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.
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.
The leading correction-to-scaling exponent ω for the three-dimensional (3D) dilute Ising model is calculated in the framework of the field theoretic renormalization group approach. Both in the minimal subtraction scheme and in the massive field theory (resummed four-loop expansion) excellent agreement with recent Monte Carlo calculations (H. G. Ballesteros et al., Phys. Rev. B 58, 2740 (1998)) is achieved. The expression of ω as a series in a ε expansion up to O(ɛ2) does not permit a reliable estimate for d=3.
The dynamical critical exponent z for 2d and 3d Ising models from five-loop ϵ expansion
We calculate the dynamical critical exponent z for 2d and 3d Ising universality classes by means of minimally subtracted five-loop ε 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 2.14(2) and 2.0235(8) respectively. The numbers found are in good agreement with recent results obtained using different approaches.
This paper is a field theoretic description of the critical dynamics of spin systems with frozen nonmagnetic impurities. For three-dimensional systems the dynamical critical exponent is found directly by employing the three-loop approximation with the Padé-Borel summation technique. The results are compared with those obtained by calculating the dynamical exponent for homogeneous systems in the four-loop approximation, and with the values obtained by computer simulation of the critical dynamics by Monte Carlo methods. Calculations of the dynamical exponent for the two-dimensional Ising model in the four-loop approximation are also presented.