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Physical review. B, Solid state

Ground-State Theorem for Free Polarons

L. F. Lemmens, J. DE SITTER, J. T. Devreese

Published Sep 15, 1973• 24 citations
Publisher

Abstract

A ground-state theorem for the Fr\"ohlich polaron is derived. We show that the zeroth moment of the current-current correlation function is proportional to the kinetic energy of the polaron. While the electron-phonon interaction does not depend on the electron mass, the coupling constant $\ensuremath{\alpha}$ does; this allows us to relate the kinetic energy with the ground-state energy using the Feynman-Helmann theorem. As an example we compare the polaron ground-state energy on variational grounds with the ground-state energy obtained from the function $\ensuremath{\chi}(\ensuremath{\omega})$ calculated by Feynman, Hellwarth, Iddings, and Platzman. For $\ensuremath{\alpha}<5$ both ground-state energies turn out to be identical, as can be concluded from numerical calculations.

DOI
10.1103/physrevb.8.2717

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

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

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 matter•Mar 15, 1985•122 citations
A ground-state theorem for the Fr"ohlich polaron is derived. We show that the zeroth moment of the current-current correlation function is proportional to the kinetic energy of the polaron. While the electron-phonon interaction does not depend on the electron mass, the coupling constant \ensuremathα\ensuremath{\alpha}\ensuremathα does; this allows us to relate the kinetic energy with the ground-state energy using the Feynman-Helmann theorem. As an example we compare the polaron ground-state energy on variational grounds with the ground-state energy obtained from the function \ensuremathχ(\ensuremathω)\ensuremath{\chi}(\ensuremath{\omega})\ensuremathχ(\ensuremathω) calculated by Feynman, Hellwarth, Iddings, and Platzman. For \ensuremathα<5\ensuremath{\alpha}<5\ensuremathα<5 both ground-state energies turn out to be identical, as can be concluded from numerical calculations.
In the expansion of the ground-state energy of a polaron in a weak magnetic field, the zeroth-order term is the polaron self-energy, while the first-order term is inversely proportional to the polaron effective mass. The effective mass so obtained is exactly equivalent to the freepolaron effective mass as defined by Fr\"ohlich. This equivalence principle is used to approximate the polaron effective mass by employing an approximate expression for the ground-state energy of a polaron in a weak magnetic field obtained by applying Feynman's path-integral variational method. The resultant polaron effective mass is found to be higher than Feynman's result by less than 1%.
Taking the same trial wave function, the ground-state energy of Fröhlich polaron is investigated by variational method and coherent-state expansion (CSE) one, respectively. Within the accuracy to α2α^{2}α2(the electron-phonon coupling constant), both methods can give the same analytic expressions of polaron ground-state energy as the function of electron-phonon coupling constant. We find that the CSE method can much simplify the calculation and shows more advantages in higher order approximations.
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.
Partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Fröhlich Hamiltonian is performed using the Feynman-Pines diagram technique.Renormalized spectral parameters of ground and first excited (phonon repeat) polaron state are accurately calculated for a weak electron-phonon coupling at T = 0 K.It is shown that the stronger electron-phonon interaction shifts the energy of both states into low-energy region of the spectra.The ground state stays stationary and the excited one decays at a bigger coupling constant.
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 E2D{E}_{2\mathrm{D}}E2D​(\ensuremath{\alpha})=(2/3)E3D{E}_{3\mathrm{D}}E3D​(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where E2D{E}_{2\mathrm{D}}E2D​ (E3D{E}_{3\mathrm{D}}E3D​) is the ground-state energy of the 2D (3D) polaron.