Exact and approximate results for the ground-state energy of a Fröhlich polaron in two dimensions

Abstract

The ground-state energy of a two-dimensional (2D) Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and gives E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-(\ensuremath{\pi}/2)\ensuremath{\alpha}-0.063 97${\ensuremath{\alpha}}^{2}$ with \ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$ the surface optical-phonon energy. In the strong-coupling limit the adiabatic approximation is used and E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-0.4047${\ensuremath{\alpha}}^{2}$ is found to leading order in \ensuremath{\alpha}. The Feynman path-integral approximation, the Gaussian approximation, and the modified Lee-Low-Pines unitary transformation approximation to the polaron ground-state energy all satisfy the scaling relation ${E}_{2\mathrm{D}}$(\ensuremath{\alpha})=(2/3)${E}_{3\mathrm{D}}$(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where ${E}_{2\mathrm{D}}$ (${E}_{3\mathrm{D}}$) is the ground-state energy of the 2D (3D) polaron.