Critical exponents and the~pseudo-$\varepsilon$-expansion

M. A. Nikitina ; A. I. Sokolov

Teoretičeskaâ i matematičeskaâ fizika, Tome 186 (2016) no. 2, pp. 230-242

Résumé

We present the pseudo-$\varepsilon$-expansions ($\tau$-series) for the critical exponents of a $\lambda\phi^4$-type three-dimensional $O(n)$-symmetric model obtained on the basis of six-loop renormalization-group expansions. We present numerical results in the physically interesting cases $n=1$, $n=2$, $n=3$, and $n=0$ and also for $4\le n\le32$ to clarify the general properties of the obtained series. The pseudo-$\varepsilon$-expansions or the exponents $\gamma$ and $\alpha$ have coefficients that are small in absolute value and decrease rapidly, and direct summation of the $\tau$-series therefore yields quite acceptable numerical estimates, while applying the Padé approximants allows obtaining high-precision results. In contrast, the coefficients of the pseudo-$\varepsilon$-expansion of the scaling correction exponent $\omega$ do not exhibit any tendency to decrease at physical values of $n$. But the corresponding series are sign-alternating, and to obtain reliable numerical estimates, it also suffices to use simple Padé approximants in this case. The pseudo-$\varepsilon$-expansion technique can therefore be regarded as a distinctive resummation method converting divergent renormalization-group series into expansions that are computationally convenient.

Détail

Keywords: three-dimensional $O(n)$-symmetric model, critical exponent, pseudo-$\varepsilon$-expansion, numerical result.
Mots-clés : Padé approximant