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

Universal sextic effective interaction at criticality

A. I. Sokolov, E. V. Orlov, V. A. Ul’kov

Published Jan 1, 1998• 15 citations
OpenAlex

Abstract

The renormalization group approach in three dimensions is used to estimate the universal critical value of dimensionless sextic effective coupling constant for the Ising model. Four–loop expansion for g6 is calculated and resummed resulting in g ∗ 6 = 1.596, while the most accurate estimate for g ∗ 6 is argued to be equal to 1.61. 1 The critical thermodynamics of the three–dimensional Ising model is known to be described by Euclidean scalar field theory with the Hamiltonian H =

Related papers

Renormalization group and sextic effective interaction for the two-dimensional Ising model

Aleksandr Ivanovich Sokolov, E. V. Orlov

The universal critical value g 6 of the renormalized sextic coupling constant entering the equation of state of the two{dimensional Ising model is estimated within the renormalization{group approach.Four{loop RG expansion for g6 is calculated and resummed by means of the Pad e{Borel{Leroy technique.The estimated g 6 = 1:10 is obtained which agree quite well with that deduced recently by Zinn, Lai, and Fisher from the high{temperature expansions.

Journal of Physical Studies•Jan 1, 1999•0 citations

Renormalized sextic coupling constant for the two-dimensional Ising model from field theory

Aleksandr Ivanovich Sokolov, E. V. Orlov

The field-theoretical renormalization-group approach is used to estimate the universal critical value ${g}_{6}^{*}$ of the renormalized sextic coupling constant for the two-dimensional Ising model. A four-loop perturbative expansion for ${g}_{6}$ is calculated and resummed by means of the Pad\'e-Borel-Leroy technique. Under the optimal value of the shift parameter $b,$ providing the fastest convergence of the iteration procedure, the estimates ${g}_{6}^{*}=1.10,$ ${g}_{6}^{*}{/g}_{4}^{*2}=2.94$ are obtained, which agree quite well with those deduced recently by Zinn, Lai, and Fisher [Phys. Rev. E 54, 1176 (1996)] from the high-temperature expansions.

arXiv:cond-mat/9804008•Physical review. B, Condensed matter•Aug 1, 1998•21 citations

On free energy of three-dimensional ising model at criticality

Aleksandr Ivanovich Sokolov, V. A. Ul’kov, E. V. Orlov

Higher{order vertices at zero external momenta for the scalar eld theory, describing the critical behaviour of the Ising model, are studied within the eld{theoretical renormalization group (RG) approach in three dimensions.Dimensionless six{point g6 and eight{point g8 eective coupling constants are calculated in the three{loop approximation.Their numerical values, universal at criticality, are estimated by means of the Pad e and Pad e{Borel summation of the RG expansions found and by putting the renormalized quartic coupling constant equal to its universal xed{point value known from six{loop RG calculations.The values of g 6 obtained are compared with their analogs resulting from the {expansion, Monte Carlo simulations, the Wegner{Houghton equations and the linked cluster expansion series.The eld{theoretical estimates for g 6 are shown to be in a good agreement with each other, diering considerably from the values given by other methods.

arXiv:cond-mat/9803352•Journal of Physical Studies•Jan 1, 1997•12 citations

Critical behavior of the weakly disordered Ising model: Six-loop ɛ expansion study

Mikhail Vladimirovich Kompaniets, Andrey Kudlis, Aleksandr Ivanovich Sokolov

The critical behavior of three-dimensional weakly diluted quenched Ising model is examined on the base of six-loop renormalization group expansions obtained within the minimal subtraction scheme in 4-ɛ space dimensions. For this purpose the ϕ^{4} field theory with cubic symmetry was analyzed in the replica limit n→0. Along with renormalization group expansions in terms of renormalized couplings the sqrt[ɛ] expansions of critical exponents are presented. Corresponding numerical estimates for the physical, three-dimensional system are obtained by means of different resummation procedures applied both to the sqrt[ɛ] series and to initial renormalization group expansions. The results given by the latter approach are in a good agreement with their counterparts obtained experimentally and within the Monte Carlo simulations, while resumming of sqrt[ɛ] series themselves turned out to be disappointing.

arXiv:2011.10782•Physical review. E•Feb 19, 2021•9 citations

2n-point renormalized coupling constants in the three-dimensional Ising model:Estimates by high temperature series to order Β17

P. Butera, M. Comi

We compute the 2n-point renormalized coupling constants in the symmetric phase of the three-dimensional (3D) Ising model on the simple cubic lattice in terms of the high temperature expansions O(${\mathrm{\ensuremath{\mathrm{B}}}}^{17}$) of the Fourier transformed 2n-point connected correlation functions at zero momentum. Our high temperature estimates of these quantities, which enter into the small field expansion of the effective potential for a 3D scalar field at the infrared fixed point or, equivalently, in the critical equation of state of the 3D Ising model universality class, are compared with recent results obtained by renormalization group methods, strong coupling, stochastic simulations, as well as previous high temperature expansions.

arXiv:hep-lat/9703017•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 1997•40 citations

Effective and asymptotic critical exponents of a weakly diluted quenched Ising model: Three-dimensional approach versus ɛ expansion

