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/9805210 · Physical review. B, Condensed matter

Wilson-renormalization-group approach of the principal chiral model around two dimensions

Bertrand Delamotte, D. Mouhanna, P. Lecheminant

Published Mar 1, 1999• 3 citations
arXivHTMLPDF

Abstract

We study the principal chiral Ginzburg-Landau-Wilson model around two dimensions within the local potential approximation of an exact renormalization-group equation. This model, relevant for the long-distance physics of classical frustrated spin systems, exhibits a fixed point of the same universality class as does the corresponding nonlinear sigma model. This allows to shed light on the long-standing discrepancy between the different perturbative approaches of frustrated spin systems.

DOI
10.1103/physrevb.59.6006

Related papers

Nonperturbative approach of the principal chiral model between two and four dimensions

Matthieu Tissier, D. Mouhanna, Bertrand Delamotte

We investigate the principal chiral model between two and four dimensions by means of a nonperturbative Wilson-like renormalization-group equation. We are thus able to follow the evolution of the effective coupling constants within this whole range of dimensions without having recourse to any kind of small parameter expansion. This allows us to identify its three-dimensional critical physics and to solve the long-standing discrepancy between the different perturbative approaches that characterizes the class of models to which the principal chiral model belongs.

arXiv:cond-mat/9908352•Physical review. B, Condensed matter•Jun 1, 2000•35 citations

Chiral critical behavior of frustrated spin systems in two dimensions from five-loop renormalization-group expansions

Pasquale Calabrese, E. V. Orlov, Pietro Parruccini +1 more

We analyze the critical behavior of two-dimensional N-vector spin systems with noncollinear order within the five-loop renormalization-group (RG) approximation. The structure of the RG flow is studied for different N leading to the conclusion that the chiral fixed point governing the critical behavior of physical systems with $N=2$ and $N=3$ does not coincide with that given by the $1/N$ expansion. We show that the stable chiral fixed point for $N<~{N}^{*},$ including $N=2$ and $N=3,$ turns out to be a focus. We give a complete characterization of the critical behavior controlled by this fixed point, also evaluating the subleading crossover exponents. The spiral-like approach of the chiral fixed point is argued to give rise to unusual crossover and near-critical regimes that may imitate varying critical exponents seen in numerous physical and computer experiments.

arXiv:cond-mat/0207187•Physical review. B, Condensed matter•Jan 16, 2003•23 citations

Global fixed point potential approach to frustrated antiferromagnets

Shunsuke Yabunaka, Bertrand Delamotte

Abstract We revisit the critical behavior of classical frustrated systems using the nonperturbative renormalization group (NPRG) equation. Our study is performed within the local potential approximation of this equation to which is added the flow of the field renormalization. Our flow equations are functional to avoid possible artifacts coming from the field expansion of the fixed point potential which consists in keeping only a limited number of coupling constants. We explain in detail our numerical implementation, its advantages and the difficulties encountered in the vicinity of d = 2. For N-component spins, the function N c ( d ) separating the regions of first and second order transitions in the (d, N) plane is computed for d between 4 and 2.3. Our results confirm what was previously found with cruder approximations of the NPRG equation and contradict both the fixed dimension perturbative approach and some of the results obtained within the conformal bootstrap approach.

arXiv:2409.17897•Journal of Statistical Mechanics Theory and Experiment•Feb 3, 2025•1 citation

The Local Potential Approach to frustrated antiferromagnets

Shunsuke Yabunaka, Bertrand Delamotte

We revisit the critical behavior of classical frustrated systems using the nonperturbative renormalization group (NPRG) equation. Our study is performed within the local potential approximation of this equation to which is added the flow of the field renormalization. Our flow equations are functional to avoid possible artifacts coming from field expansions which consists in keeping only a limited number of coupling constants. We present a simple numerical method to follow the fixed point solution of our equations by changing gradually the dimension d and the number N of spin-components. We explain in details the advantage of this method as well as the numerical difficulties we encounter, which become severe close to d = 2. The function N_c(d) separating the regions of first and second order in the (d,N) plane is computed for d between 4 and 2.2. Our results confirm what was previously found within cruder approximation of the NPRG equation and contradict both the fixed dimension perturbative approach and the results obtained within the conformal bootstrap approach.

arXiv:1604.01916•Apr 7, 2016•0 citations

Renormalization group and 1/N expansion for the three-dimensional Ginzburg-Landau-Wilson model

Ian D. Lawrie, Dominic J. Lee

A renormalization-group scheme is developed for the three-dimensional $O(2N)$-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a $1/N$ expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at $z=0,$ where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at $z=0.$ This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of $\mathrm{ln}y.$

arXiv:cond-mat/0107401•Physical review. B, Condensed matter•Oct 16, 2001•1 citation

Perturbative renormalization of the Ginzburg-Landau model revisited

J. Kaupuÿzs

The perturbative renormalization of the Ginzburg–Landau model is reconsidered based on the Feynman diagram technique. We derive renormalization group (RG) flow equations, exactly calculating all vertices appearing in the perturbative renormalization of the ' 4 model up to the 3 order of the -expansion. In this case, the ' 2 , ' 4 , ' 6 , and ' 8 vertices appear. All these are relevant in the sense that they do not vanish at the fixed point. We have tested the expected basic properties of the RG flow, such as the semigroup property. Under repeated RG transformation Rs, appropriately represented RG flow on the critical surface converges to certain s–independent fixed point. The Fourier–transformed two–point correlation function G(k) has been considered. Although the -expansion of X(k) = 1=G(k) is well defined on the critical surface, we have revealed a fundamental inconsistency of the perturbative method with the exact rescaling of X(k), represented as an expansion in powers of k at k ! 0. We discuss also certain paradox, related to the -expansion of X(k) above the critical point. Apart from the –expansion, we have tested also a modified approach, where the ' 4 coupling constant u is the expansion parameter at a fixed spatial dimensionality d. Our tests point to some internal inconsistency of such a method.

