arXiv:cond-mat/0510088 · Physics of the Solid State

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

A. I. Sokolov

Published Nov 1, 2005• 14 citations

Abstract

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.

DOI: 10.1134/1.2131160

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arXiv:cond-mat/0510088 · Physics of the Solid State

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

A. I. Sokolov

Published Nov 1, 2005• 14 citations

arXiv HTML PDF

Abstract

DOI: 10.1134/1.2131160

Related papers

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

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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.