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Physical review. B, Condensed matter

Comment on ‘‘Monte Carlo calculation of the ground-state energy of an optical polaron’’

François Maria Peeters, J. T. Devreese

Published May 15, 1985• 6 citations
Publisher

Abstract

Becker, Gerlach, and Schliffke [Phys. Rev. B 28, 5735 (1983)] calculated the free energy of a three-dimensional optical polaron at nonzero temperature (i.e., \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=25) by the Monte Carlo method. In the present Comment (1) we show that the free energy of the Feynman polaron for \ensuremath{\alpha}\ensuremath{\lesssim}2 and \ensuremath{\alpha}>4 is lower than the corresponding Monte Carlo results of Becker, Gerlach, and Schliffke (but lies within the 1% error of the Monte Carlo results) and (2) we study (numerically) the polaron free energy as function of the temperature within the Feynman variational approach.

DOI
10.1103/physrevb.31.6826

Related papers

Reply to ‘‘Comment on ‘Monte Carlo calculation of the ground-state energy of an optical polaron.’ ’’

W. Becker, B. Gerlach, H. Schliffke

In the preceding Comment Peeters and Devreese criticize our recent work [Phys. Rev. B 28, 5735 (1983)]. In particular, they show via a variational ansatz that the free energy of the optical polaron may be lower than our Monte Carlo results (thier improvement being at most 0.8%). Moreover, they find it ``incorrect to compare the Monte Carlo results for \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=25 [finite temperature] with the ground-state energy (i.e., \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=\ensuremath{\infty}) of other theories . . . .'' Our reply is as follows: In our original paper, we specified two different possibilities for numerical errors. Firstly, the application of the ``law of large numbers'' causes a statistical uncertainty, which we estimated to be on a 1% level; secondly, we induce a systematical error due to integral approximations and the fact that we evaluated ${F}_{\ensuremath{\beta}}$ for \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=25 instead of \ensuremath{\infty}---this error is also on a 1% level. It seems that Peeters and Devreese do not realize the significance of these error estimates---e.g., their variational results are fully consistent with our results within the quoted 1% error. In fact, the main purpose of our paper was not to improve the high-standard accuracy of variational calculations, but to test the applicability of the Monte Carlo method to such a formidable problem as that of the Fr\"ohlich polaron.

Physical review. B, Condensed matter•May 15, 1985•3 citations

Monte Carlo calculation of the ground-state energy of an optical polaron

W. Becker, B. Gerlach, H. Schliffke

We consider a (band) electron---LO-phonon system described by Fr\"ohlich's Hamiltonian. Making use of functional-integration methods, one can eliminate the phonon terms exactly leading to an effective one-particle system with the same spectral properties as the original one. The (formal) free energy of this effective system is calculated numerically by means of the Monte Carlo technique. We give a detailed description and first-error estimates of our numerical procedure. Furthermore, we draw a comparison with previous work. Our results compare favorably with all earlier results of variational type of which we are aware.

Physical review. B, Condensed matter•Nov 15, 1983•28 citations

Free Energy of the Fröhlich Polaron in Two and Three Dimensions

John Tatini Titantah, Carlo Pierleoni, Sergio Ciuchi

We present a novel path integral Monte Carlo scheme to solve the Fröhlich polaron model. At intermediate and strong electron-phonon couplings, the polaron self-trapping is properly taken into account at the level of an effective action obtained by a preaveraging procedure with a retarded trial action. We compute the free energy at several couplings and temperatures in three and two dimensions. Our results show that the accuracy of the Feynman variational upper bound for the free energy is always within a few percent. The method can be generalized to the N-polarons case.

arXiv:cond-mat/0010386•Physical Review Letters•Oct 30, 2001•26 citations

Functional-integral approach to the polaron mass

B. Gerlach, Hartmut Löwen, H. Schliffke

Making use of functional-integration techniques, we calculate the formal free energy of a polaron in the subspace of fixed total momentum Q. Utilizing this result, we prove an analytical expression for the polaron mass. We include numerical results of variational and Monte Carlo types. As a by-product, we comment on the position of the continuum edge of the polaronic energy spectrum for fixed Q.

Physical review. B, Condensed matter•Oct 15, 1987•20 citations

Exact and approximate results for the ground-state energy of a Fröhlich polaron in two dimensions

Xiaoguang Wu, François Maria Peeters, J. T. Devreese

The ground-state energy of a two-dimensional (2D) Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and gives E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-(\ensuremath{\pi}/2)\ensuremath{\alpha}-0.063 97${\ensuremath{\alpha}}^{2}$ with \ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$ the surface optical-phonon energy. In the strong-coupling limit the adiabatic approximation is used and E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-0.4047${\ensuremath{\alpha}}^{2}$ is found to leading order in \ensuremath{\alpha}. The Feynman path-integral approximation, the Gaussian approximation, and the modified Lee-Low-Pines unitary transformation approximation to the polaron ground-state energy all satisfy the scaling relation ${E}_{2\mathrm{D}}$(\ensuremath{\alpha})=(2/3)${E}_{3\mathrm{D}}$(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where ${E}_{2\mathrm{D}}$ (${E}_{3\mathrm{D}}$) is the ground-state energy of the 2D (3D) polaron.

