Self-similar variational perturbation theory for critical exponents

We extend field theoretic variational perturbation theory by self-similar approximation theory, which greatly accelerates convergence. This is illustrated by recalculating the critical exponents of $O\left(N\right)$-symmetric ${\varphi }^4$ theory. From only three-loop perturbation expansions in $4-ϵ$ dimensions, we obtain analytic results for the exponents, which are close to those derived recently from ordinary field-theoretic variational perturbational theory to seventh order. In particular, the specific-heat exponent is found to be in good agreement with best-measured exponent $\alpha \approx -0.0127$ of the specific-heat peak in superfluid helium, found in a satellite experiment. In addition, our analytic expressions reproduce also the exactly known large-$N$ behavior of the exponents.

Figure 1

Solid curves show our third-order approximations to $\omega ,\nu ,\gamma ,\eta$. Short-dashed is second-, long-dashed curve is third-order approximation. Thin dots show sixth-order approximation of the textbook [1], fat dots the extrapolations to infinite order. The dash-dotted lines in the second and third figures are interpolations to the Padé-Borel resummations of [27, 28] (where $\omega$ was not calculated). Their data for $\eta$ scatter too much to be represented in this way—they are indicated by small circles in the fourth figure. The dotted curves show $1∕N$ expansions of all four quantities. Note that our results lie closer to these than those of S.A. Antonenko and A.I. Sokolov. The solid $\eta$ curve was calculated in the textbook [1] (see Fig. 20.2). The exact large-$N$ limits are ${\omega }_{N=\infty }=4-D$, ${\nu }_{N=\infty }=1∕(D-2)$, ${\gamma }_{N=\infty }=2∕(D-2)$, and ${\eta }_{N=\infty }=0$. The exact values at $N=-2$ are ${\nu }_{N=-2}=1∕2$, ${\gamma }_{N=-2}=1$, and ${\eta }_{N=-2}=0$ for all $D$.