Physical review. B, Solid state

Intermediate Coupling Theory: Padé Approximants for Polarons

Ping Sheng, John D. Dow

Published Aug 15, 1971• 39 citations

Abstract

A general method is presented for obtaining an accurate intermediate-coupling theory from weak- and strong-coupling perturbation theory. The method uses two-point Pad\'e approximants to extrapolate (low-order) expansions about the weak- and the strong-coupling limits into the intermediate-coupling regime. The method is used to evaluate the ground-state energy and effective mass of the polaron with gratifying success. In addition, the weak-coupling perturbation theory of the polaron dispersion relation, ground-state energy, and effective mass are extended to fourth, sixth, and fourth order, respectively. A scheme based on two-point Pad\'e approximants is used to obtain an optimal polaron dispersion relation.

DOI: 10.1103/physrevb.4.1343

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Direct path-integral treatment of the polaron problem

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Physical review. B, Solid state

Intermediate Coupling Theory: Padé Approximants for Polarons

Ping Sheng, John D. Dow

Published Aug 15, 1971• 39 citations

Publisher

Abstract

A general method is presented for obtaining an accurate intermediate-coupling theory from weak- and strong-coupling perturbation theory. The method uses two-point Pad'e approximants to extrapolate (low-order) expansions about the weak- and the strong-coupling limits into the intermediate-coupling regime. The method is used to evaluate the ground-state energy and effective mass of the polaron with gratifying success. In addition, the weak-coupling perturbation theory of the polaron dispersion relation, ground-state energy, and effective mass are extended to fourth, sixth, and fourth order, respectively. A scheme based on two-point Pad'e approximants is used to obtain an optimal polaron dispersion relation.

DOI: 10.1103/physrevb.4.1343

Related papers

Direct path-integral treatment of the polaron problem

Arkady L. Kholodenko, Karl F. Freed

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.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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The ground-state characterization of the polaron problem is retrieved within the framework of a variational scheme proposed previously by Devreese et al for the bound polaron. The formulation is based on the standard canonical transformation of the strong coupling ansatz and consists of a variationally determined perturbative extension serving for the theory to interpolate in the overall range of the coupling constant. Specializing our considerations to the bulk and strict two-dimensional polaron models we see that the theory yields significantly improved energy upper bounds in the strong coupling regime and, moreover, extrapolates itself successfully towards the well-established weak coupling limits for all polaron quantities of general interest.The ground-state characterization of the polaron problem is retrieved within the framework of a variational scheme proposed previously by Devreese et al for the bound polaron. The formulation is based on the standard canonical transformation of the strong coupling ansatz and consists of a variationally determined perturbative extension serving for the theory to interpolate in the overall range of the coupling constant. Specializing our considerations to the bulk and strict two-dimensional polaron models we see that the theory yields significantly improved energy upper bounds in the strong coupling regime and, moreover, extrapolates itself successfully towards the well-established weak coupling limits for all polaron quantities of general interest.

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Petr A. Khomyakov

The one-dimensional optical polaron is treated on the basis of perturbation theory in the weak-coupling limit. A special matrix diagrammatic technique is developed. It is shown how to evaluate all terms of the perturbation theory for the ground-state energy of a polaron to any order by means of this technique. The ground-state energy is calculated up to eighth order of perturbation theory. The effective mass of an electron is obtained up to sixth order of perturbation theory. The radius of convergence of the series obtained is estimated.The one-dimensional optical polaron is treated on the basis of perturbation theory in the weak-coupling limit. A special matrix diagrammatic technique is developed. It is shown how to evaluate all terms of the perturbation theory for the ground-state energy of a polaron to any order by means of this technique. The ground-state energy is calculated up to eighth order of perturbation theory. The effective mass of an electron is obtained up to sixth order of perturbation theory. The radius of convergence of the series obtained is estimated.

