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Physical review. B, Condensed matter

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

W. Becker, B. Gerlach, H. Schliffke

Published May 15, 1985• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.31.6829

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

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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.

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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\ensuremathβ{F}_{\ensuremath{\beta}}F\ensuremathβ​ 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.
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.
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 develop and compare several analytical approximations for the ground state and zero-temperature polaron dispersion in finite-width, nonparabolic conduction bands. The main focus of the work is an extension of the Feynman variational method to a tight-binding lattice, where the effective-mass approximation is no longer applicable. The resulting variational formulation is not restricted to a specific phonon dispersion or electron-phonon interaction and provides a uniform description across weak-, intermediate-, and strong-coupling regimes. In addition, we revisit and generalize other analytical approaches traditionally formulated for continuum polarons, including canonical transformations and self-consistent Wigner-Brillouin-type approximations. For lattice polarons, these methods exhibit qualitative features absent in the continuum case, such as a nontrivial connection between weak- and strong-coupling limits. We show that an improved Wigner-Brillouin scheme yields a momentum-dependent polaron self-energy free of resonances and in good agreement with numerically exact results. All methods are applied to the Holstein model on a tight-binding lattice and are benchmarked against numerically exact calculations, including diagrammatic Monte Carlo (both our calculations and preceding works) and exact diagonalization results. Furthermore, the analytical approaches are extended to polarons with Rashba-type spin-orbit coupling, providing a stringent test of their applicability in systems with nontrivial band structure. Our results demonstrate that the modified Feynman variational method yields ground-state energies and dispersions with accuracy comparable to, and in many cases exceeding, that of other established analytical approaches. The developed framework offers a versatile and reliable analytical description of lattice polarons beyond the continuum approximation.
Abstract The still-maturing noisy intermediate-scale quantum technology faces strict limitations on the algorithms that can be implemented efficiently. In the realm of quantum chemistry, the variational quantum eigensolver (VQE) algorithm has become ubiquitous, with many variations. Alternatively, a promising new avenue has been unraveled by the quantum variants of techniques grounded on expansions of the moments of the Hamiltonian, notably the connected moments expansion (CMX) and the Peeters–Devreese–Soldatov (PDS) energy functional. Common to those approaches is that, upon preparing an approximate ground state used to compute the necessary moments, the accuracy of the estimated ground state energy depends on the degree of overlap between the prepared state and the true ground state. Thus, we use the ADAPT-VQE algorithm to test shallow circuit construction strategies for the purpose of increasing the overlap with the exact ground state, validated by the sizable accuracy improvement herein reported in the PDS and CMX ground state energies. We also show that we can take advantage of the fact that the terms to be measured are highly recurring in different moments, incurring a substantial reduction in the number of necessary measurements. By coupling this measurement caching with a threshold that determines whether a given term is to be measured based on its associated scalar coefficient, we observe a further reduction in the number of circuit implementations while allowing for tunable accuracy.