Effective and Asymptotic Critical Exponents of Weakly Diluted Quenched Ising Model: 3d Approach Versus $\epsilon^{1/2}$-Expansion

TL;DR

This paper compares the 3d renormalization group approach and the $oldsymbol{ ext{ extit{ extepsilon}}^{1/2}}$-expansion in analyzing the critical behavior of the weakly diluted 3d Ising model, providing numerical estimates for critical exponents.

Contribution

It offers a detailed comparison of the 3d approach and $ ext{ extit{ extepsilon}}^{1/2}$-expansion for the weakly diluted 3d Ising model, including resummed critical exponents and convergence analysis.

Findings

3d approach yields results consistent with Monte Carlo simulations

$ ext{ extit{ extepsilon}}^{1/2}$-expansion unreliable for d=3

Resummed numerical values for effective and asymptotic critical exponents provided

Abstract

We present a field-theoretical treatment of the critical behavior of three-dimensional weakly diluted quenched Ising model. To this end we analyse in a replica limit n=0 5-loop renormalization group functions of the $\phi^4$-theory with O(n)-symmetric and cubic interactions (H.Kleinert and V.Schulte-Frohlinde, Phys.Lett. B342, 284 (1995)). The minimal subtraction scheme allows to develop either the $\epsilon^{1/2}$-expansion series or to proceed in the 3d approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents providing a local measure for the degree of singularity of different physical quantities in the critical region. We report resummed numerical values for the effective and asymptotic critical exponents. Obtained within the 3d approach results agree pretty well with recent Monte Carlo simulations. $\epsilon^{1/2}$-expansion does not allow reliable estimates for d=3.