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arXiv:cond-mat/0606530 · Theoretical and Mathematical Physics

Calculations of the dynamical critical exponent using the asymptotic series summation method

A. S. Krinitsyn, V. V. Prudnikov, Pavel Vladimirovich Prudnikov

Published Apr 1, 2006• 34 citations
arXivHTMLPDF
DOI
10.1007/s11232-006-0063-z

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The influence of nonequilibrium initial values of the order parameter on its evolution at a critical point is described using a renormalization group approach of the field theory. The dynamic critical exponent θ of the short time evolution of a system with an n -component order parameter is calculated within a dynamical dissipative model using the method of Σ-expansion in a three-loop approximation. Numerical values of θ for three-dimensional systems are determined using the Padé-Borel method for the summation of asymptotic series.
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.
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.