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arXiv:cond-mat/0312111 · Physical Review Letters

Optical Conductivity of the Fröhlich Polaron

Andrei S. Mishchenko, Naoto Nagaosa, Nikolay V. Prokof'ev, Akira Sakamoto, Boris V. Svistunov

Published Dec 3, 2003• 70 citations
arXivHTMLPDF

Abstract

We present accurate results for optical conductivity of the three dimensional Fröhlich polaron in all coupling regimes. The systematic-error free diagrammatic quantum Monte Carlo method is employed where the Feynman graphs for the momentum-momentum correlation function in imaginary time are summed up. The real-frequency optical conductivity is obtained by the analytic continuation with stochastic optimization. We compare numerical data with available perturbative and nonperturbative approaches to the optical conductivity and show that the picture of sharp resonances due to relaxed excited states in the strong-coupling regime is "washed out" by large broadening of these states. As a result, the spectrum contains only a single-maximum broad peak with peculiar shape and a shoulder.

DOI
10.1103/physrevlett.91.236401

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We present accurate results for optical conductivity of the three dimensional Fröhlich polaron in all coupling regimes. The systematic-error free diagrammatic quantum Monte Carlo method is employed where the Feynman graphs for the momentum-momentum correlation function in imaginary time are summed up. The real-frequency optical conductivity is obtained by the analytic continuation with stochastic optimization. We compare numerical data with available perturbative and nonperturbative approaches to the optical conductivity and show that the picture of sharp resonances due to relaxed excited states in the strong-coupling regime is "washed out" by large broadening of these states. As a result, the spectrum contains only a single-maximum broad peak with peculiar shape and a shoulder.
The polaron optical conductivity at zero temperature is derived within the strong-coupling approximation accounting for internal nonadiabaticity. The polaron optical conductivity spectrum is provided by the multiphonon optical transitions. The polaron optical conductivity spectra calculated within our analytic strong-coupling approach and the numerically accurate diagrammatic quantum Monte Carlo (DQMC) data are in a good agreement with each other at large $\ensuremath{\alpha}\ensuremath{\gtrsim}9$.
The optical absorption of the Fröhlich polaron model is obtained by an approximation-free diagrammatic Monte Carlo method and compared with two new approximate approaches that treat lattice relaxation effects in different ways. We show that: (i) a strong coupling expansion, based on the Franck-Condon principle, well describes the optical conductivity for large coupling strengths (alpha > 10); (ii) a memory function formalism with phonon broadened levels reproduces the optical response for weak coupling strengths (alpha < 6) taking the dynamic lattice relaxation into account. In the coupling regime 6 < alpha < 10, the optical conductivity is a rapidly changing superposition of both Franck-Condon and dynamic contributions.
In the present work, the problem of an all-coupling analytic description for the optical conductivity of the Fr\"ohlich polaron is treated, with the goal being to bridge the gap in the validity range that exists between two complementary methods: on the one hand, the memory-function formalism and, on the other hand, the strong-coupling expansion based on the Franck-Condon picture for the polaron response. At intermediate coupling, both methods were found to fail as they do not reproduce diagrammatic quantum Monte Carlo results. To resolve this, we modify the memory-function formalism with respect to the Feynman-Hellwarth-Iddings-Platzman approach in order to take into account a nonquadratic interaction in a model system for the polaron. The strong-coupling expansion is extended beyond the adiabatic approximation by including in the treatment nonadiabatic transitions between excited polaron states. The polaron optical conductivity that we obtain at T=0T=0T=0 by combining the two extended methods agrees well, both qualitatively and quantitatively, with the diagrammatic quantum Monte Carlo results in the whole available range of the electron-phonon coupling strength.
The momentum average approximation is used to derive a new kind of nonperturbational analytical expression for the optical conductivity (OC) of a Holstein polaron at zero temperature. This provides insight into the shape of the OC, by linking it to the structure of the polaron's phonon cloud. Our method works in any dimension, properly enforces selection rules, can be systematically improved, and also generalizes to momentum-dependent couplings. Its accuracy is demonstrated by a comparison with the first detailed set of three-dimensional numerical OC results, obtained using the approximation-free diagrammatic Monte Carlo method.
A detailed study of the Fr\"ohlich polaron model is performed on the basis of diagrammatic quantum Monte Carlo method [N. V. Prokof'ev and B. V. Svistunov, Phys. Rev. Lett. 81,81,81, 2514 (1998)]. The method is further developed both quantitatively (performance) and qualitatively (new estimators), and is enhanced by spectral analysis of the polaron Green's function, within an approach developed in the present paper. We present up to date results for the binding energy, and make available precise data for the effective mass, including the region of intermediate and strong couplings. We look at the structure of the polaron cloud and answer such questions as the average number of phonons in the cloud and their number/momentum distribution. The spectral analysis reveals nontrivial structure of the spectral density at intermediate and large coupling: the spectral continuum features pronounced peaks that we attribute to unstable excited states of the polaron.
A detailed study of the Frohlich polaron model is performed on the basis of diagrammatic quantum Monte Carlo method. The method is further developed both quantitatively (performance) and qualitatively (new estimators), and is enhanced by spectral analysis of the polaron Green function, within a novel approach. We present the best up to date results for the binding energy, and for the first time make available precise data for the effective mass, including the region of intermediate and strong couplings. We look at the structure of the polaron cloud and answer such questions as the average number of phonons in the cloud and their number/momentum distribution. The spectral analysis reveals non-trivial structure of the spectral density at intermediate and large coupling: the spectral continuum features pronounced peaks that we attribute to unstable excited states of the polaron.