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Critical exponents and the pseudo-$\varepsilon$-expansion
M. A. Nikitinaab,
A. I. Sokolova a St. Petersburg State University,
Fock Research Institute of Physics, St. Petersburg, Russia
b St. Petersburg National Research University for Information
Technologies, Mechanics, and Optics, St. Petersburg, Russia
Abstract:
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.
Keywords:
three-dimensional $O(n)$-symmetric model, critical exponent, pseudo-$\varepsilon$-expansion, Padé approximant, numerical result.
PACS:
05.10.Cc,
05.70.Jk,
64.60.ae, 64.60.Fr
Received: 14.05.2015
DOI:
10.4213/tmf8966
Fulltext:
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