The Fröhlich Polaron at Strong Coupling -- Part II: Energy-Momentum Relation and Effective Mass
Morris Brooks, Robert Seiringer
TL;DR
This work analyzes the Fröhlich polaron in three dimensions at strong coupling, deriving a sharp energy-momentum relation by connecting the many-body quantum problem to the Landau–Pekar (Pekar) effective description. The authors prove a parabolic lower bound for the truncated model, identify a Bogoliubov-type quadratic structure, and show that the ground-state energy comprises the Pekar energy, a quantum correction term, and a parity-bounded quadratic term in momentum, yielding $E_\alpha(P)=e^{Pek}-\frac{1}{2\alpha^2}\mathrm{Tr}[1-\sqrt{H^{Pek}}]+\min\{\frac{|P|^2}{2\alpha^4 m}, \alpha^{-2}\}+O_{\alpha\to\infty}(\alpha^{-(2+w)})$. The Landau–Pekar effective mass $m$ emerges as the semi-latus rectum of the parabolic energy-momentum relation in the strong-coupling limit, with $M_{\mathrm{eff}}(\alpha) \lesssim \alpha^4 m$. The construction relies on condensation in the Pekar minimizer, ultraviolet regularization, and careful operator inequalities to control radiative and phonon-number effects, culminating in a robust parabolic approximation below the continuum threshold. This provides a rigorous footing for the Landau–Pekar mass picture and quantifies the quantum corrections to the semi-classical mass in the Fröhlich polaron model.
Abstract
We study the Fröhlich polaron model in $\mathbb{R}^3$, and prove a lower bound on its ground state energy as a function of the total momentum. The bound is asymptotically sharp at large coupling. In combination with a corresponding upper bound proved earlier, it shows that the energy is approximately parabolic below the continuum threshold, and that the polaron's effective mass (defined as the semi-latus rectum of the parabola) is given by the celebrated Landau--Pekar formula. In particular, it diverges as $α^4$ for large coupling constant $α$.