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Physical Review

Green's Function Approximation Method. II. The Polaron

David S. Falk

Published Aug 15, 1959• 4 citations
Publisher

Abstract

A method previously derived for the approximate construction of the nucleon Green's function is here applied to that of the polaron. After making an arbitrary translation of the phonon variables, a linear integral equation for the Green's function is derived by means of a symmetrical treatment and a noncorrelation assumption, in complete analogy to the nucleon problem. This equation is solved through the introduction of a spectral representation in the special case of zero total momentum. The lowest energy state of the system is calculated in terms of the arbitrary translations and then minimized with respect to them. Using the simplest nontrivial cutoff procedure to obtain the variational equation and its solution, results are obtained for values of the coupling parameter $\ensuremath{\alpha}<~3$, and are compared with those of Feynman.

DOI
10.1103/physrev.115.1074

Related papers

New Dynamical Theory of the Large Polaron. I. Mathematical Formulation

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We have developed a new equation of motion Green's function theory for the large polaron, that is completely nonperturbative in origin, and therefore gives a unified treatment of the polaron ground state energy for all interaction strengths. The theory also incorporates translational invariance rigorously, a priori. We find the following new results. (a) Previous theories that are valid for all coupling strengths are based on a nonperturbative summation of infinite subsets of entire diagrams (equations of motion), rather than on recoil parts of all diagrams. We can thus recover lowest order results of Feynman for example, via a translationally noninvariant form of the Fock approximation, which resembles that reported in a previous work. (b) The total ground state energy we obtain in the strong coupling limit is E0 = −0.426α2 to lowest order in the coupling constant α, a result substantially below that of previous calculations, E0 = −0.106α2 (using gaussian trial functions in both cases). Our result is also below the lower bound of Lieb and Yamazaki, although we show in a following paper that our result is an upper bound. (c) The origin of the difference of our results compared to that of other strong coupling theories is several fold, and will be presented in detail in subsequent papers. However, the essence of the difference is that the phonons in the field, via their interaction with the electron, assume the role of independent degrees of freedom, in the limit of an infinite number of phonons. We then have a field theoretic problem with an infinite number of degrees of freedom which can exhibit a totally different physical behavior compared to one with a finite number of degrees of freedom. This aspect of the problem has not been given explicit recognition in all other theories, including that of Lieb and Yamazaki.

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Sum rules satisfied by Feynman-approximation solutions to the polaron problem

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Direct path-integral treatment of the polaron problem

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Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that $n\mathrm{th}$-order terms in the polaron coupling constant are obtained from $n\mathrm{th}$-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a $2n\mathrm{th}$-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.

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Study of polaron properties using the momentum average approximation

Glen L. Goodvin

In this work we present a highly efficient and accurate analytical approximation for the Green's function of a polaron: the momentum average (MA) approximation. It is obtained by summing all of the self-energy diagrams, but with each diagram averaged over the momenta of its free propagators. The result becomes exact for both zero bandwidth and for zero electron-phonon coupling, and is accurate everywhere in the parameter space. The approximation is first used to investigate the Holstein model. A detailed analysis of its accuracy is provided through diagrammatics and spectral weight sum rules. It is shown that the resulting Green's function satisfies exactly the first six spectral weight sum rules, and all higher order sum rules are satisfied with great accuracy. Comparison with numerical data also confirms this accuracy. We then show how to improve the MA approximation by systematically improving the accuracy of the self-energy diagrams in such a way that they can still all be summed efficiently. This allows us to fix some of the problems of the zeroth-order MA approximation. The quantitative agreement with numerical data is also improved. Next, we generalize the MA approximation to study the properties of models with momentum-dependent electron-phonon coupling, and then show that further improvements can be obtained based on variational considerations, using the 1D breathing-mode Hamiltonian as a specific example. For example, by using this variational MA, we obtain ground state energies within at most 0.3% error of the numerical data. Finally, we study the effects of a nearby surface on the spectral weight of a Holstein polaron. The broken translational symmetry is accounted for without any additional approximations, and the resulting inhomogeneous MA approximation continues to be accurate for all coupling strengths. We show that the surface changes properties significantly, with bulk values being recovered only very far away from it. We find that the electron-phonon coupling gives rise to an additional surface potential which is responsible for binding surface states even when they are not normally expected. These results demonstrate that interpretation in terms of bulk properties of spectroscopic data sensitive only to a few surface layers is not straightforward.

