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Journal of the Physical Society of Japan

Strong-Coupling Limit of the Polaron Ground State

Satoru J. Miyake

Published Jan 15, 1975• 105 citations
Publisher

Abstract

In the strong-coupling limit, the polaron ground state energy has been known to be proportional to α 2 , α being the electron-phonon coupling constant. The coefficient of proportionality is found by numerical calculation to be 0.108513 in an appropriate unit. Pekar's variational function is found to give an excellent approximate solution; the energy differs by less than 0.01%, and the error in wave function is less than 1% where the value of the wave function is appreciable.

DOI
10.1143/jpsj.38.181

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In the strong-coupling limit, the polaron ground state energy has been known to be proportional to α 2 , α being the electron-phonon coupling constant. The coefficient of proportionality is found by numerical calculation to be 0.108513 in an appropriate unit. Pekar's variational function is found to give an excellent approximate solution; the energy differs by less than 0.01%, and the error in wave function is less than 1% where the value of the wave function is appreciable.
The next lowest order term of the ground state energy has its origin in lowering of frequencies of phonons localized around the polaron. An eigenvalue equation is derived, the solution of which gives phonon eigenmodes and frequencies in the presence of the polaron. Several eigenvalues are obtained by numerical integration. From the results, the ground state energy is estimated as E0/ℏω0=−0.108513α2−2.836+O(1/α2)E_{0}/\hbar\omega_{0}{=}-0.108513\alpha^{2}-2.836+O(1/\alpha^{2})E0​/ℏω0​=−0.108513α2−2.836+O(1/α2), where ω 0 is the frequency of the optical phonon and α is the electron-phonon coupling constant. The numerical value of the effective mass is 0.0227019α 4 .
The ground-state energy of the optical polaron is calculated by the generalized path-integral formulation. The ground-state approximation is made to simplify the complicated expression, and the Ritz variational and direct integration methods are used to obtain the ground-state energy in the whole range of coupling strength. The results agree with previous work, and it is found that there is a transition for coupling constant ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}9.2$. The ground-state energy obtained by harmonic approximation, which is equivalert to taking Gaussian-like trial wave functions, is compared with those obtained by Pekar's trial wave function.
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.
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.
A novel variational approach is presented for the calculation of the ground-state energy of the polaron in arbitrary N dimensions in the strong-coupling limit. By using the phonon coherent state to represent the wavefunction of phonons, a self-consistent integro-differential equation for the electron wavefunction is derived. The calculated results of the ground-state energy for N = 1, 2 and 3 agree well with the best results in the literature. It is also found that, for arbitrary N, the present results are less than the Feynman path integral ones by small percentages. It is proposed that this approach should be universal for systems involving polarons in the strong-coupling regime.
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.