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Canadian Journal of Physics

Momentum average approximations for the Holstein polaron

Mona Berciu

Published Apr 1, 2008• 4 citations
Publisher

Abstract

We present a novel approach for obtaining simple yet highly accurate approximations for interacting problems. We use the Holstein polaron as an example and show that the predictions of the so-called Momentum Average approximation are in good agreement with results of numerical simulations over most of the parameter space, and become exact in various asymptotic limits. 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. Furthermore, the accuracy can be improved systematically, at a slightly increased computational cost. PACS Nos.: 71.38.–k, 72.10.Di, 63.20.Kr

DOI
10.1139/p07-160

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We present a novel approach for obtaining simple yet highly accurate approximations for interacting problems. We use the Holstein polaron as an example and show that the predictions of the so-called Momentum Average approximation are in good agreement with results of numerical simulations over most of the parameter space, and become exact in various asymptotic limits. 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. Furthermore, the accuracy can be improved systematically, at a slightly increased computational cost. PACS Nos.: 71.38.–k, 72.10.Di, 63.20.Kr
We present a highly efficient yet accurate analytical approximation for the Green's function of a Holstein polaron. It is obtained by summing all the self-energy diagrams, but with each self-energy 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 resulting Green's function satisfies exactly the first six spectral weight sum rules. All higher sum rules are satisfied with great accuracy, becoming asymptotically exact for coupling both much larger and much smaller than the free particle bandwidth. Comparison with existing numerical data also confirms this accuracy. We use this approximation to analyze in detail the redistribution of the spectral weight as the coupling strength varies.
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 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.
We present a new, highly efficient yet accurate approximation for the Green's functions of dressed particles, using the Holstein polaron as an example. Instead of summing a subclass of self-energy diagrams (e.g., the noncrossed ones, in the self-consistent Born approximation), we sum all the diagrams, but with each diagram averaged over its free propagators' momenta. The resulting Green's function satisfies exactly the first six spectral weight sum rules. All higher sum rules are satisfied with great accuracy, becoming asymptotically exact for coupling both much larger and much smaller than the free particle bandwidth. Possible generalizations to other models are also discussed.
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.
It is generally accepted that the dynamical mean field theory gives a good solution of the Holstein model, but only in dimensions greater than two. Here, we show that this theory, which becomes exact in the weak coupling and in the atomic limit, provides an excellent, numerically cheap, approximate solution for the spectral function of the Holstein model in the whole range of parameters, even in one dimension. To establish this, we make a detailed comparison with the spectral functions that we obtain using the newly developed momentum-space numerically exact hierarchical equations of motion method, which yields electronic correlation functions directly in real time. We crosscheck these conclusions with our path integral quantum Monte Carlo and exact diagonalization results, as well as with the available numerically exact results from the literature.