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Progress of Theoretical Physics

1/ n Expansion for Weakly Random System with Impurity Correlation

Kentaro Yamamoto, R. Abe

Published Nov 1, 1981• 1 citation
Publisher

Abstract

Critical behavior of weakly random system with a specific impurity correlation is studied by means of 1/ n expansion. The correlation function and the equation of state are derived up to order a2 (a being a measure of randomness) in the limit n →∞. Critical exponents η, β and γ are calculated up to this order. It is shown that these exponents are nonuniversal but satisfy a scaling law relation.

DOI
10.1143/ptp.66.1604

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Critical behavior of weakly random system with a specific impurity correlation is studied by means of 1/ n expansion. The correlation function and the equation of state are derived up to order a2 (a being a measure of randomness) in the limit n →∞. Critical exponents η, β and γ are calculated up to this order. It is shown that these exponents are nonuniversal but satisfy a scaling law relation.
In the scheme of 1/n expansion the effects of weak randomness on critical exponents are investigated. By extending the lowest-order argument of Abe, the higher-order terms of the O(a3) and of the O(1/n) are studied(a being a measure of the randomness). Though each diagram constituting higher-order terms shows the logarithmic dependence, which has the possibility of modifying critical exponents, they are canceled out when summed up. It is concluded that random impurities are irrelevant to critical exponents.
The theoretic renormalization group approach is applied to the study of the critical behavior of non-interacting system with long-range correlated quenched impurities, which has a power-like correlations r-(d-ρ). Totwo-loop order, the asymptotic scaling laws and the critical exponents are studied in the frame of a double (e, ρ)expansion with ρ of order e = 4 - d. In d ＜ 4, it is argued that the initial slip exponent θ = 0 together with the dynamicexponent z ＜ 2 is exact in this kind of random system.
The effects of random impurities on the various thermodynamic functions are studied with the aid of the two-dimensional Ising model. Of special interest are the limiting forms of these functions near the critical point. This problem is approached by a Green's function formulation as developed by Kadanoff. Since experiments only measure the average effects of impurities, we consider the average of the Green's function over all distributions of impurities and calculate it by perturbation theory. From this function we find the location of the new critical temperature and the temperature dependence of the coherence length. Thermodynamic and correlation functions are calculated by means of perturbation expansions and their functional dependence on the coherence length investigated. It is found that the magnetic properties such as the magnetization, susceptibility, and spin-spin correlations retain the same functional dependence on the coherence length. The specific heat is found to have a finite value at the critical point, and the T = Tc form of the energy-energy correlation c function is changed. The thermodynamic functions exhibit new critical indices very close to Tc. These renormalized exponents have been predicted c in literature and are found to satisfy the scaling laws predictions.
The renormalization-group functions governing the critical behavior of the three-dimensional weakly-disordered Ising model are calculated in the five-loop approximation. The random fixed point location and critical exponents for impure Ising systems are estimated by means of the Padé-Borel-Leroy resummation of the renormalization-group expansions derived. The asymptotic critical exponents are found to be γ=1.325 ± 0.003, η=0.025 ± 0.01, ν= 0.671 ± 0.005, α=−0.0125 ± 0.008, β=0.344 ± 0.006, while for the correction-to-scaling exponent, a less accurate estimate ω=0.32 ± 0.06 is obtained.
The theoretic renormalization-group approach is applied to the study of the critical behavior of the -dimensional Ising model with long-range correlated quenched impurities, which has a power-like correlations . The asymptotic scaling law is studied in the framework of the expansion in . In , the dynamic exponent is calculated up to the second order in with . The shape function is obtained in one-loop calculation. When , the logarithmic corrections to the critical behavior are found. The finite size effect on the order parameter relaxation rate is also studied.