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Physical Review

Ground-State Energy and Effective Mass of the Polaron

Élliott H. Lieb, Kazuo Yamazaki

Published Aug 1, 1958• 79 citations
Publisher

Abstract

The polaron Hamiltonian would be easily soluble were it not for the quartic term appearing therein. It is proposed to substitute for the quartic term a quadratic term having roughly the same properties, and in such a way that the ground-state energy of the new Hamiltonian is rigorously a lower bound for the true energy. With a very small amount of work one can obtain a lower bound as a continuous function of $\ensuremath{\alpha}$ for all values of $\ensuremath{\alpha}$. The result agrees fairly well with the results obtained by other methods. Using the equivalent Hamiltonian one can also obtain an analytic expression for the effective mass, although one cannot say it is a bound for the true effective mass. Futhermore, once one has obtained a lower bound for the energy as a continuous function of the parameters of the Hamiltonian, one can rigorously derive upper and lower bounds for the ground-state expectation values of various operators. For example, it can be shown that for large $\ensuremath{\alpha}$ and large k, $〈{{a}_{\mathrm{k}}}^{*}{a}_{\mathrm{k}}〉\ensuremath{\sim}{\mathrm{k}}^{\ensuremath{-}6}$ and not ${\mathbf{k}}^{\ensuremath{-}2}$ exp(-${\mathbf{k}}^{\ensuremath{-}2}$) as in Pekar's solution. Because of its simplicity, it is possible that this method may have application to other ground-state problems.

DOI
10.1103/physrev.111.728

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Using a variational ansatz in which pair correlations between wave vectors of virtually emitted phonons are taken into account, we have obtained better upper bounds to the polaron ground-state energy than have been available heretofore for $\ensuremath{\alpha}\ensuremath{\le}3.5$. The same variational trial functions are also used to obtain lower bounds for $\ensuremath{\alpha}\ensuremath{\le}2.5$. These lower bounds, although not completely rigorous, represent a considerable improvement over the Lieb-Yamazaki values, which are, to the author's knowledge, the only other lower bounds in the literature. The variational ansatz chosen is also suitable for practical calculations of the polaron effective mass.

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The polaron effective mass using the path integral approach was first introduced by Feynman. He has shown that from the partition function, he obtained the ground state energy and the effective mass mF of the polaron. By using the partition function, Krivoglaz and Pekar proposed another definition of the effective mass mKP. Sa-yakanit has shown that using the density matrix Feynmam's path integral representation, it is possible to obtain both Feynman's and Krivoglaz and Pekar's results. In this thesis we propose a consistent definition of effective mass that mF and mKP should be equal. We have shown that the wave functions obtained from the density matrix should be normalized. We have also shown that the dynamic mass of the polaron can be obtained from the density matrix. The comparison of our results with recent work by Wang et.al.are given.

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The last unsolved problem about the many-polaron system, in the Pekar–Tomasevich approximation, is the case of bosons with the electron-electron Coulomb repulsion of strength exactly 1 (the “neutral case”). We prove that the ground state energy, for large N, goes exactly as −N7/5, and we give upper and lower bounds on the asymptotic coefficient that agree to within a factor of 22/5.

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The polaron Hamiltonian would be easily soluble were it not for the quartic term appearing therein. It is proposed to substitute for the quartic term a quadratic term having roughly the same properties, and in such a way that the ground-state energy of the new Hamiltonian is rigorously a lower bound for the true energy. With a very small amount of work one can obtain a lower bound as a continuous function of \ensuremathα\ensuremath{\alpha}\ensuremathα for all values of \ensuremathα\ensuremath{\alpha}\ensuremathα. The result agrees fairly well with the results obtained by other methods. Using the equivalent Hamiltonian one can also obtain an analytic expression for the effective mass, although one cannot say it is a bound for the true effective mass. Futhermore, once one has obtained a lower bound for the energy as a continuous function of the parameters of the Hamiltonian, one can rigorously derive upper and lower bounds for the ground-state expectation values of various operators. For example, it can be shown that for large \ensuremathα\ensuremath{\alpha}\ensuremathα and large k, 〈ak∗ak〉\ensuremath∼k\ensuremath−6〈{{a}_{\mathrm{k}}}^{*}{a}_{\mathrm{k}}〉\ensuremath{\sim}{\mathrm{k}}^{\ensuremath{-}6}〈ak​∗ak​〉\ensuremath∼k\ensuremath−6 and not k\ensuremath−2{\mathbf{k}}^{\ensuremath{-}2}k\ensuremath−2 exp(-k\ensuremath−2{\mathbf{k}}^{\ensuremath{-}2}k\ensuremath−2) as in Pekar's solution. Because of its simplicity, it is possible that this method may have application to other ground-state problems.
Using a variational ansatz in which pair correlations between wave vectors of virtually emitted phonons are taken into account, we have obtained better upper bounds to the polaron ground-state energy than have been available heretofore for $\ensuremath{\alpha}\ensuremath{\le}3.5$. The same variational trial functions are also used to obtain lower bounds for $\ensuremath{\alpha}\ensuremath{\le}2.5$. These lower bounds, although not completely rigorous, represent a considerable improvement over the Lieb-Yamazaki values, which are, to the author's knowledge, the only other lower bounds in the literature. The variational ansatz chosen is also suitable for practical calculations of the polaron effective mass.
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%.
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 polaron effective mass using the path integral approach was first introduced by Feynman. He has shown that from the partition function, he obtained the ground state energy and the effective mass mF of the polaron. By using the partition function, Krivoglaz and Pekar proposed another definition of the effective mass mKP. Sa-yakanit has shown that using the density matrix Feynmam's path integral representation, it is possible to obtain both Feynman's and Krivoglaz and Pekar's results. In this thesis we propose a consistent definition of effective mass that mF and mKP should be equal. We have shown that the wave functions obtained from the density matrix should be normalized. We have also shown that the dynamic mass of the polaron can be obtained from the density matrix. The comparison of our results with recent work by Wang et.al.are given.
The method of moments is employed to determine the ground state of the phonon-polaron system. The trial function employed in this calculation is a bare-electron wave function. It appears that the method is convergent for this problem. Although the energy is in general a complicated function of the parameter and must be determined numerically, it is possible to formally construct a power series expansion for the energy about the unperturbed energy in powers of the coupling constant. The first term in this series has been determined and agrees with the result of conventional perturbation theory. The second term has not yet been evaluated to all orders but it appears that it will have the opposite sign from the corresponding term in perturbation theory. The second and higher terms in some orders of approximation are singular for small values of the cutoff.