•Jan 1, 2009•1 citation
A method previously derived for the approximate construction of the nucleon Green's function is here applied to that of the polaron. After making an arbitrary translation of the phonon variables, a linear integral equation for the Green's function is derived by means of a symmetrical treatment and a noncorrelation assumption, in complete analogy to the nucleon problem. This equation is solved through the introduction of a spectral representation in the special case of zero total momentum. The lowest energy state of the system is calculated in terms of the arbitrary translations and then minimized with respect to them. Using the simplest nontrivial cutoff procedure to obtain the variational equation and its solution, results are obtained for values of the coupling parameter \ensuremathα< 3\ensuremath{\alpha}<~3\ensuremathα< 3, and are compared with those of Feynman.
We have developed a new equation of motion Green's function theory for the large polaron, that is completely nonperturbative in origin, and therefore gives a unified treatment of the polaron ground state energy for all interaction strengths. The theory also incorporates translational invariance rigorously, a priori. We find the following new results. (a) Previous theories that are valid for all coupling strengths are based on a nonperturbative summation of infinite subsets of entire diagrams (equations of motion), rather than on recoil parts of all diagrams. We can thus recover lowest order results of Feynman for example, via a translationally noninvariant form of the Fock approximation, which resembles that reported in a previous work. (b) The total ground state energy we obtain in the strong coupling limit is E0 = −0.426α2 to lowest order in the coupling constant α, a result substantially below that of previous calculations, E0 = −0.106α2 (using gaussian trial functions in both cases). Our result is also below the lower bound of Lieb and Yamazaki, although we show in a following paper that our result is an upper bound. (c) The origin of the difference of our results compared to that of other strong coupling theories is several fold, and will be presented in detail in subsequent papers. However, the essence of the difference is that the phonons in the field, via their interaction with the electron, assume the role of independent degrees of freedom, in the limit of an infinite number of phonons. We then have a field theoretic problem with an infinite number of degrees of freedom which can exhibit a totally different physical behavior compared to one with a finite number of degrees of freedom. This aspect of the problem has not been given explicit recognition in all other theories, including that of Lieb and Yamazaki.
A free-energy theorem for polarons is derived as an extension to arbitrary lattice temperature of the ground-state theorem derived by Lemmens, deSitter, and and Devreese for zero lattice temperature. These theorems relate the exact free energy of the polaron to the exact absorption spectrum integrated over frequency. With approximate solutions in general, and with approximate solutions to many-body problems in particular, it is of interest to know how closely such sum rules are satisfied. We prove that the free-energy theorem is valid for any Feynman-approximation solution to the polaron problem. Using this analytic result, we are able to determine how sensitively satisfying the free-energy theorem depends upon the details of the absorption spectrum. Considering examples of two quite different spectra at zero temperature, we conclude that the ground-state theorem is of little value in discriminating between different calculated absorption spectra. A more sensitive alternative is discussed which makes use of the fluctuation-dissipation theorem for comparing the calculated absorption with the fluctuation energy at each frequency. Since this alternative method requires equality between two functions for each value of their argument (frequency), rather than merely between two numbers as with sum rules, the method easily distinguishes between spectra, each of which may satisfy the ground-state rule very closely. This method is closely related to a self-consistent procedure developed previously. The latter procedure removed the ambiguity of which approximate influence functional to use in path-integral treatments of polaronlike problems.
Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that nthn\mathrm{th}nth-order terms in the polaron coupling constant are obtained from nthn\mathrm{th}nth-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a 2nth2n\mathrm{th}2nth-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.
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.
Abstract A general method for studying the properties of free and bound polarons at arbitrary strength of electron‐phonon coupling is given from a unified point of view by first performing a suitable canonical transformation of the electron and phonon field operators and subsequently applying the double‐time Green's function formalism to the transformed polaron‐phonon Hamiltonian. Calculating the one‐polaron Green's function in the lowest approximation and employing the variational principle for the free energy, a pair of coupled differential equations is derived for the effective wave function and a trial function as introduced by the canonical transformation. From these equations, the weakly excited polaron energy levels at low temperatures can be found in the entire range of coupling constants when external fields are present. In this paper, detailed calculations are confined to the optical‐polaron problem in the zero‐field case. Intermediate‐ and strong‐coupling formulas for temperature‐dependent polaron self‐energy and effective mass are given. The strong‐coupling regime appears to increase with temperature.
In this work we present a highly efficient and accurate analytical approximation for the Green's function of a polaron: the momentum average (MA) approximation. It is obtained by summing all of the self-energy diagrams, but with each diagram averaged over the momenta of its free propagators. The result becomes exact for both zero bandwidth and for zero electron-phonon coupling, and is accurate everywhere in the parameter space. The approximation is first used to investigate the Holstein model. A detailed analysis of its accuracy is provided through diagrammatics and spectral weight sum rules. It is shown that the resulting Green's function satisfies exactly the first six spectral weight sum rules, and all higher order sum rules are satisfied with great accuracy. Comparison with numerical data also confirms this accuracy. We then show how to improve the MA approximation by systematically improving the accuracy of the self-energy diagrams in such a way that they can still all be summed efficiently. This allows us to fix some of the problems of the zeroth-order MA approximation. The quantitative agreement with numerical data is also improved. Next, we generalize the MA approximation to study the properties of models with momentum-dependent electron-phonon coupling, and then show that further improvements can be obtained based on variational considerations, using the 1D breathing-mode Hamiltonian as a specific example. For example, by using this variational MA, we obtain ground state energies within at most 0.3% error of the numerical data. Finally, we study the effects of a nearby surface on the spectral weight of a Holstein polaron. The broken translational symmetry is accounted for without any additional approximations, and the resulting inhomogeneous MA approximation continues to be accurate for all coupling strengths. We show that the surface changes properties significantly, with bulk values being recovered only very far away from it. We find that the electron-phonon coupling gives rise to an additional surface potential which is responsible for binding surface states even when they are not normally expected. These results demonstrate that interpretation in terms of bulk properties of spectroscopic data sensitive only to a few surface layers is not straightforward.