Highly Accurate Critical Exponents from Self-Similar Variational Perturbation Theory

Abstract

We extend field theoretic variational perturbation theory by self-similar approximation theory, which greatly accelerates convergence. This is illustrated by re-calculating the critical exponents of O(N)-symmetric $\vp^4$ theory. From only three-loop perturbation expansions in $4-\epsilon$ dimensions we obtain {\em analytic results for the exponents, with practically the same accuracy as those derived recently from ordinary field-theoretic variational perturbational theory to seventh order. In particular, the theory explains the best-measured exponent $\al\approx-0.0127$ of the specific heat peak in superfluid helium, found in a satellite experiment with a temperature resolution of nanoKelvin. In addition, our analytic expressions reproduce also the exactly known large-N behaviour of the exponents $\nu$ and $\gamma =\nu (2-\eta )$ with high precision.

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• Calculation of critical exponents by self-similar factor approximants
• Continued functions and perturbation series: Simple tools for convergence of diverging series in $O(n)$-symmetric ${\varphi }^4$ field theory at weak coupling limit
• Reliability of the optimized perturbation theory in the 0-dimensional $O(N)$ scalar field model
• Wilson-Polchinski exact renormalization group equation for O(N) systems: Leading and next-to-leading orders in the derivative expansion
• Interplay Between Approximation Theory and Renormalization Group
• Self-similar extrapolation in quantum field theory
• Asymptotically Free Theory with Scale Invariant Thermodynamics
• Numerical diagonalization analysis of the criticality of the (2+1)-dimensional XY model: Off-diagonal Novotny's method
• Addressing energy density functionals in the language of path-integrals I: Comparative study of diagrammatic techniques applied to the (0+0)-D $O(N)$-symmetric ${\phi }^4$-theory
• Extrapolation of perturbation-theory expansions by self-similar approximants
• From Asymptotic Series to Self-Similar Approximants