Physical review. B, Condensed matter•Mar 15, 1985•122 citations

Polaron Problem by Diagrammatic Quantum Monte Carlo

Nikolay V. Prokof'ev, Boris V. Svistunov

We present a numerical solution of the polaron problem by a novel Monte Carlo method. Starting from a conventional diagrammatic expansion for the polaron Green function $G(\mathbf{k},\ensuremath{\tau})$, we construct a process which generates continuous random variables $\mathbf{k}$ and $\ensuremath{\tau}$, with a distribution function coinciding exactly with $G(\mathbf{k},\ensuremath{\tau})$. The polaron spectrum is extracted from the asymptotic behavior of the Green function. We compare our results for the polaron energy with the variational treatment of Feynman, and present an accurate dispersion curve which features an end point at finite momentum.

arXiv:cond-mat/9804097•Physical Review Letters•Sep 21, 1998•331 citations
Becker, Gerlach, and Schliffke [Phys. Rev. B 28, 5735 (1983)] calculated the free energy of a three-dimensional optical polaron at nonzero temperature (i.e., \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=25) by the Monte Carlo method. In the present Comment (1) we show that the free energy of the Feynman polaron for \ensuremath{\alpha}\ensuremath{\lesssim}2 and \ensuremath{\alpha}>4 is lower than the corresponding Monte Carlo results of Becker, Gerlach, and Schliffke (but lies within the 1% error of the Monte Carlo results) and (2) we study (numerically) the polaron free energy as function of the temperature within the Feynman variational approach.
In the preceding Comment Peeters and Devreese criticize our recent work [Phys. Rev. B 28, 5735 (1983)]. In particular, they show via a variational ansatz that the free energy of the optical polaron may be lower than our Monte Carlo results (thier improvement being at most 0.8%). Moreover, they find it ``incorrect to compare the Monte Carlo results for \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=25 [finite temperature] with the ground-state energy (i.e., \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=\ensuremath{\infty}) of other theories . . . .'' Our reply is as follows: In our original paper, we specified two different possibilities for numerical errors. Firstly, the application of the ``law of large numbers'' causes a statistical uncertainty, which we estimated to be on a 1% level; secondly, we induce a systematical error due to integral approximations and the fact that we evaluated ${F}_{\ensuremath{\beta}}$ for \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=25 instead of \ensuremath{\infty}---this error is also on a 1% level. It seems that Peeters and Devreese do not realize the significance of these error estimates---e.g., their variational results are fully consistent with our results within the quoted 1% error. In fact, the main purpose of our paper was not to improve the high-standard accuracy of variational calculations, but to test the applicability of the Monte Carlo method to such a formidable problem as that of the Fr\"ohlich polaron.
We consider a (band) electron---LO-phonon system described by Fr\"ohlich's Hamiltonian. Making use of functional-integration methods, one can eliminate the phonon terms exactly leading to an effective one-particle system with the same spectral properties as the original one. The (formal) free energy of this effective system is calculated numerically by means of the Monte Carlo technique. We give a detailed description and first-error estimates of our numerical procedure. Furthermore, we draw a comparison with previous work. Our results compare favorably with all earlier results of variational type of which we are aware.
We present a novel path integral Monte Carlo scheme to solve the Fröhlich polaron model. At intermediate and strong electron-phonon couplings, the polaron self-trapping is properly taken into account at the level of an effective action obtained by a preaveraging procedure with a retarded trial action. We compute the free energy at several couplings and temperatures in three and two dimensions. Our results show that the accuracy of the Feynman variational upper bound for the free energy is always within a few percent. The method can be generalized to the N-polarons case.
The ground-state energy of a two-dimensional (2D) Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and gives E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-(\ensuremath{\pi}/2)\ensuremath{\alpha}-0.063 97${\ensuremath{\alpha}}^{2}$ with \ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$ the surface optical-phonon energy. In the strong-coupling limit the adiabatic approximation is used and E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-0.4047${\ensuremath{\alpha}}^{2}$ is found to leading order in \ensuremath{\alpha}. The Feynman path-integral approximation, the Gaussian approximation, and the modified Lee-Low-Pines unitary transformation approximation to the polaron ground-state energy all satisfy the scaling relation E2D{E}_{2\mathrm{D}}E2D​(\ensuremath{\alpha})=(2/3)E3D{E}_{3\mathrm{D}}E3D​(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where E2D{E}_{2\mathrm{D}}E2D​ (E3D{E}_{3\mathrm{D}}E3D​) is the ground-state energy of the 2D (3D) polaron.
We present a numerical solution of the polaron problem by a novel Monte Carlo method. Starting from a conventional diagrammatic expansion for the polaron Green function $G(\mathbf{k},\ensuremath{\tau})$, we construct a process which generates continuous random variables k\mathbf{k}k and $\ensuremath{\tau}$, with a distribution function coinciding exactly with $G(\mathbf{k},\ensuremath{\tau})$. The polaron spectrum is extracted from the asymptotic behavior of the Green function. We compare our results for the polaron energy with the variational treatment of Feynman, and present an accurate dispersion curve which features an end point at finite momentum.