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:cond-mat/9908352 · Physical review. B, Condensed matter

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

Matthieu Tissier, D. Mouhanna, Bertrand Delamotte

Published Jun 1, 2000• 35 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.61.15327

Related papers

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

Bertrand Delamotte, D. Mouhanna, P. Lecheminant

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.

arXiv:hep-th/9805210•Physical review. B, Condensed matter•Mar 1, 1999•3 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

Tensor renormalization group study of 3D principal chiral model

Judah Unmuth-Yockey, Shinichiro Akiyama, Raghav G. Jha

We study the three-dimensional $SU(2)$ principal chiral model using different tensor renormalization group methods based on the triad and anisotropic decomposition of the tensor. The tensor network representation is formulated based on the character expansion of the Boltzmann weight. We compare the average action obtained using these two tensor network algorithms and confirm that the resulting critical coupling and exponent are comparable with the recent estimations from the Monte Carlo methods.

International Symposium on Lattice Field Theory (LATTICE)•Dec 27, 2023•1 citation

Tensor renormalization group study of 3D principal chiral model

Shinichiro Akiyama, Raghav G. Jha, Judah Unmuth-Yockey

We study the three-dimensional $SU(2)$ principal chiral model (PCM) using different tensor renormalization group methods based on the triad and anisotropic decomposition of the tensor. The tensor network representation is formulated based on the character expansion of the Boltzmann weight. We compare the average action obtained using these two tensor network algorithms and confirm that the resulting critical coupling and exponent are comparable with the recent estimations from the Monte Carlo methods.

arXiv:2312.11649•Dec 18, 2023•0 citations

S U ( 2 ) principal chiral model with tensor renormalization group on a cubic lattice

Shinichiro Akiyama, Raghav G. Jha, Judah Unmuth-Yockey

We study the continuous phase transition and thermodynamic observables in the three-dimensional Euclidean S U ( 2 ) principal chiral field model with the triad tensor renormalization group and the anisotropic tensor renormalization group methods. Using these methods, we find results that are consistent with previous Monte Carlo estimates and the predicted renormalization group scaling of the magnetization close to criticality. These results bring us one step closer to studying finite-density quantum chromodynamics in four dimensions using tensor network methods. Published by the American Physical Society 2024

arXiv:2406.10081•Physical review. D/Physical review. D.•Aug 30, 2024•8 citations

The renormalization group (RG) approach: The critical theory near four dimensions

Jean Zinn‐Justin

Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.

•Apr 15, 2021•0 citations
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.
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.
We study the three-dimensional SU(2)SU(2)SU(2) principal chiral model using different tensor renormalization group methods based on the triad and anisotropic decomposition of the tensor. The tensor network representation is formulated based on the character expansion of the Boltzmann weight. We compare the average action obtained using these two tensor network algorithms and confirm that the resulting critical coupling and exponent are comparable with the recent estimations from the Monte Carlo methods.
We study the three-dimensional SU(2)SU(2)SU(2) principal chiral model (PCM) using different tensor renormalization group methods based on the triad and anisotropic decomposition of the tensor. The tensor network representation is formulated based on the character expansion of the Boltzmann weight. We compare the average action obtained using these two tensor network algorithms and confirm that the resulting critical coupling and exponent are comparable with the recent estimations from the Monte Carlo methods.
We study the continuous phase transition and thermodynamic observables in the three-dimensional Euclidean S U ( 2 ) principal chiral field model with the triad tensor renormalization group and the anisotropic tensor renormalization group methods. Using these methods, we find results that are consistent with previous Monte Carlo estimates and the predicted renormalization group scaling of the magnetization close to criticality. These results bring us one step closer to studying finite-density quantum chromodynamics in four dimensions using tensor network methods. Published by the American Physical Society 2024
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.