Fisher Exponent from Pseudo-$ε$ Expansion

Abstract

Critical exponent $η$ for three-dimensional systems with $n$-vector order parameter is evaluated in the frame of pseudo-$ε$ expansion approach. Pseudo-$ε$ expansion ($τ$-series) for $η$ found up to $τ^7$ term for $n$ = 0, 1, 2, 3 and within $τ^6$ order for general $n$ is shown to have a structure rather favorable for getting numerical estimates. Use of Padé approximants and direct summation of $τ$-series result in iteration procedures rapidly converging to the asymptotic values that are very close to most reliable numerical estimates of $η$ known today. The origin of this fortune is discussed and shown to lie in general properties of the pseudo-$ε$ expansion machinery interfering with some peculiarities of the renormalization group expansion of $η$.