R. Folk, Yurij V. Holovatch, Taras Yavors’kii

We present a field-theoretical treatment of the critical behavior of a three-dimensional weakly diluted quenched Ising model. To this end we analyze in the replica limit $\stackrel{\ensuremath{\rightarrow}}{n}0$ the five-loop renormalization-group functions of the ${\ensuremath{\varphi}}^{4}$ theory with $O(n)$-symmetric and cubic interactions [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)]. The minimal subtraction scheme allows one to develop either the $\sqrt{\ensuremath{\varepsilon}}$-expansion series or to proceed within the three-dimensional approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents. We report resummed numerical values for the effective and asymptotic critical exponents. The results obtained within the three-dimensional approach agree pretty well with recent Monte Carlo simulations. $\sqrt{\ensuremath{\varepsilon}}$ expansion does not allow reliable estimates for $d=3.$

arXiv:cond-mat/9909121•Physical review. B, Condensed matter•Jun 1, 2000•74 citations
The renormalization group approach in three dimensions is used to estimate the universal critical value of dimensionless sextic effective coupling constant for the Ising model. Four–loop expansion for g6 is calculated and resummed resulting in g ∗ 6 = 1.596, while the most accurate estimate for g ∗ 6 is argued to be equal to 1.61. 1 The critical thermodynamics of the three–dimensional Ising model is known to be described by Euclidean scalar field theory with the Hamiltonian H =
The field-theoretical renormalization-group approach is used to estimate the universal critical value g6∗{g}_{6}^{*}g6∗​ of the renormalized sextic coupling constant for the two-dimensional Ising model. A four-loop perturbative expansion for g6{g}_{6}g6​ is calculated and resummed by means of the Pad\'e-Borel-Leroy technique. Under the optimal value of the shift parameter b,b,b, providing the fastest convergence of the iteration procedure, the estimates g6∗=1.10,{g}_{6}^{*}=1.10,g6∗​=1.10, g6∗/g4∗2=2.94{g}_{6}^{*}{/g}_{4}^{*2}=2.94g6∗​/g4∗2​=2.94 are obtained, which agree quite well with those deduced recently by Zinn, Lai, and Fisher [Phys. Rev. E 54, 1176 (1996)] from the high-temperature expansions.
Higher{order vertices at zero external momenta for the scalar eld theory, describing the critical behaviour of the Ising model, are studied within the eld{theoretical renormalization group (RG) approach in three dimensions.Dimensionless six{point g6 and eight{point g8 eective coupling constants are calculated in the three{loop approximation.Their numerical values, universal at criticality, are estimated by means of the Pad e and Pad e{Borel summation of the RG expansions found and by putting the renormalized quartic coupling constant equal to its universal xed{point value known from six{loop RG calculations.The values of g 6 obtained are compared with their analogs resulting from the {expansion, Monte Carlo simulations, the Wegner{Houghton equations and the linked cluster expansion series.The eld{theoretical estimates for g 6 are shown to be in a good agreement with each other, diering considerably from the values given by other methods.
The critical behavior of three-dimensional weakly diluted quenched Ising model is examined on the base of six-loop renormalization group expansions obtained within the minimal subtraction scheme in 4-ɛ space dimensions. For this purpose the ϕ^{4} field theory with cubic symmetry was analyzed in the replica limit n→0. Along with renormalization group expansions in terms of renormalized couplings the sqrt[ɛ] expansions of critical exponents are presented. Corresponding numerical estimates for the physical, three-dimensional system are obtained by means of different resummation procedures applied both to the sqrt[ɛ] series and to initial renormalization group expansions. The results given by the latter approach are in a good agreement with their counterparts obtained experimentally and within the Monte Carlo simulations, while resumming of sqrt[ɛ] series themselves turned out to be disappointing.
We compute the 2n-point renormalized coupling constants in the symmetric phase of the three-dimensional (3D) Ising model on the simple cubic lattice in terms of the high temperature expansions O(${\mathrm{\ensuremath{\mathrm{B}}}}^{17}$) of the Fourier transformed 2n-point connected correlation functions at zero momentum. Our high temperature estimates of these quantities, which enter into the small field expansion of the effective potential for a 3D scalar field at the infrared fixed point or, equivalently, in the critical equation of state of the 3D Ising model universality class, are compared with recent results obtained by renormalization group methods, strong coupling, stochastic simulations, as well as previous high temperature expansions.
We present a field-theoretical treatment of the critical behavior of a three-dimensional weakly diluted quenched Ising model. To this end we analyze in the replica limit $\stackrel{\ensuremath{\rightarrow}}{n}0$ the five-loop renormalization-group functions of the ${\ensuremath{\varphi}}^{4}$ theory with O(n)O(n)O(n)-symmetric and cubic interactions [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)]. The minimal subtraction scheme allows one to develop either the $\sqrt{\ensuremath{\varepsilon}}$-expansion series or to proceed within the three-dimensional approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents. We report resummed numerical values for the effective and asymptotic critical exponents. The results obtained within the three-dimensional approach agree pretty well with recent Monte Carlo simulations. $\sqrt{\ensuremath{\varepsilon}}$ expansion does not allow reliable estimates for d=3.d=3.d=3.