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arXiv:1111.5253 · Physical Review B

Optical conductivity of polarons: Double phonon cloudconcept verified by diagrammatic Monte Carlo simulations

G. De Filippis, V. Cataudella, Andrei S. Mishchenko, Naoto Nagaosa

Published Mar 9, 2012• 27 citations
arXivHTMLPDF

Abstract

We study theoretically a fundamental issue in solids: the evolution of the optical spectra of polaron as the electron-phonon coupling increases. By comparing the exact results obtained by the diagrammatic Monte Carlo method and the data obtained through exact diagonalization within an appropriate subspace of the phononic wavefunctions, the physical nature of the crossover from weak to strong coupling is revealed. The optical spectra are well understood by the quantum mechanical superposition of states with light and heavy phonon clouds corresponding to large and small polarons, respectively. It is also found that the strong coupling Franck-Condon regime is not accessible at realistic values of the coupling constant.

DOI
10.1103/physrevb.85.094302

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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 $\ensuremath{\alpha}\ensuremath{\gtrsim}9$.

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We study theoretically a fundamental issue in solids: the evolution of the optical spectra of polaron as the electron-phonon coupling increases. By comparing the exact results obtained by the diagrammatic Monte Carlo method and the data obtained through exact diagonalization within an appropriate subspace of the phononic wavefunctions, the physical nature of the crossover from weak to strong coupling is revealed. The optical spectra are well understood by the quantum mechanical superposition of states with light and heavy phonon clouds corresponding to large and small polarons, respectively. It is also found that the strong coupling Franck-Condon regime is not accessible at realistic values of the coupling constant.
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.
Many condensed matter problems involve a particle coupled to its environment. The polaron, originally introduced to describe electrons in a polarizable medium, describes a particle coupled to a bosonic field. The Holstein polaron model, although simple, including only optical Einstein phonons and an interaction that couples them to the electron density, captures almost all of the standard polaronic properties. We herein investigate polarons that differ significantly from this behaviour. We study a model with phonon-modulated hopping, and find a radically different behaviour at strong couplings. We report a sharp transition, not a crossover, with a diverging effective mass at the critical coupling. We also look at a model with acoustic phonons, away from the perturbative limit, and again discover unusual polaron properties. Our work relies on the Bold Diagrammatic Monte Carlo (BDMC) method, which samples Feynman diagrammatic expansions efficiently, even those with weak sign problems. Proposed by Prokof'ev and Svistunov, it is extended to lattice polarons for the first time here. We also use the Momentum Average (MA) approximation, an analytical method proposed by Berciu, and find an excellent agreement with the BDMC results. A novel MA approximation able to treat dispersive phonons is also presented, along with a new exact solution for finite systems, inspired by the same formalism.
In most cases, as the strength of electron-phonon coupling increases, the effective mass of polarons typically increases. However, in this paper, we uncover a fascinating phenomenon: the presence of light polarons even within the strong coupling regime, where electron-phonon coupling includes both Holstein and bond types (electron-phonon coupling occurring on the hopping bonds). To investigate this, we employ a diagrammatic Monte Carlo method based on the path-integral formulation of the particle sector and the Fock-state representation of the phonon sector. Our study centers on analyzing the impact of bond electron-phonon coupling on the Holstein polaron's essential properties, including its effective mass, ground-state energy, and the average number of phonons. We examine two distinct scenarios: one where the phonon frequency of both Holstein and bond types is the same, and another where the phonon frequency of the Holstein type is twice as large as that of the bond type. In both of these cases, when the Holstein coupling falls within the light mass regime, we observe minimal changes in the effective mass as a function of bond coupling, compared to the case of the bare bond polaron. However, the behavior of the effective mass undergoes a significant change in the heavy mass regime. Most intriguingly, we discover a nonmonotonic dependence of the effective mass on the bond coupling gB{g}_{B}gB​ when the Holstein coupling gH{g}_{H}gH​ falls within the heavy mass regime. This finding holds promise, particularly in the context of bipolarons, where achieving a light effective mass and a large binding energy to get a compact size is crucial.
We present numeric results for ground state and angle resolved photoemission spectra (ARPES) for a single hole in the t-J model coupled to optical phonons. The systematic-error-free diagrammatic Monte Carlo method is employed where the Feynman graphs for the Matsubara Green function in imaginary time are summed up completely with respect to phonon variables, while magnetic variables are subjected to the noncrossing approximation. We obtain that at electron-phonon coupling constants relevant for high T(c) cuprates the polaron undergoes a self-trapping crossover to the strong-coupling limit and theoretical ARPES demonstrate features observed in experiment: A broad peak in the bottom of the spectra has momentum dependence which coincides with that of a hole in the pure t-J model.
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 $\ensuremath{\alpha}\ensuremath{\gtrsim}9$.