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.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.

Physical review. B, Condensed matterMar 15, 1985122 citations

Polaron ground-state energy inddimensions

Gurjav Ganbold, Гарий Владимирович Ефимов

The polaron path integral introduced by Feynman is generalized to the case of an electron moving in d space dimensions. A scheme of systematic calculations of the obtained path integral is developed to estimate the ground-state energy of the polaron in the same way for different values of the electron-phonon coupling constant \ensuremath{\alpha}. The leading-order term in this approach yields an upper bound to the polaron self-energy and improves Feynman's variational estimates for d=2 and d=3. A scaling relation between polaron self-energies for different space dimensions is obtained for this term. The next correction to this estimation is calculated by numerical integration for d=2 and d=3.The polaron path integral introduced by Feynman is generalized to the case of an electron moving in d space dimensions. A scheme of systematic calculations of the obtained path integral is developed to estimate the ground-state energy of the polaron in the same way for different values of the electron-phonon coupling constant \ensuremath{\alpha}. The leading-order term in this approach yields an upper bound to the polaron self-energy and improves Feynman's variational estimates for d=2 and d=3. A scaling relation between polaron self-energies for different space dimensions is obtained for this term. The next correction to this estimation is calculated by numerical integration for d=2 and d=3.

Physical review. B, Condensed matterAug 1, 199411 citations

On the ground state energy of a polaron

K.G. Chakraborty

Chemical Physics LettersAug 1, 19740 citations

Ground-state energy of a polaron inndimensions

François Maria Peeters, Xiaoguang Wu, J. T. Devreese

The Fr\"ohlich Hamiltonian is generalized to the case of an electron moving in n space dimensions. For n=2 and n=3 the familiar Fr\"ohlich Hamiltonian is reobtained. The polaron ground-state energy is calculated up to fourth order in perturbation theory. We found that within the Feynman two-particle polaron model approximation the polaron ground-state energy satisfies the scaling relation ${E}_{n\mathrm{D}(\mathrm{\ensuremath{\alpha}})}$=(n/3) ${\mathrm{E}}_{3\mathrm{D}}$ ({\ensuremath{\Gamma}[ (n-1)/2]3 \ensuremath{\surd}\ensuremath{\pi} /2n\ensuremath{\Gamma}(n/2)}\ensuremath{\alpha}), where ${E}_{n\mathrm{D}}$ is the Feynman polaron ground-state energy for the polaron in n dimensions and ${E}_{3\mathrm{D}}$ the energy in three dimensions.The Fr\"ohlich Hamiltonian is generalized to the case of an electron moving in n space dimensions. For n=2 and n=3 the familiar Fr\"ohlich Hamiltonian is reobtained. The polaron ground-state energy is calculated up to fourth order in perturbation theory. We found that within the Feynman two-particle polaron model approximation the polaron ground-state energy satisfies the scaling relation ${E}_{n\mathrm{D}(\mathrm{\ensuremath{\alpha}})}$=(n/3)${\mathrm{E}}_{3\mathrm{D}}$({\ensuremath{\Gamma}[ (n-1)/2]3 \ensuremath{\surd}\ensuremath{\pi} /2n\ensuremath{\Gamma}(n/2)}\ensuremath{\alpha}), where ${E}_{n\mathrm{D}}$ is the Feynman polaron ground-state energy for the polaron in n dimensions and ${E}_{3\mathrm{D}}$ the energy in three dimensions.

Physical review. B, Condensed matterMar 15, 1986148 citations

Exact ground state properties of polarons in the limit of large dimensions

Sergio Ciuchi, FERDINANDO DE PASQUALE, D. Feinberg

Physica C SuperconductivityDec 1, 19944 citations

{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.

{\mathrm{E}}_{3\mathrm{D}}

({\ensuremath{\Gamma}[ (n-1)/2]3 \ensuremath{\surd}\ensuremath{\pi} /2n\ensuremath{\Gamma}(n/2)}\ensuremath{\alpha}), where

{E}_{n\mathrm{D}}

is the Feynman polaron ground-state energy for the polaron in n dimensions and

{E}_{3\mathrm{D}}

the energy in three dimensions.