The critical exponent $\eta$ for three-dimensional systems with an $n$-vector order parameter is evaluated in the framework of the pseudo-$\epsilon$ expansion approach. The pseudo-$\epsilon$ expansion ($\tau$ series) for $\eta$ found up to the ${\tau }^7$ term for $n$ = 0, 1, 2, 3 and within the ${\tau }^6$ order for general $n$ is shown to have a structure that is rather favorable for getting numerical estimates. The use of Padé approximants and direct summation of the $\tau$ series result in iteration procedures rapidly converging to the asymptotic values that are very close to the most reliable numerical estimates of $\eta$ known today. The origin of such an efficiency is discussed and shown to lie in the general properties of the pseudo-$\epsilon$ expansion machinery interfering with some peculiarities of the renormalization group expansion of $\eta$.

• Received 13 April 2014

DOI:https://doi.org/10.1103/PhysRevE.90.012102