© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics A Mathematical and General

Critical exponents of the dilute Ising model from four-loop expansions

I O Mayer

Published Jul 21, 1989• 95 citations

Abstract

Four-loop expansions of the coefficients of the Callan-Symanzik equation for 3D and 2D dilute Ising models are calculated. Fixed-point coordinates and critical exponents are estimated by two methods of summation of double series: a generalisation of the Pade-Borel approximation and the first confluent form of the in algorithm of Wynn. Summation of the double series for the 2D dilute Ising model gives exponents very close to exponents of the pure Ising model, in accordance with the exact solution. The two methods are also applied to summation of single-variable series of pure Ising and polymer models, both in 3D and 2D cases. Both methods are shown to provide close agreement between present estimates and results obtained earlier either exactly in conformal invariant theories (2D) or numerically with high accuracy (3D). An additional test is provided by estimation of critical exponents of the 2D O(n) model for n=-1. The methods tested this way are used to calculate critical exponents of the 3D dilute Ising model. The values obtained are consistent with recent experimental results.

DOI: 10.1088/0305-4470/22/14/028

Related papers

On the crossover to universal criticality in dilute Ising systems

H. K. Janssen, Klaus Oerding, E Sengespeick

Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.

Journal of Physics A Mathematical and General•Nov 7, 1995•57 citations

The dynamical critical exponent $z$ for 2d and 3d Ising models from five-loop $\epsilon$ expansion

L. Ts. Adzhemyan, Daniil A. Evdokimov, Michal Hnatič +4 more

We calculate the dynamical critical exponent $z$ for 2d and 3d Ising universality classes by means of minimally subtracted five-loop $\varepsilon$ expansion obtained for the A model. This breakthrough turns out to be possible through the successful adaptation of the Sector Decomposition technique to the problems of critical dynamics. The obtained fifth perturbative order accompanied by the use of advanced resummation techniques for asymptotic series allows us to find highly accurate numerical estimates of $z$: for two- and three-dimensional cases we obtain $\boldsymbol{2.14(2)}$ and $\boldsymbol{2.0235(8)}$ respectively. The numbers found are in good agreement with recent results obtained using different approaches.

•Nov 8, 2021•0 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.

arXiv:cond-mat/0510088•Physics of the Solid State•Nov 1, 2005•14 citations

Accurate critical exponents from field theory

J. C. Le Guillou, Jean Zinn‐Justin

This is the third article in a sequence in which we reexamine the calculation of the critical exponents of the N-vector model from field theory. In the first article we have applied summation methods to the ε-expansion for which new terms have been calculated (up to order ε5). In the second article we have incorporated in the summation method, in the case of Ising-like systems (N = 1), the knowledge of the exact values of exponents in d = 2 dimensions. Since exact values are now also known in the case of polymers (N = 0), we apply here the same method to this case. Moreover, since for the exponent η for N = 1 the results coming from the ε-expansion and perturbation series at fixed dimension are slightly different, we have recalculated the exponents in the latter case using the variant of the summation method developed for the ε-expansion. The results do not change significantly showing both the reliability and the limits of the method.

Journal de physique•Jan 1, 1989•122 citations

Five-loop critical exponents of the weakly diluted Ising model: 3D approach versus √ε-expansion

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

Based on the recently obtained ve{loop renormalization group functions of the 4 {theory with O(n){symmetric and cubic interactions, we analyse these functions for the weakly diluted Ising model case n = 0 both by means of the p "{expansion and in the xed d = 3 technique.We apply dierent resummation procedures and report numerical values for the critical exponents dependent on the order of approximation.We compare the convergence of the results obtained in the framework of the two schemes.

Journal of Physical Studies•Jan 1, 1998•13 citations

The dynamic critical exponent of dilute and pure Ising systems

Klaus Oerding

A massive dynamic field theory is employed to calculate the critical exponent z of the three-dimensional dilute Ising model to two-loop order without expansion in square root 4-d (with result z approximately=2.191). Applying the same method to the pure Ising model at three-loop order we find z=2.022 for d=3 and z=2.124 for d=2.

Journal of Physics A Mathematical and General•Dec 21, 1995•6 citations

Four-loop expansions of the coefficients of the Callan-Symanzik equation for 3D and 2D dilute Ising models are calculated. Fixed-point coordinates and critical exponents are estimated by two methods of summation of double series: a generalisation of the Pade-Borel approximation and the first confluent form of the in algorithm of Wynn. Summation of the double series for the 2D dilute Ising model gives exponents very close to exponents of the pure Ising model, in accordance with the exact solution. The two methods are also applied to summation of single-variable series of pure Ising and polymer models, both in 3D and 2D cases. Both methods are shown to provide close agreement between present estimates and results obtained earlier either exactly in conformal invariant theories (2D) or numerically with high accuracy (3D). An additional test is provided by estimation of critical exponents of the 2D O(n) model for n=-1. The methods tested this way are used to calculate critical exponents of the 3D dilute Ising model. The values obtained are consistent with recent experimental results.

Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.

The dynamical critical exponent

z

for 2d and 3d Ising models from five-loop

\epsilon

expansion

We calculate the dynamical critical exponent

z

for 2d and 3d Ising universality classes by means of minimally subtracted five-loop

\varepsilon

expansion obtained for the A model. This breakthrough turns out to be possible through the successful adaptation of the Sector Decomposition technique to the problems of critical dynamics. The obtained fifth perturbative order accompanied by the use of advanced resummation techniques for asymptotic series allows us to find highly accurate numerical estimates of

z

: for two- and three-dimensional cases we obtain

\boldsymbol{2.14(2)}

and

\boldsymbol{2.0235(8)}

respectively. The numbers found are in good agreement with recent results obtained using different approaches.

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.

This is the third article in a sequence in which we reexamine the calculation of the critical exponents of the N-vector model from field theory. In the first article we have applied summation methods to the ε-expansion for which new terms have been calculated (up to order ε5). In the second article we have incorporated in the summation method, in the case of Ising-like systems (N = 1), the knowledge of the exact values of exponents in d = 2 dimensions. Since exact values are now also known in the case of polymers (N = 0), we apply here the same method to this case. Moreover, since for the exponent η for N = 1 the results coming from the ε-expansion and perturbation series at fixed dimension are slightly different, we have recalculated the exponents in the latter case using the variant of the summation method developed for the ε-expansion. The results do not change significantly showing both the reliability and the limits of the method.