3 \le d < 4 3 ≤ d < 4 , using numerical conformal-bootstrap techniques. A detailed comparison is made with results by the resummed epsilon expansion in varying dimension, the analytic bootstrap, Monte Carlo and non-perturbative renormalization-group methods, finding very good overall agreement. Precise conformal field theory data of scaling dimensions and structure constants are obtained as functions of dimension, improving on earlier findings, and providing benchmarks in 3 \le d < 4 3 ≤ d < 4 .
We compute the critical exponents ν, η and ω of O(N) models for various values of N by implementing the derivative expansion of the nonperturbative renormalization group up to next-to-next-to-leading order [usually denoted O(∂^{4})]. We analyze the behavior of this approximation scheme at successive orders and observe an apparent convergence with a small parameter, typically between 1/9 and 1/4, compatible with previous studies in the Ising case. This allows us to give well-grounded error bars. We obtain a determination of critical exponents with a precision which is similar or better than those obtained by most field-theoretical techniques. We also reach a better precision than Monte Carlo simulations in some physically relevant situations. In the O(2) case, where there is a long-standing controversy between Monte Carlo estimates and experiments for the specific heat exponent α, our results are compatible with those of Monte Carlo but clearly exclude experimental values.
The functional renormalization group (FRG) is a powerful tool that has facilitated the investigation of various strongly interacting theories, encompassing systems characterized by both bosonic and fermionic variables. Specifically, this project focuses on the analysis of theories with O(N) symmetry with bosonic scalar fields. These theories are of particular interest due to their broad applicability in real physical systems. For example, Z2 symmetry (the O(1) model) describes the well-known Ising universality class, which in turn models the liquid-gas transition. On the other hand, the O(2) model belongs to the universality class of the XY model, used to describe the transition from fluid to superfluid in 4 He, while the O(3) model, known as the Heisenberg model, describes the ferromagnetic transition in isotropic materials. Finally the O(0) model is related to the self-avoiding walk (SAW). To characterize these theories, various methods have been implemented within the framework of the renormalization group. Perturbation theory has been applied since the early days of the subject, specifically in the form of the ε-expansion, which has reached high loop order in recent years. Alternatively, a non-perturbative approach can be chosen, which is known as the non-perturbative renormalization group (NPRG). Within the NPRG, there are several approximation schemes, with the derivative expansion being particularly noteworthy. The ultimate goal of these efforts is to calculate the critical exponents, which define the given universality class. The standard non-perturbative method involves solving an equation that depends on a regulator or cut-off, which represents one of the main challenges of this approach. Although the regulators are designed so that the theory does not depend on them at the scales of interest, the results for the critical exponents can vary depending on the regulator used once approximations are made. In this project, we propose an approach that eliminates the need for an explicit mass cut-off significantly simplifying the calculation of critical exponents while mitigating the regulator dependence of the results. In particular, we show that it is possible to use dimensional regularization (DR) beyond the ε-expansion in the context of RG calculations of critical properties. Based on this we propose a new functional RG scheme called Functional Dimensional Regularization (FDR) and apply it to the O(N) model in three dimension, finding excellent agreement with state-of-the-art computations
We show that it is possible to use dimensional regularization (DR) beyond the usual ε\varepsilonε-expansion in the context of renormalization group (RG) calculations in Critical Phenomena. Based on this fact, we propose a new functional RG scheme - Functional Dimensional Regularization (FDR) - and apply it to a scalar theory in three dimensions. We compute the critical exponents of the Ising universality class directly in d=3d=3d=3 under various typical approximations. The method that emerges combines the agility typical of DR with the generality proper of functional RG. Moreover, at a given order of approximation, FDR seems to provide faster convergence and better estimates than other functional RGs.
Thanks to the impressive progress of conformal bootstrap methods we have now very precise estimates of both scaling dimensions and operator product expansion coefficients for several 3D universality classes. We show how to use this information to obtain similarly precise estimates for off-critical correlators using conformal perturbation. We discuss in particular the $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$, $⟨\ensuremath{\epsilon}(r)\ensuremath{\epsilon}(0)⟩$ and $⟨\ensuremath{\sigma}(r)\ensuremath{\epsilon}(0)⟩$ two-point functions in the high and low temperature regimes of the 3D Ising model and evaluate the leading and next to leading terms in the $s={\mathrm{tr}}^{{\mathrm{\ensuremath{\Delta}}}_{t}}$ expansion, where ttt is the reduced temperature. Our results for $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$ agree both with Monte Carlo simulations and with a set of experimental estimates of the critical scattering function.