Journal of Physical Studies

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

Aleksandr Ivanovich Sokolov, E. V. Orlov

Published Jan 1, 1999• 0 citations

Abstract

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.

DOI: 10.30970/jps.03.280

Related papers

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

Aleksandr Ivanovich Sokolov, E. V. Orlov

arXiv:cond-mat/9804008Physical review. B, Condensed matterAug 1, 199821 citations

Universal sextic effective interaction at criticality

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

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 =

Jan 1, 199815 citations

On free energy of three-dimensional ising model at criticality

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

arXiv:cond-mat/9803352Journal of Physical StudiesJan 1, 199712 citations

Pseudo-ε-expansion and the two-dimensional Ising model

A. I. Sokolov

arXiv:cond-mat/0510088Physics of the Solid StateNov 1, 200514 citations

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

P. Butera, M. Comi

arXiv:hep-lat/9703017Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJun 1, 199740 citations

The critical equation of state of the two-dimensional Ising model

Michele Caselle, Martin Hasenbusch B, Andrea Pelissetto C +1 more

arXiv:cond-mat/0011305Journal of Physics A Mathematical and GeneralMar 30, 200132 citations

Journal of Physical Studies

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

Aleksandr Ivanovich Sokolov, E. V. Orlov

Published Jan 1, 1999• 0 citations

Publisher

Abstract

DOI: 10.30970/jps.03.280

Related papers

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.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/9804008Physical review. B, Condensed matterAug 1, 199821 citations

Universal sextic effective interaction at criticality

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

Jan 1, 199815 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.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/9803352Journal of Physical StudiesJan 1, 199712 citations

Pseudo-ε-expansion and the two-dimensional Ising model

A. I. Sokolov

The pseudo-ε-expansions for the coordinate of the fixed point g * , the critical exponents, and the sextic effective coupling constant g 6 are determined for the two-dimensional Ising model on the basis of the five-loop renormalization group series. It is found that the pseudoe-expansions for the coordinate of the fixed point g * , the inverse exponent γ −1 , and the constant g 6 possess a remarkable property, namely, the higher terms of these series are so small that reliable numerical results can be obtained without invoking Borel summation.The pseudo-ε-expansions for the coordinate of the fixed point g * , the critical exponents, and the sextic effective coupling constant g 6 are determined for the two-dimensional Ising model on the basis of the five-loop renormalization group series. It is found that the pseudoe-expansions for the coordinate of the fixed point g * , the inverse exponent γ −1 , and the constant g 6 possess a remarkable property, namely, the higher terms of these series are so small that reliable numerical results can be obtained without invoking Borel summation.

arXiv:cond-mat/0510088Physics of the Solid StateNov 1, 200514 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.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/9703017Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJun 1, 199740 citations

The critical equation of state of the two-dimensional Ising model

Michele Caselle, Martin Hasenbusch B, Andrea Pelissetto C +1 more

We compute the 2 n -point coupling constants in the high-temperature phase of the two-dimensional Ising model by using transfer-matrix techniques. This provides the first few terms of the expansion of the effective potential (Helmholtz free energy) and of the equation of state in terms of the renormalized magnetization. By means of a suitable parametric representation, we determine an analytic extension of these expansions, providing the equation of state in the whole critical region in the t , h plane.We compute the 2 n -point coupling constants in the high-temperature phase of the two-dimensional Ising model by using transfer-matrix techniques. This provides the first few terms of the expansion of the effective potential (Helmholtz free energy) and of the equation of state in terms of the renormalized magnetization. By means of a suitable parametric representation, we determine an analytic extension of these expansions, providing the equation of state in the whole critical region in the t , h plane.

arXiv:cond-mat/0011305Journal of Physics A Mathematical and GeneralMar 30, 200132 citations

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.