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Phase transitions in two dimensions and multiloop renormalization group expansions
A. I. Sokolov Saint Petersburg State University, St. Petersburg, Russia
Abstract:
We discuss using the field theory renormalization group (RG) to study the critical behavior of two-dimensional (2D) models. We write the RG functions of the 2D
$\lambda\phi^4$ Euclidean
$n$-vector theory up to five-loop terms, give numerical estimates obtained from these series by Padé–Borel–Leroy resummation, and compare them with their exact counterparts known for
$n=1,0,-1$. From the RG series, we then derive pseudo-
$\epsilon$-expansions for the Wilson fixed point location
$g^*$, critical exponents, and the universal ratio
$R_6=g_6/g^2$, where
$g_6$ is the effective sextic coupling constant. We show that the obtained expansions are “friendler” than the original RG series: the higher-order coefficients of the pseudo-
$\epsilon$-expansions for
$g^*$,
$R_6$, and
$\gamma^{-1}$ turn out to be considerably smaller than their RG analogues. This allows resumming the pseudo-
$\epsilon$-expansions using simple Padé approximants without the Borel–Leroy transformation. Moreover, we find that the numerical estimates obtained using the pseudo-
$\epsilon$-expansions for
$g^*$ and
$\gamma^{-1}$ are closer to the known exact values than those obtained from the five-loop RG series using the Padé–Borel–Leroy resummation.
Keywords:
renormalization group, two-dimensional Ising model, $n$-vector model,
five-loop expansion, critical exponent, pseudo-$\epsilon$-expansion. Received: 19.12.2012
DOI:
10.4213/tmf8496
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