Field Theoretic Calculation of the Universal Amplitude Ratio of Correlation Lengths in 3D-Ising Systems

Abstract

In three-dimensional systems of the Ising universality class the ratio of correlation length amplitudes for the high- and low-temperature phases is a universal quantity. Its field theoretic determination apart from the $ε$-expansion represents a gap in the existing literature. In this article we present a method, which allows to calculate this ratio by renormalized perturbation theory in the phases with unbroken and broken symmetry of a one-component $ϕ^4$-theory in fixed dimensions $D=3$. The results can be expressed as power series in the renormalized coupling constant of either of the two phases, and with the knowledge of their fixed point values numerical estimates are obtainable. These are given for the case of a two-loop calculation.