•Sep 30, 2006•0 citations
We study the principal chiral Ginzburg-Landau-Wilson model around two dimensions within the local potential approximation of an exact renormalization-group equation. This model, relevant for the long-distance physics of classical frustrated spin systems, exhibits a fixed point of the same universality class as does the corresponding nonlinear sigma model. This allows to shed light on the long-standing discrepancy between the different perturbative approaches of frustrated spin systems.
We investigate the principal chiral model between two and four dimensions by means of a nonperturbative Wilson-like renormalization-group equation. We are thus able to follow the evolution of the effective coupling constants within this whole range of dimensions without having recourse to any kind of small parameter expansion. This allows us to identify its three-dimensional critical physics and to solve the long-standing discrepancy between the different perturbative approaches that characterizes the class of models to which the principal chiral model belongs.
We analyze the critical behavior of two-dimensional N-vector spin systems with noncollinear order within the five-loop renormalization-group (RG) approximation. The structure of the RG flow is studied for different N leading to the conclusion that the chiral fixed point governing the critical behavior of physical systems with N=2N=2N=2 and N=3N=3N=3 does not coincide with that given by the 1/N1/N1/N expansion. We show that the stable chiral fixed point for N< N∗,N<~{N}^{*},N< N∗, including N=2N=2N=2 and N=3,N=3,N=3, turns out to be a focus. We give a complete characterization of the critical behavior controlled by this fixed point, also evaluating the subleading crossover exponents. The spiral-like approach of the chiral fixed point is argued to give rise to unusual crossover and near-critical regimes that may imitate varying critical exponents seen in numerous physical and computer experiments.
Abstract We revisit the critical behavior of classical frustrated systems using the nonperturbative renormalization group (NPRG) equation. Our study is performed within the local potential approximation of this equation to which is added the flow of the field renormalization. Our flow equations are functional to avoid possible artifacts coming from the field expansion of the fixed point potential which consists in keeping only a limited number of coupling constants. We explain in detail our numerical implementation, its advantages and the difficulties encountered in the vicinity of d = 2. For N-component spins, the function N c ( d ) separating the regions of first and second order transitions in the (d, N) plane is computed for d between 4 and 2.3. Our results confirm what was previously found with cruder approximations of the NPRG equation and contradict both the fixed dimension perturbative approach and some of the results obtained within the conformal bootstrap approach.
We revisit the critical behavior of classical frustrated systems using the nonperturbative renormalization group (NPRG) equation. Our study is performed within the local potential approximation of this equation to which is added the flow of the field renormalization. Our flow equations are functional to avoid possible artifacts coming from field expansions which consists in keeping only a limited number of coupling constants. We present a simple numerical method to follow the fixed point solution of our equations by changing gradually the dimension d and the number N of spin-components. We explain in details the advantage of this method as well as the numerical difficulties we encounter, which become severe close to d = 2. The function N_c(d) separating the regions of first and second order in the (d,N) plane is computed for d between 4 and 2.2. Our results confirm what was previously found within cruder approximation of the NPRG equation and contradict both the fixed dimension perturbative approach and the results obtained within the conformal bootstrap approach.
A renormalization-group scheme is developed for the three-dimensional O(2N)O(2N)O(2N)-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a 1/N1/N1/N expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at z=0,z=0,z=0, where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at z=0.z=0.z=0. This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of lny.\mathrm{ln}y.lny.
The perturbative renormalization of the Ginzburg–Landau model is reconsidered based on the Feynman diagram technique. We derive renormalization group (RG) flow equations, exactly calculating all vertices appearing in the perturbative renormalization of the ' 4 model up to the 3 order of the -expansion. In this case, the ' 2 , ' 4 , ' 6 , and ' 8 vertices appear. All these are relevant in the sense that they do not vanish at the fixed point. We have tested the expected basic properties of the RG flow, such as the semigroup property. Under repeated RG transformation Rs, appropriately represented RG flow on the critical surface converges to certain s–independent fixed point. The Fourier–transformed two–point correlation function G(k) has been considered. Although the -expansion of X(k) = 1=G(k) is well defined on the critical surface, we have revealed a fundamental inconsistency of the perturbative method with the exact rescaling of X(k), represented as an expansion in powers of k at k ! 0. We discuss also certain paradox, related to the -expansion of X(k) above the critical point. Apart from the –expansion, we have tested also a modified approach, where the ' 4 coupling constant u is the expansion parameter at a fixed spatial dimensionality d. Our tests point to some internal inconsistency of such a method.