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Journal of Qufu Normal University

A Comparison of two Methods to Calculate the Ground-state Energy of a Polaron

Wen Liu, Su Xiyu, Meng Fan-hua

Published Jan 1, 2006• 0 citations
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Abstract

Based on LLP canonical transformation,a different polaron ground-state which includes the two-phonon correlation is adopted to calculate the ground-state energy of polaron.The energy created by two-phonon correlation is obtained.The result shows that the ground-state energy obtained including the correlation effect is smaller than that obtained by perturbative theory.

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Based on LLP canonical transformation,a different polaron ground-state which includes the two-phonon correlation is adopted to calculate the ground-state energy of polaron.The energy created by two-phonon correlation is obtained.The result shows that the ground-state energy obtained including the correlation effect is smaller than that obtained by perturbative theory.
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 ground state energy of large polarons is investigated by means of squeezed states. First the authors introduced single-mode squeezed states and used the variational method to calculate this energy. This gives a result that is valid in the weak-coupling regime and only for very small values of the coupling constant. In order to improve the result they have considered two-mode squeezed states in which the correlation between phonons is involved. In this way the ground state energy is found to be lower than the Feynman result in the intermediate-coupling regime for alpha < 3.7.
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.