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Physical review. B, Condensed matter

Intermediate region between large and small polarons in many-electron systems

H. Zheng, M. Avignon

Published Jul 10, 2003• 4 citations
Publisher

Abstract

An analytical approach has been developed for a many-electron Holstein model in one, two, and three dimensions. We show that our approach is flexible and physically clear because it is based on perturbation theory. When the dimensionless ratio ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}$ $(2\mathrm{Dt}$ is the half bandwidth in D dimensions) is small, ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}\ensuremath{\ll}1,$ our result is similar to that of the Migdal-Eliashberg theory. When ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}>1,$ our result approaches that of the Lang-Firsov transformation and small polaron theory. We have calculated the Green's function, the effective mass of electrons at the Fermi surface, the number of electrons ${n}_{\mathbf{k}}$ in a momentum state $\mathbf{k},$ and the jump of ${n}_{\mathbf{k}}$ at the Fermi surface to show that the perturbation approach works well in the intermediate region in which ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}\ensuremath{\sim}1.$

DOI
10.1103/physrevb.68.024301

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An analytical approach has been developed for a many-electron Holstein model in one, two, and three dimensions. We show that our approach is flexible and physically clear because it is based on perturbation theory. When the dimensionless ratio \ensuremathωp/2Dt{\ensuremath{\omega}}_{p}/2\mathrm{Dt}\ensuremathωp​/2Dt (2Dt(2\mathrm{Dt}(2Dt is the half bandwidth in D dimensions) is small, \ensuremathωp/2Dt\ensuremath≪1,{\ensuremath{\omega}}_{p}/2\mathrm{Dt}\ensuremath{\ll}1,\ensuremathωp​/2Dt\ensuremath≪1, our result is similar to that of the Migdal-Eliashberg theory. When \ensuremathωp/2Dt>1,{\ensuremath{\omega}}_{p}/2\mathrm{Dt}>1,\ensuremathωp​/2Dt>1, our result approaches that of the Lang-Firsov transformation and small polaron theory. We have calculated the Green's function, the effective mass of electrons at the Fermi surface, the number of electrons nk{n}_{\mathbf{k}}nk​ in a momentum state k,\mathbf{k},k, and the jump of nk{n}_{\mathbf{k}}nk​ at the Fermi surface to show that the perturbation approach works well in the intermediate region in which \ensuremathωp/2Dt\ensuremath∼1.{\ensuremath{\omega}}_{p}/2\mathrm{Dt}\ensuremath{\sim}1.\ensuremathωp​/2Dt\ensuremath∼1.
An analytical approach has been developed for the electron–phonon coupling in the Holstein–Hubbard model. The Hubbard U is treated approximately by the slave-boson approach. We show that our approach is flexible and physically clear because it is based on perturbation theory. When the dimensionless ratio ω p /2 Dt is small, ω p /2 Dt ≪ 1, our result is similar to that of the Migdal–Eliashberg theory. When ω p /2 Dt > 1, our result approaches that of the Lang–Firsov transformation and small polaron theory. We have calculated the effective mass of electrons at the Fermi surface to show that the new perturbation approach works well in the intermediate region where ω p /2 Dt ∼ 1.
According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupling term in the Hamiltonian into the form of an effective hopping integral and derive the single-particle Green's function describing propagation of the small Frohlich polaron. One and two dimensional spectral functions are studied by expanding the Green's function perturbatively. Numerical calculations of the spectral functions are produced. Remarkably, the coherent spectral weight (Z) and effective mass (Z') renormalisation exponents are found to be different with Z'>>Z, which can explain a small coherent spectral weight and a relatively moderate mass enhancement in oxides.
We derive an effective polaronic interaction Hamiltonian, exact to second order in perturbation, for the spinless one-dimensional Holstein model. The small parameter is given by the ratio of the hopping term (t)(t)(t) to the polaronic energy $({g}^{2}{\ensuremath{\omega}}_{0})$ in the whole region of validity for our perturbation; however, the exception is the regime of extreme antiadiabaticity $(t∕{\ensuremath{\omega}}_{0}\ensuremath{\leqslant}0.1)$ and small electron-phonon coupling (g<1)(g<1)(g<1) where the small parameter is $t∕{\ensuremath{\omega}}_{0}$. We map our polaronic Hamiltonian onto a next-to-nearest-neighbor interaction anisotropic Heisenberg spin model. By studying the mass gap and the power-law exponent of the spin-spin correlation function for our Heisenberg spin model, we analyze the Luttinger liquid to charge-density wave transition at half filling in the effective polaronic Hamiltonian. We calculate the structure factor at all fillings and find that the spin-spin correlation length decreases as one deviates from half filling. We also extend our derivation of polaronic Hamiltonian to ddd dimensions.
We look for an analytical method which can describe the Holstein model over the entire region of electron–phonon (e–ph) coupling strength. Considering a toy model of two sites the perturbation expansions are developed for the Holstein Hamiltonian using different standard phonon bases. It is found that the modified Lang Firsov phonon basis shows satisfactory convergence for almost the entire range of e–ph coupling from the antiadiabatic limit to the intermediate range of hopping. This method is used to study the magnetic transition and the small to large polaron crossover in a double exchange model with e–ph interaction. Implications of the results in the context of manganites are discussed.
According to recent quantum Monte Carlo simulations, the small polaron theory is practically exact in a wide range of the long-range (Fröhlich) electron-phonon coupling and adiabatic ratio. Applying Lang-Firsov transformation, we derive a single-particle Green's function describing the propagation of small Fröhlich polaron. One- and two-dimensional spectral functions are studied by expanding Green's function perturbatively. Remarkably, the coherent spectral weight ( Z ) and effective mass ( Z ′) renormalisation exponents are found to be different with Z ′ ≫ Z , which can explain a small coherent spectral weight and a relatively moderate mass enhancement in oxides.
The normal state of the Holstein model is studied at half-filling in infinite dimensions and in the adiabatic regime. The dynamical mean-field equations are solved using perturbation expansions around the extremal paths of the effective action for the atoms. We find that the Migdal-Eliashberg expansion breaks down in the metallic state if the electron-phonon coupling $\ensuremath{\lambda}$ exceeds a value of about 1.3 in spite of the fact that the formal expansion parameter $\ensuremath{\lambda}{\ensuremath{\omega}}_{0}{/E}_{F} ({\ensuremath{\omega}}_{0}$ is the phonon frequency, EF{E}_{F}EF​ the Fermi energy) is much smaller than 1. The breakdown is due to the appearance of more than one extremal path of the action. We present numerical results which illustrate in detail the evolution of the local Green's function, the self-energy, and the effective atomic potential as a function of $\ensuremath{\lambda}.$