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

On the ground state energy of a polaron

K.G. Chakraborty

Chemical Physics LettersAug 1, 19740 citations

Weak coupling limit for the ground state energy of the 2D Fermi polaron

David Mitrouskas

We analyze the ground state energy for N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We allow for mass ratios M > 1.225 between the impurity mass and the mass of a fermion and consider arbitrarily large box sizes while keeping the Fermi energy fixed. Our main result shows that the ground state energy in the limit of weak coupling is given by the polaron energy. The polaron energy is an energy estimate based on trial states up to first order in particle-hole expansion, which was proposed by Chevy in the physics literature. For the proof we apply a Birman-Schwinger principle that was recently obtained by Griesemer and Linden. One main new ingredient is a suitable localization of the polaron energy.We analyze the ground state energy for N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We allow for mass ratios M > 1.225 between the impurity mass and the mass of a fermion and consider arbitrarily large box sizes while keeping the Fermi energy fixed. Our main result shows that the ground state energy in the limit of weak coupling is given by the polaron energy. The polaron energy is an energy estimate based on trial states up to first order in particle-hole expansion, which was proposed by Chevy in the physics literature. For the proof we apply a Birman-Schwinger principle that was recently obtained by Griesemer and Linden. One main new ingredient is a suitable localization of the polaron energy.

arXiv:2012.09621Dec 17, 20200 citations

A concise approach to the calculation of the polaron ground-state energy

K. Wang, Qing‐Hu Chen, Shaolong Wan

Physics Letters AFeb 1, 199424 citations

A Two‐Particle Model for the Ground State Energy of the Polaron

G. Bader, Yves Lépine

Abstract The dielectric polaron is treated using the Gross unitary transform and the Matz‐Green's function techniques. A two‐particle Feynman‐like model is introduced to compare these methods with the best Feynman path integral formulation. It is found that the introduction of a second particle in the model problem does not improve on the usual result for a one‐particle model. However, a comparison of these different approaches leads to the conclusion that no vertex contributions are contained in the Feynman approximation. It is also concluded that it is the very special treatment of the kinetic energy of the electron in Feynman's theory that makes his results lower in energy.Abstract The dielectric polaron is treated using the Gross unitary transform and the Matz‐Green's function techniques. A two‐particle Feynman‐like model is introduced to compare these methods with the best Feynman path integral formulation. It is found that the introduction of a second particle in the model problem does not improve on the usual result for a one‐particle model. However, a comparison of these different approaches leads to the conclusion that no vertex contributions are contained in the Feynman approximation. It is also concluded that it is the very special treatment of the kinetic energy of the electron in Feynman's theory that makes his results lower in energy.

physica status solidi (b)Jan 1, 19811 citation

{\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.