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

The ground-state energy of a two-dimensional (2D) Fröhlich polaron is calculated to second order in the coupling constant (α) and gives E/ħ${\omega }_s$=-(π/2)α-0.063 97${\alpha }^2$ with ħ${\omega }_s$ the surface optical-phonon energy. In the strong-coupling limit the adiabatic approximation is used and E/ħ${\omega }_s$=-0.4047${\alpha }^2$ is found to leading order in α. 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 $E_{2\mathrm{D}}$(α)=(2/3)$E_{3\mathrm{D}}$(( 3π/4)α), where $E_{2\mathrm{D}}$ ($E_{3\mathrm{D}}$) is the ground-state energy of the 2D (3D) polaron.