Absence of ground states at large total momentum for translation-invariant polarons

Prove the absence of ground states at large total momentum P for translation-invariant polaron models, including the Fröhlich polaron and the Bose polaron, without ultraviolet regularization; currently the absence at large P is only established for the ultraviolet-regularized Fröhlich polaron.

Background

The authors highlight broader questions about spectral phase transitions in non-relativistic quantum field models beyond the spin–boson case. For translation-invariant polarons at fixed total momentum P, ground states are known to exist at small P, but the large-P regime lacks a general absence result.

They note that only the ultraviolet-regularized Fröhlich polaron currently admits a proof of absence at large momentum. Their Ising-duality methods may help, motivating the explicit formulation of this open problem.

References

Especially, when one turns to translation-invariant polaron models such as the Fr\"ohlich polaron or the Bose polaron at fixed total momentum $P$, then the existence of ground states at small total momentum is well-known and the absence of ground states at large total momentum is at least conjectured, but except for the ultraviolet regularized Fr\"ohlich polaron not proven.

— On the Ising Phase Transition in the Infrared-Divergent Spin Boson Model  (2501.19362 - Betz et al., 31 Jan 2025) in Introduction (Outlook to future research)