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arXiv:0810.1498 · Physical Review B

Momentum average approximation for models with electron-phonon coupling dependent on the phonon momentum

Glen L. Goodvin, Mona Berciu

Published Dec 18, 2008• 38 citations
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Abstract

We generalize the momentum average (MA) approximation to study the properties of models with ${g}_{\mathbf{q}}$ momentum-dependent electron-phonon coupling. As in the case of the application of the original MA to the Holstein model, the results are analytical, numerically trivial to evaluate, exact for both zero bandwidth and for zero electron-phonon coupling, and accurate everywhere in parameter space. Comparison with available numerical data confirms this accuracy. We then show that further improvements can be obtained based on variational considerations using the one-dimensional 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.

DOI
10.1103/physrevb.78.235120

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We generalize the momentum average (MA) approximation to study the properties of models with gq{g}_{\mathbf{q}}gq​ momentum-dependent electron-phonon coupling. As in the case of the application of the original MA to the Holstein model, the results are analytical, numerically trivial to evaluate, exact for both zero bandwidth and for zero electron-phonon coupling, and accurate everywhere in parameter space. Comparison with available numerical data confirms this accuracy. We then show that further improvements can be obtained based on variational considerations using the one-dimensional 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.
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.
We investigate the effects of coupling to multiple-phonon modes on the properties of a Holstein polaron. To this end, we generalize the Momentum Average approximations MA (0) and MA (1) to deal with multiple-phonon modes. As for a single-phonon mode, these approximations are found to be numerically very efficient. They become exact for very weak or very strong couplings, and are highly accurate in the intermediate regimes, e.g. the spectral weights obey exactly the first six, respectively eight, sum rules. Our results show that the effect on ground-state properties is cumulative in nature. As a result, if the effective coupling to one mode is much larger than to all the others, this mode effectively determines the ground-state properties. However, even very weak coupling to a second phonon mode has important non-perturbational effects on the higher-energy spectrum, in particular on the dispersion and the phonon statistics of the polaron band. This has important consequences on the analysis and interpretation of data for real materials.
We show how to systematically improve the momentum average (MA) approximation for the Green's function of a Holstein polaron 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 MA approximation, e.g., we now find the expected polaron+phonon\text{polaron}+\text{phonon}polaron+phonon continuum at the correct location, and a momentum-dependent self-energy. The quantitative agreement with numerical data is further improved as expected since the number of exactly satisfied spectral weight sum rules is increased. The corrections are found to be larger in lower dimensional systems.
A concise variational approach is to propose to calculate the ground-state properties of none-dimensional Holstein model. The results in the weak and strong coupling limit can be easily recovered analytically. It is shown that, in the nontrivial intermediate electron-phonon coupling regime, the present results are in good agreement with those by density-matrix renormalization group and numerical exact diagonalizations. The present approach is more concise than any other analytical ones in this field, and can be easily generalized to Holstein models in higher dimensions and with more electrons.
We demonstrate that the ground-state energy band of an extended one-dimensional Holstein model, featuring a specific type of nonlinear electron-phonon coupling for a single electron in the band, is exactly solvable. Numerical results indicate that, despite the significant differences in electron-phonon coupling between the extended Holstein model and the original Holstein model, the ground-state energy of the Holstein model can be accurately reproduced by the extended model through appropriate rescaling of the coupling parameter. Additionally, for an NNN-site lattice, there are ${2}^{N\ensuremath{-}1}$ distinct choices for the set of parameters in the nonlinear electron-phonon coupling, all yielding the same ground-state energy but with varying ground-state properties. This flexibility allows for the selection of the optimal parameter set in the extended model to best simulate the ground-state properties of the system.
A variational approach is developed to study the groundstate (GS) of the two-site Holstein model. By the extended coherent state, where the more phonon correlations are easily incorporated, we can get the very accurate ground state energy for all electron-phonon coupling range in typical values of hopping integral t = 0.5,1.1, and 2.1 (in units of phonon frequency ω 0 ), which covers the crossover region from antiadiabatic limit to the adiabatic limit. Within a very wide t range [0,2.7], the exact results for the GS energy are obtained with the twelfth (fourteenth) order corrections to the zeroth order wave function. Moreover, the present approach is more concise than any other analytical ones in this field, and hopefully can be easily generalized to many other Holstein models.