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Two- and three-dimensional polarons with extended coherent states

Chen Qinghu, Fang Minghu, Zhang Qirui +2 more

The properties of two- and three-dimensional polarons are investigated with the use of extended coherent states. In the weak-coupling limit, the obtained expansions for the ground-state energy and effective mass to the second order in the coupling constant \ensuremath{\alpha} are the same as the previous exact ones. Moreover, in the intermediate-coupling regime, the numerical results for the ground-state energy and effective mass agree with the results by the Feynman path-integral method, and perhaps closer to the exact results than the Feynman ones. It is suggested that the present approach should be an effective one in polaron physics. \textcopyright{} 1996 The American Physical Society.The properties of two- and three-dimensional polarons are investigated with the use of extended coherent states. In the weak-coupling limit, the obtained expansions for the ground-state energy and effective mass to the second order in the coupling constant \ensuremath{\alpha} are the same as the previous exact ones. Moreover, in the intermediate-coupling regime, the numerical results for the ground-state energy and effective mass agree with the results by the Feynman path-integral method, and perhaps closer to the exact results than the Feynman ones. It is suggested that the present approach should be an effective one in polaron physics. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matterMay 1, 199617 citations

Polaron effective mass inddimensions within and beyond the generalized Gaussian approximation

Gurjav Ganbold, Гарий Владимирович Ефимов

The effective mass of the Fröhlich polaron model in d spatial dimensions is obtained within the Gaussian-equivalent representation, that is, within a suitably rearranged perturbative path-integral approach, whose leading order already takes care of all Gaussian fluctuations. In doing so, the authors extend their earlier investigations ( Phys. Rev. B 50 3733 (1994)) of the polaron ground-state energy. Analytical expressions for the effective mass are derived in the weak- and strong-coupling limit. In the intermediate-coupling regime, the effective mass is calculated numerically for the cases of two and three dimensions. The new results are compared with previously known data.The effective mass of the Fröhlich polaron model in d spatial dimensions is obtained within the Gaussian-equivalent representation, that is, within a suitably rearranged perturbative path-integral approach, whose leading order already takes care of all Gaussian fluctuations. In doing so, the authors extend their earlier investigations ( Phys. Rev. B 50 3733 (1994)) of the polaron ground-state energy. Analytical expressions for the effective mass are derived in the weak- and strong-coupling limit. In the intermediate-coupling regime, the effective mass is calculated numerically for the cases of two and three dimensions. The new results are compared with previously known data.

Journal of Physics Condensed MatterJun 8, 19986 citations

Intermediate-Coupling Polaron Effective Mass

David M. Larsen

We have calculated the polaron effective mass ${m}^{*}$ by extending the variational ansatz of a previous paper, where, in the Lee-Low-Pines representation, pair correlations between wave vectors of virtually emitted phonons are taken into account. Our results suggest that (1) effective masses calculated by fourth-order perturbation theory lie below the true values but are quite accurate for $\ensuremath{\alpha}<~3$ and (2) the Feynman theory gives effective masses which are somewhat too high in the intermediate-coupling region. For $\ensuremath{\alpha}\stackrel{\ensuremath{\sim}}{<}4.5$ we propose the estimate $\frac{{m}^{*}}{m}=1+\frac{\ensuremath{\alpha}}{6}+0.02363{\ensuremath{\alpha}}^{2}+0.0014{\ensuremath{\alpha}}^{3}$.We have calculated the polaron effective mass ${m}^{*}$ by extending the variational ansatz of a previous paper, where, in the Lee-Low-Pines representation, pair correlations between wave vectors of virtually emitted phonons are taken into account. Our results suggest that (1) effective masses calculated by fourth-order perturbation theory lie below the true values but are quite accurate for $\ensuremath{\alpha}<~3$ and (2) the Feynman theory gives effective masses which are somewhat too high in the intermediate-coupling region. For $\ensuremath{\alpha}\stackrel{\ensuremath{\sim}}{<}4.5$ we propose the estimate $\frac{{m}^{*}}{m}=1+\frac{\ensuremath{\alpha}}{6}+0.02363{\ensuremath{\alpha}}^{2}+0.0014{\ensuremath{\alpha}}^{3}$.

Physical ReviewOct 15, 196834 citations

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.

We have calculated the polaron effective mass

{m}^{*}

by extending the variational ansatz of a previous paper, where, in the Lee-Low-Pines representation, pair correlations between wave vectors of virtually emitted phonons are taken into account. Our results suggest that (1) effective masses calculated by fourth-order perturbation theory lie below the true values but are quite accurate for $\ensuremath{\alpha}<~3$ and (2) the Feynman theory gives effective masses which are somewhat too high in the intermediate-coupling region. For $\ensuremath{\alpha}\stackrel{\ensuremath{\sim}}{<}4.5$ we propose the estimate $\frac{{m}^{*}}{m}=1+\frac{\ensuremath{\alpha}}{6}+0.02363{\ensuremath{\alpha}}^{2}+0.0014{\ensuremath{\alpha}}^{3}$.