Determine whether the Holstein-polaron convergence radius is finite

Determine whether the weak-coupling series of the three-dimensional Holstein polaron has a finite radius of convergence for nonzero hopping t>0, rather than an infinite radius of convergence.

Background

The appendix extends the fourth-order series analysis to the three-dimensional Holstein polaron, parameterized by the adiabaticity ratio ω₀/t and coupling λ. The computed effective-mass coefficients are positive through fourth order for the cases considered, while the ratios suggest a finite-radius singularity for ω₀/t≲2.

Unlike the Fröhlich polaron, the Holstein polaron can have exponential strong-coupling growth, as in the atomic limit. Therefore, positivity of the known coefficients does not by itself force a finite convergence radius: an infinite-radius series remains mathematically possible. The available four coefficients do not settle which behavior occurs when t>0.

References

An infinite radius of convergence with coefficients of one sign is not excluded for the Holstein model, however, because the mass grows exponentially rather than as a power of the coupling at strong coupling; the atomic limit is an explicit example. Whether the radius is finite at $t>0$ is not settled by four coefficients, although the growing ratios for $\omega_0\lesssim2t$ suggest that it is.

— 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 Appendix, Section “Series analysis and convergence properties,” subsection “Sign of the coefficients”