Critical Exponents in Two Dimensions and Pseudo-\epsilon\ Expansion

Abstract

The critical behavior of two-dimensional $n$-vector $\lambda\phi^4$ field model is studied within the framework of pseudo-$\epsilon$ expansion approach. Pseudo-$\epsilon$ expansions for Wilson fixed point location $g^*$ and critical exponents originating from five-loop 2D renormalization group series are derived. Numerical estimates obtained within Pad\'e and Pad\'e-Borel resummation procedures as well as by direct summation are presented for $n = 1$, $n = 0$ and $n = -1$, i. e. for the models which are exactly solvable. The pseudo-$\epsilon$ expansions for $g^*$, critical exponents $\gamma$ and $\nu$ have small lower-order coefficients and slow increasing higher-order ones. As a result, direct summation of these series with optimal cut off provides numerical estimates that are no worse than those given by the resummation approaches mentioned. This enables one to consider the pseudo-$\epsilon$ expansion technique itself as some specific resummation method.