Construct a constrained weak-to-strong-coupling interpolant

Construct an interpolant for the three-dimensional Fröhlich polaron effective mass that reproduces the weak- and strong-coupling endpoint data, incorporates the nearest-singularity location |α_c|≈8.5, remains analytic on the positive real α axis, and is tested against diagrammatic Monte Carlo data.

Background

The weak-coupling Rayleigh–Schrödinger series for the Fröhlich polaron has a finite radius of convergence, with the nearest complex singularities estimated to have modulus approximately 8.5. Consequently, ordinary Padé approximants and simple two-parameter forms describe the weak- and intermediate-coupling regimes but cannot analytically continue through the singularity pair to the strong-coupling regime.

The paper notes that imposing only the weak-coupling coefficients and the Landau–Pekar strong-coupling asymptote does not determine the crossover: additional terms can preserve both endpoint constraints while changing the crossover arbitrarily. The proposed unresolved task is therefore to develop a more constrained interpolant that also respects the complex-singularity information, analyticity on the positive real axis, and numerical Monte Carlo benchmarks.

References

The results of this paper suggest a more constrained strategy. An interpolant should reproduce not only the endpoint data but also the location $|\alpha_c|\simeq8.5$ of the nearest singularities, should be analytic on the positive real axis, and should be tested against the DMC data of Table~\ref{tab:dmc} rather than assumed. We leave this to future work.

— Fourth-order perturbation theory for the Frohlich polaron: Analytic structure of the weak-coupling series and the crossover to strong coupling  (2610.10030 - Sarma, 7 Oct 2026) in Section 6.2, “Strong coupling, Borel methods, and two-point interpolation”