Phase Transition and Padé Approximants for Fröhlich Polarons

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Abstract
To extrapolate the weak‐ and the strong‐coupling expansions into the intermediate‐coupling regime two‐point nondiagonal Padé approximants are used. It is observed that there exist two alternative versions of approximants, leading to upper and lower bounds whose properties differ from those proven for Stieltjes series. Taking the Feynman polaron as an example, its characteristics are calculated in higher orders of the weak‐ and the strong‐coupling expansions and the corresponding Padé approximants are constructed. This makes possible to tighten the gap between the bounds considerably. Applying the method to the actual Fröhlich polaron the exact result are used of three‐loops calculations of its ground state energy (E3 = −0.806070048 × 10−3). The polaron characteristics seem to lie between smooth functions of the coupling constant — the fact that restricts their hypothetic jumps. This indicates the absence of a phase transition at least in those numerical limits where it is suspected to exist.

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