arXiv:cond-mat/9804008 · Physical review. B, Condensed matter

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

Aleksandr Ivanovich Sokolov, E. V. Orlov

Published Aug 1, 1998• 21 citations

Abstract

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.

DOI: 10.1103/physrevb.58.2395

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arXiv:cond-mat/9804008 · Physical review. B, Condensed matter

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

Aleksandr Ivanovich Sokolov, E. V. Orlov

Published Aug 1, 1998• 21 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.58.2395

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Journal of Physical StudiesJan 1, 19990 citations

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

Jan 1, 199815 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.

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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.

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Critical thermodynamics of the two-dimensional systems in five-loop renormalization-group approximation

E. V. Orlov, Aleksandr Ivanovich Sokolov

The RG functions of the 2D $n$ -vector $ϕ^4$ model are calculated in the five-loop approximation. Perturbative series for the $β$ function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case $n = 1$ as well as in the others ( $n = 0$ , $n = -1$ , $n = 2, 3,...32$ ) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the $β$ function that can not be found perturbatively. The evaluation of the critical exponents for $n = 1$ , $n = 0$ and $n = -1$ in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.The RG functions of the 2D $n$-vector $ϕ^4$ model are calculated in the five-loop approximation. Perturbative series for the $β$ function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case $n = 1$ as well as in the others ($n = 0$, $n = -1$, $n = 2, 3,...32$) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the $β$ function that can not be found perturbatively. The evaluation of the critical exponents for $n = 1$, $n = 0$ and $n = -1$ in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.

arXiv:hep-th/0003140Mar 16, 20001 citation

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 RG functions of the 2D

n

-vector

ϕ^4

model are calculated in the five-loop approximation. Perturbative series for the

β

function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case

n = 1

as well as in the others (

n = 0

n = -1

n = 2, 3,...32

) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the

β

function that can not be found perturbatively. The evaluation of the critical exponents for

n = 1

n = 0

and

n = -1

in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.